\documentclass[a4paper, 11pt, oneside]{article} % A4 paper size, default 11pt font size and oneside for equal margins
\usepackage{geometry}
\geometry{left=2.5cm,right=2.5cm,top=2.5cm,bottom=2.5cm}
\usepackage{booktabs} % For formal tables
\usepackage{mathptmx} % This is Times font
\usepackage{fancyhdr}
\usepackage[normalem]{ulem}
\usepackage[utf8]{inputenc}
%\usepackage{microtype}
\usepackage{graphicx}
\usepackage{colortbl}
\usepackage{multirow}
\usepackage{amsmath}
\usepackage{bm}
\usepackage{epsfig}
\usepackage{authblk}
\usepackage{amssymb,enumitem}
\usepackage[hyphens]{url}
\usepackage{hyperref}
\usepackage[hypcap = false]{caption}
\usepackage{color}
\usepackage{soul}
\usepackage{bbding}
\usepackage{graphicx, subfig}
\usepackage{listings}
\usepackage{fontawesome}
\usepackage{longtable}
\usepackage{algorithm}
\usepackage{algorithmic}
\usepackage{mdsymbol}
\usepackage{color}
\usepackage{multirow}
\usepackage{booktabs}
\usepackage[version=4]{mhchem}
\usepackage{chemfig}

\newcommand{\tabincell}[2]{\begin{tabular}{@{}#1@{}}#2\end{tabular}}
% \newcommand{\comp}{\hspace{0pt}\nolinebreak-\hspace{0pt}}
% \definecolor{mygray}{gray}{.88}

\begin{document} 

%%%%封面内容编辑%%%%
\begin{titlepage} % Suppresses headers and footers on the title page

	\centering % Centre everything on the title page
	
	%\scshape % Use small caps for all text on the title page
	
	\vspace*{\baselineskip} % White space at the top of the page
	
	%------------------------------------------------
	%	Title
	%------------------------------------------------
	
	\rule{\textwidth}{1.6pt}\vspace*{-\baselineskip}\vspace*{2pt} % Thick horizontal rule
	\rule{\textwidth}{0.4pt} % Thin horizontal rule
	
	\vspace{0.75\baselineskip} % Whitespace above the title
	
	{\LARGE AxiaBench: Verifying Effect-uncovering Methodologies in Artificial Intelligence for Sciences} % Title
	
	\vspace{0.75\baselineskip} % Whitespace below the title
	
	\rule{\textwidth}{0.4pt}\vspace*{-\baselineskip}\vspace{3.2pt} % Thin horizontal rule
	\rule{\textwidth}{1.6pt} % Thick horizontal rule
	
	\vspace{2\baselineskip} % Whitespace after the title block
	
	%------------------------------------------------
	%	Subtitle
	%------------------------------------------------
	
	%Subtitle here % Subtitle or further description
	
	\vspace*{3\baselineskip} % Whitespace under the subtitle
	
	%------------------------------------------------
	%	Editor(s)
	%------------------------------------------------
	
	Edited By
	
	\vspace{0.5\baselineskip} % Whitespace before the editors
	
	{\scshape\Large Hongxiao Li\\ Chenxi Wang\\ Fanda Fan\\ Lei Wang\\ Wanling Gao\\ Jianfeng Zhan\\ }
	
	%{\scshape\Large Wanling Gao\\ Fei Tang\\ Lei Wang\\Jianfeng Zhan\\Chunxin Lan\\ Chunjie Luo\\Yunyou Huang\\Jiahui Dai\\Hainan Ye\\Zheng Cao\\Daoyi Zheng\\Haoning Tang\\Kent Zhan\\Biao Wang\\Defei Kong\\Shimin Gong\\Minghe Yu\\Chongkang Tan\\Yabin Huang\\Xinhui Tian\\Yatao Li\\Junchao Shao\\Xiaoyu Wang\\Zhenyu Wang\\ } % Editor list
	
	\vspace{0.5\baselineskip} % Whitespace below the editor list
	
	
	
	\vfill % Whitespace between editor names and publisher logo
	
	%------------------------------------------------
	%	Publisher
	%------------------------------------------------
	
	%\plogo % Publisher logo
	%\def\BUlogo{\epsfig{file=ICT.pdf,height=3cm}}
	%\includegraphics[scale=0.135]{ICT.pdf}
	\epsfig{file=bc-blue.pdf,height=2cm}
	\textit{\\BenchCouncil: International Open Benchmark Council\\Chinese Academy of Sciences\\Beijing, China\\https://www.benchcouncil.org} % Editor affiliation
	\vspace{5\baselineskip} % Whitespace under the publisher logo
	
	
	Technical Report No. BenchCouncil-AxiaBench-2026 % Publication year
	
	{\large April 20, 2026} % Publisher

\end{titlepage}

%----------------------------------------------------------------------------------------

%%%title here%%%
\title{AxiaBench: Verifying Effect-uncovering Methodologies in Artificial Intelligence for Sciences}

\author[1,2]{Hongxiao Li}
\author[1,2]{Chenxi Wang}
\author[1,3]{Fanda Fan}
\author[1,3]{Lei Wang}
\author[1,3]{Wanling Gao}
\author[1,3]{Jianfeng Zhan\thanks{Lei Wang, Wanling Gao, and Jianfeng Zhan are the corresponding authors.}}



\affil[1]{Research Center for Advanced Computer Systems, State Key Lab of Processors, Institute of Computing Technology, Chinese Academy of Sciences\\ \{wangchenxi21s, fanfanda, wanglei\_2011, gaowanling, zhanjianfeng\}@ict.ac.cn}
\affil[2]{University of Chinese Academy of Sciences\\ lihongxiao19@mails.ucas.ac.cn}
\affil[3]{BenchCouncil (International Open Benchmark Council)}




\date{Apr 20, 2026}
\maketitle


\begin{abstract}

Uncovering the effects of a cause, i.e., an object or thing, plays a fundamental role in modern social and natural sciences and engineering. While effect-uncovering methodologies such as design of experiments (DoE), randomized controlled trials (RCTs), do-calculus, structural causal models (SCMs), quasi-experiments, and observational designs have shaped modern scientific and engineering practice, there has been little systematic inquiry into their validity and efficiency. In this article, we introduce a formal methodology and framework for verifying effect-uncovering methodologies, which we call AxiaBench. AxiaBench enables the first large-scale, quantitative comparison of nine widely used effect-uncovering methodologies across six representative application domains in Artificial Intelligence for Sciences. Our verification reveals a fundamental limitation: existing methodologies are valid only in certain scenarios, and only a few can achieve acceptable efficiency under the validity requirement.

\end{abstract}

\section{Introduction}

Uncovering the effects of a cause, i.e., an object or thing, is the cornerstone of modern science and engineering~\cite{zhan2025evaluatology}. From physics to medicine, from economics to computer science, scientific progress has been enabled not only by discoveries themselves but also by the effect-uncovering methodologies. Across history, diverse approaches have emerged to serve this role: the design of experiments (DoE)~\cite{jain1990art,douglas2001design} in agricultural science, randomized controlled trials (RCTs)~\cite{chalmers1981method} in medicine, causal inference frameworks~\cite{imbens2015causal} in social and environmental studies, and quasi-experiments~\cite{thyer2012quasi} in economics and the behavioral sciences. At the very least, observational designs~\cite{zalta2003stanford} permeate nearly every empirical discipline. Recent Nobel Prizes in the sciences underscore the centrality of effect-uncovering in shaping rigorous and trustworthy knowledge~\cite{jumper2021highly}.

%Unfortunately, the formal definition and pragmatic paradigm have not been unified in consensus, let alone the evaluation of evaluation -- meta evaluation. The concept of meta-evaluation was first proposed in the work of Scriven~\cite{scriven1991evaluation}. However, this work provides only basic concepts and rules, rather than concrete design and practical utility for meta-evaluation. Besides, there are existing studies on some mature evaluation methods, including observation, design of experiments (DoE), randomized controlled trials (RCTs), reporting possible effectiveness and limitations of different methods, not including the comparison of their performances with several definite indexes in any concrete scenarios. For example, there is a CPU performance evaluation work~\cite{wang2024achieving} that involves similar concerns.

The methodologies discussed above yield influential conclusions in physics, chemistry, biology, finance, and other areas. However, such conclusions are only correct in certain specific evaluation conditions, 
%depending on different system and environment configurations, 
and later studies confirmed their failure in other evaluation conditions. Surprisingly, the evaluation conditions under which the validity of the above methodologies holds are still insufficiently theorized. In this article, we define validity as the accuracy of the inferred effect relative to the true effect of a cause, and efficiency as the cost to achieve the goal.

%While methodological traditions have crystallized within individual domains, the formal study of validating evaluation effects has scarcely advanced beyond conceptual sketches. Scriven’s early articulation~\cite{scriven1991evaluation} provided only definitions and guiding principles, without offering systematic design, operational frameworks, or practical toolsets.

% (real-world or artifact concrete conditions)

Existing studies typically highlight the strengths and weaknesses of individual methods -- such as DoE~\cite{jain1990art,douglas2001design}, RCTs~\cite{chalmers1981method}, or observational studies~\cite{zalta2003stanford}. Recent attempts refocus on the validity of evaluation methods from the perspective of evaluation, such as Zhan et al.'s ``Evaluatology''~\cite{zhan2024evaluatology,zhan2025evaluatology} and a CPU performance evaluation work~\cite{wang2024achieving}. However, they failed to explicitly formulate the general problem, propose a universal methodology, and design and implement a unified framework for verifying the effect-uncovering methodologies, thereby allowing different effect-uncovering methodologies to be verified on common ground. Also, they rarely offer quantitative comparisons across contexts. Consequently, methodological choices are often dictated by disciplinary conventions rather than their validity and efficiency, leading to fragmented practices. 

%space in which evaluation methods can be analyzed on a common ground. 
In a nutshell, these studies fall short of tackling the core challenge: how can we systematically test or verify the validity and efficiency of methodologies across diverse scientific and engineering domains? Throughout the rest of this article, the terms test and verify are used interchangeably. 

%In one word, these studies stop short of addressing the core challenge: how can we systematically test or verify the validity and efficiency of multiple methodologies across diverse scientific and engineering domains? In the rest of this article, the two words testing or verifying are used interchangeably. 
%As a result, the choices of methodologies are often dictated by disciplinary convention rather than their validity or efficiency, and the practices remain fragmented. 

% However, 

%Evaluatology by Zhan et al.~\cite{zhan2024evaluatology} is the first unified theory of evaluation that aims to formally introduce the discipline of science and engineering valuation, focusing on the validity and cost-effectiveness of the evaluation. It defines the evaluation condition (EC) and minimal evaluation system (MES), referring to the entire evaluating design space that includes lots of variables concerned in the evaluation process. The ground-truth performance of an evaluation method on a concrete task is then defined as the distribution of evaluation performances in the entire configuration space, measured by given indexes (e.g., accuracy, cost, and so on).

%问题定义
We reuse the terminologies from Evaluatology~\cite{zhan2025evaluatology} to formulate the problem of verifying the validity and efficiency of effect-uncovering methodologies within the domains of artificial intelligence for sciences (AI4S). Any effect-uncovering problem focuses on one or more objects that have causal effects on other objects. We call the former object(s) the EO (or cause). The causal effects are uncovered, based on a set of quantities tested or measured on finite quantities of objects. We call the minimal set of objects based on which quantities can be directly obtained as AO. For example, the disparity of quantities tested or measured between the presence and absence of an EO is the causal effect. We first define the problem of this research as: what is the disparity between the inferred effect using an effect-uncovering method and the ground-truth effect? That is, given an effect-uncovering method and a task, we study the validity of the method, which is decided by the disparity between the inferred effect and the ground-truth effect. The ground-truth effect is defined as the average effect of an effect-uncovering method in an entire configuration space of a certain task. The configuration space is an essential part of the AO, which spans a mathematical space of several dimensions of variables. By taking different variable values, the quantities tested and measured are different, which influences the inference of EO's effects.

%方法
We advance this agenda by introducing a rigorous verification methodology for effect uncovering. The entire methodology includes scientific problem investigation, problem structure extraction, simplified task definition, configuration space and test oracle definition, and validity verification. To the best of our knowledge, our work is the first one to verify the effect-uncovering methodology. As the baseline, we also propose a na\"ive method that only includes the process of task definition and application of an ad hoc effect-uncovering method. In comparison with the na\"ive one, our methodology has two merits. First, it studies the problem structure and verification of validity, which is not considered in the na\"ive one.
Second, a na\"ive one uses an arbitrary configuration of a single point in the space, resulting in insufficient effect uncovering and biased attribution to the EO. On the contrary, our method allows a more reliable result analysis. The overview of our method and a na\"ive method is presented in Figure~\ref{fig:methodology}. The complete illustration of our method is in Section~\cite{methodology}.

\begin{figure}
	\centering
		\includegraphics[scale=.5]{figs/methodology.png}
	\caption{\textbf{The comparison of our methodology and a na\"ive methodology.} Our methodology extracts common structures of vast scientific problems, analyzes on a full space, and verifies the validity of effect-uncovering methods. On the contrary, a na\"ive methodology is not only missing two steps of the study, but also works on a single point, resulting in probable biases and misleading conclusions.}
	\label{fig:methodology}
\end{figure}

%框架
With the above methodology, our study forms a verification framework —AxiaBench—which consists of four modules: method, task, verification, and execution. The method module includes ten effect-uncovering methods. We adopted multiple state-of-the-art effect-uncovering methods in respective fields. The task module includes six effect-uncovering tasks in the AI4S fields. Considering the above internal disparities of problems, we abstract and simplify six problems of effect-uncovering among massive problems, to represent different problem structures. We provide a detailed task definition and its configuration space illustration of every task in the latter sections. The verification module includes the definitions of validity and efficiency, determined by the accuracy and cost of every task. The execution module includes the entire pipeline execution of verification.

%These methods are not prevalently used in other fields. Our work prove the feasibility of these methods in different fields and discover practical superiority of some methods that has not been studied.

\begin{table}[ht]
\centering
\caption{\textbf{Problems of effect-uncovering (Part 1).} This table presents essential effects uncovering problems with their causes, effects, fundamental models, and corresponding simplified tasks (Part 1: Task 1 to Task 4).}\label{tab:problems_part1}
\begin{tabular}{ccccc}
\hline
Problem & Cause & Effect & Model & Simplified task \\\hline
\tabincell{c}{Turbulent\\fluid} & Initial phase & Turbulence & \tabincell{c}{Reynolds stress~\cite{reynolds1895iv},\\Law of the Wall~\cite{bradshaw1995law}} & \multirow{3}{*}{\tabincell{c}{effect-uncovering\\of the degree\\of chaos of\\celestial systems\\(Task 1)}} \\
Atmosphere & Initial phase & Weather & \tabincell{c}{Primitive equation~\cite{bjerknes1904problem},\\Barotropic model~\cite{charney1950numerical}} & \\
\tabincell{c}{Celestial\\motion} & Initial position & Trait & Lorentz system~\cite{emanuel1994atmospheric} & \\\hline
Heat wave & \tabincell{c}{Prediction\\model} & Accuracy & HWDI~\cite{radinovic2012criteria} & \multirow{3}{*}{\tabincell{c}{effect-uncovering\\of the accuracy\\of models' prediction\\on rainfall datasets\\(Task 2)}} \\
Earthquake & \tabincell{c}{Prediction\\model} & Accuracy & \tabincell{c}{Gutenberg-Richter\\theorem~\cite{gutenberg1956earthquake}} & \\
Rainfall & \tabincell{c}{Prediction\\model} & Accuracy & \tabincell{c}{Clausius-Clapeyron\\equation~\cite{clausius1850ueber}} & \\\hline
\tabincell{c}{Ecosystem\\stability} & \tabincell{c}{Ecological\\condition} & Stability & \tabincell{c}{Diversity-stability\\criterion~\cite{may1972will}} & \multirow{3}{*}{\tabincell{c}{effect-uncovering\\of the dynamic\\statistics of populations\\(Task 3)}} \\
\tabincell{c}{Population\\dynamics} & \tabincell{c}{Population\\characteristic} & \tabincell{c}{Population\\shift} & Cellular automata~\cite{wolfram1983cellular} & \\\hline
\tabincell{c}{Refraction\\rate} & \tabincell{c}{Materials,\\prediction\\model} & Accuracy & \tabincell{c}{Snell-Descartes\\theorem~\cite{joyce1976descartes}} & \multirow{4}{*}{\tabincell{c}{effect-uncovering\\of the accuracy\\of models' prediction\\on formation energies\\(Task 4)}} \\
Dielectric & \tabincell{c}{Materials,\\prediction\\model} & Accuracy & \tabincell{c}{Clausius-Mossoti\\formula~\cite{rysselberghe2002remarks}} & \\
\tabincell{c}{Formation\\energy} & \tabincell{c}{Materials,\\prediction\\model} & Accuracy & Hess's law~\cite{davik1980teaching} & \\\hline
\end{tabular}
\end{table}

\begin{table}[ht]
\centering
\caption{\textbf{Problems of effect-uncovering (Part 2).} This table presents essential effects uncovering problems with their causes, effects, fundamental models, and corresponding simplified tasks (Part 2: Task 5 to Task 6).}\label{tab:problems_part2}
\begin{tabular}{ccccc}
\hline
Problem & Cause & Effect & Model & Simplified task \\\hline
Cryptology & \tabincell{c}{Encipher\\method} & Safety & \tabincell{c}{RSA\\algorithm}~\cite{milanov2009rsa} & \multirow{2}{*}{\tabincell{c}{effect-uncovering\\of randomness test\\algorithms (Task 5)}} \\
\tabincell{c}{Pseudo-\\randomness} & \tabincell{c}{Test method} & \tabincell{c}{Correctness\\rate} & Dieharder test~\cite{brown2018dieharder} & \\\hline
Bidding & Strategy & Interest & \tabincell{c}{Vickrey\\Auction Model~\cite{ausubel1999generalized}} & \multirow{3}{*}{\tabincell{c}{effect-uncovering of\\the total income\\of commerce strategies\\in multi-round gaming\\(Task 6)}} \\
\tabincell{c}{Stock\\valuation} & \tabincell{c}{Valuation\\model} & Accuracy & Black-Scholes model~\cite{capinski2012black} & \\
\tabincell{c}{Company\\competence} & \tabincell{c}{Game\\strategy} & \tabincell{c}{Cumulative\\interest} & \tabincell{c}{Interactive\\decision model~\cite{axelrod1981evolution}} & \\\hline
\end{tabular}
\end{table}

To make a quantified comparison of different methods' validity and efficiency, we define test oracle, test case, implementation, and execution step by step. The test oracle is the ground-truth effect of an EO in a certain problem. As mentioned before, we choose the average effect of the cause in the entire configuration space. The validity is defined as the accuracy of a problem within the constraint of cost, and the efficiency is defined as low cost, where accuracy is the degree of closeness to the ground-truth, and the cost is the number of configurations taken. After that, we apply the methods, compare their results, and go through the validity verification process. The detailed verification procedure of every task on each effect-uncovering method is provided in Results and Section~\cite{framework}.

The most essential conclusion from our results is that, aside from a minority of effect-uncovering methods, the rest are invalid in most tasks. Consequently, discoveries based on these methods—including many important findings across various fields—are often unreliable. Specifically, we present our insights as follows. First, accuracy tends to increase when cost is not a constraint: classical RCTs~\cite{chalmers1981method}, and causal modeling methods~\cite{imbens2015causal,powell2018book} all exceed 95\% accuracy, with the best method achieving comparable performance at roughly half the cost. Second, observational designs~\cite{zalta2003stanford} perform markedly worse, often falling below 50\% accuracy and, critically, degrading further with repeated trials, thereby amplifying rather than reducing bias. Third, certain DoE variants (e.g., $l^k$\cite{jain1990art}) and quasi-experimental designs~\cite{thyer2012quasi} achieve acceptable accuracy ($>$ 90\%) under relaxed estimation requirements, but their validity collapses under stricter criteria, with accuracies dropping to 60\% or lower in several domains.

Taken together, these findings constitute the first systematic and cross-domain verification of effect-uncovering methodologies. Importantly, the conclusions are able to generalize to emerging AI4S fields, elevating the validity of the effect-uncovering method from a practice exercise to a cornerstone of scientific inquiry, providing both conceptual grounding and practical utility. Our main contributions include the following two parts. On the one hand, we defined the common structure of AI4S evaluation problems. On the other hand, we defined another type of evaluation problem structure, whose evaluation condition space cannot be completely reproduced. We simulated its structure with the common structure and concluded that the major referencing methods provide misleading evaluation conclusions.

Beyond its immediate utility, AxiaBench also gestures toward a broader vision: that the validity of effect-uncovering methods can be theorized, systematized, and optimized as a universal science of evidence. By releasing all tools, datasets, and results openly, we aim to foster a community-wide effort to transform the effect-uncovering methodology from a fragmented practice into a coherent framework that underwrites the validity of knowledge across disciplines. At the same time, we will closely engage with the AI4S community, leveraging state-of-the-art methods in advancing cross-disciplinary effect uncovering toward a reproducible and scalable scientific engineering practice.

\section{Results}

\subsection{Problem definition}

As illustrated in the introduction, the verification problem can be formally stated as the disparity between the inferred effect and the ground-truth effect. That is:\\
\textbf{Input}: An effect-uncovering method, a task, and its ground-truth effect,\\
\textbf{Output}: The validity of the method.

In our methodology, the ground-truth effect is represented by the average effect in different configurations in the entire configuration space decided by the task. Therefore, the problem can be induced as:\\
\textbf{Input}: An effect-uncovering method (including its implementation), a task (including its configurations and the effect for every configuration instance),\\
\textbf{Output}: The validity, computed based on the accuracy and cost of the method, which will be formalized in later sections.

\subsection{Tasks of this study}

We conducted verification experiments across six distinct tasks: celestial motion prediction~\cite{lichtenberg2013regular,cambel1993applied}, rainfall prediction~\cite{burrows1991objective}, population dynamics simulation~\cite{turing1990chemical,tilman1997spatial}, formation energy prediction~\cite{wicks1963thermodynamic,kohn1965self}, pseudo-random number verification~\cite{may1976simple,rukhin2001statistical}, and revenue evaluation of game strategies~\cite{axelrod1981evolution}.

We present different components of each task in Table~\ref{tab:tasks}: the EO, effect, AO, EXO, and SES. The EO, effect, and AO are already defined in the introduction. The EXO is defined as external objects that also have causal effects on the AO, and are not included within the EO. This factor is considered in some specific tasks. The SES is the total of EO, AO, and EXO.

\begin{table}[ht]
\centering
\caption{\textbf{Definition of each task.} This table presents the EO, effect, AO, EXO, and SES of each task of AxiaBench.}\label{tab:tasks}
\begin{tabular}{cccccc}
\hline
Task   & EO                           & Effect                                    & AO                                          & EXO                         & SES        \\\hline
Task 1 & \tabincell{c}{Attractor\\equation}           & \tabincell{c}{Disparity\\of chaos}                        & \tabincell{c}{Configurations\\and EO}                       & Not considered              & Same as AO \\\hline
Task 2 & AI model                     & \tabincell{c}{Disparity of\\accuracies}                   & \tabincell{c}{AI model,\\configurations,\\and formula}       & Not considered              & Same as AO \\\hline
Task 3 & \tabincell{c}{Simulated\\species}            & \tabincell{c}{Disparity of\\population\\statistics}        & \tabincell{c}{Configurations and\\automata rules}           & Not considered              & AO and EO  \\\hline
Task 4 & AI model                     & \tabincell{c}{Disparity of\\accuracies}                   & \tabincell{c}{AI model,\\configurations,\\and formula}       & Not considered              & Same as AO \\\hline
Task 5 & \tabincell{c}{Verification\\method}           & \tabincell{c}{Disparity of\\randomness\\testing\\abilities} & \tabincell{c}{Method,\\configurations,\\and formula}         & Not considered              & Same as AO \\\hline
Task 6 & \tabincell{c}{Game\\strategy}                & \tabincell{c}{Disparity\\of gains}                        & \tabincell{c}{Two competing\\strategies and\\configurations} & Another EO                  & Same as AO \\\hline
%Task 7 & CPU               & \tabincell{c}{Disparity of\\performances}                 & \tabincell{c}{The entire\\computer system}                  & Not considered              & Same as AO \\\hline
%Task 8 & \tabincell{c}{Large language\\models (LLMs)} & \tabincell{c}{Disparity of\\accuracies}                   & \tabincell{c}{LLM and\\math problems}                       & \tabincell{c}{System\\environments\\of LLMs} & AO and EXO \\\hline
\end{tabular}
\end{table}

The importance of these tasks cannot be overstated. In the realm of intelligent prediction, effect uncovering serves to validate essential physical laws and fine-tune model parameters for improved accuracy. They also aid in gaining a deeper insight into the mechanisms of intricate systems. In material science, establishing valid methods can significantly expedite the discovery of novel materials. In numerical and strategic simulation, it enables safeguarding the integrity of various applications and the dependability of experimental outcomes. Moreover, it helps in pinpointing robust strategies and serves as a foundation for making optimization decisions. In computer hardware and large language model domains, performance evaluations offer valuable benchmarks for stakeholders involved in development and procurement processes.

The validity of multiple methodologies are tested, including most state-of-the-art and state-of-the-practice methods across multiple domains: observational study, $2^k$, $2^{k-p}$, $l^kr$ factorial design of DoE~\cite{jain1990art} classical RCTs~\cite{chalmers1981method}, do-calculus~\cite{imbens2015causal,powell2018book}, Structural Causal Model (SCM)~\cite{imbens2015causal,powell2018book}, non-randomized controlled trials (NRCTs)~\cite{thyer2012quasi} in quasi-experiment, intervention staggering method~\cite{thyer2012quasi} in quasi-experiment, and the evaluatology method~\cite{wang2024achieving} in recent CPU evaluation. The abbreviations of them are as follows, as appearing in the latest results figures and tables. Observation: \texttt{Obs}, $2^k$: \texttt{Doe\_2\textasciicircum k}, $2^{k-p}$: \texttt{Doe\_2\textasciicircum(k-p)}, $l^k$: \texttt{Doe\_l\textasciicircum k}, RCTs: \texttt{Rct}, do-calculus: \texttt{Ci\_do}, SCM: \texttt{Ci\_scm}, NRCTs: \texttt{Qe\_nrct}, intervention staggering: \texttt{Qe\_stagger}, and evaluatology method: \texttt{Eva}.

\subsubsection{Celestial motion prediction}

This task is defined as inferring the degree of chaos of different Lorenz system equations in the atmospheric convection model~\cite{emanuel1994atmospheric}, where different strange attractors are used to simulate real-world atmosphere chaotic systems, parameters and initial values of which represent various atmospheric conditions.

This task aims at effect-uncovering of the initial values and the motion equation, whose influence is the degree of chaos. We adopt the Lorentz attractor~\cite{lorenz2017deterministic} and R\"ossler attractor~\cite{rossler1983chaotic} for simulation and uncovering under several precisions and simulation steps, using the following two indices for the degree of chaos: Lyapunov exponent~\cite{dingwell2006lyapunov}, and Kolmogorov-Smirnov (KS) statistics~\cite{fasano1987multidimensional}. This experiment aims to infer the index's accuracy estimation ability of different methods. The detailed illustration of this task is in Section~\cite{framework}.

%Our experiment results for this task are as follows. First, the $2^k$ DoE method achieves over 95\% accuracy at the cost of no more than 20 times of evaluations for both the KS statistic index and the Lyapunov exponent index. However, the $l^k$ DoE method taking $l=4$ has no accuracy improvement but a 16-times cost. On the contrary, there is a performance descend, that the accuracy is lower than 80\% within the cost of 200 times, and the maximum accuracy is no more than 95\%, no matter how many repetitions are done. Therefore, it is not a feasible evaluation method for this task. Second, the observation method achieves an unbearable low performance, while taking more evaluation cost, and repetition results in even worse performance. For the KS statistic index, the accuracy at a cost of 30 times is lower than 40\%, and there is no improvement when increasing evaluation cost. The accuracy for the Lyapunov exponent index at the cost of 100 repetitions is 90\% for the Lorentz attractor, while that of the R\"ossler attractor is only 44\%. Third, the results for the RCTs method and the evaluatology method achieve the best performances, which converge to 98\% accuracies within the cost of no more than 60 times of evaluation. The SCM method is equivalent to do-calculus if the entire EC space is evaluated through. As a complement, an experiment of the SCM method of partial space is set. The results report that it achieved 100\% accuracy for a certain object at a certain configuration, otherwise 0\%. As a conclusion, using RCTs, or the evaluatology method, should be considered as the optimal method, and taking which one has no significant impact. Using the $2^kr$ DoE method is also feasible for this task, while the $l^kr$ method that $l=4$ is not. Figure~\ref{fig:task123}(A) shows the accuracy-cost curve of different evaluation methods in this task.

Our experimental results indicate that the $2^k$ DoE method achieves over 95\% accuracy with fewer than 20 repetitions for both the KS statistic and the Lyapunov exponent. In contrast, an $l^k$ DoE approach with $l=4$ shows no improvement in accuracy despite a 16-fold increase in cost, and performance declines to below 80\% accuracy even after 200 repetitions, with a maximum accuracy not exceeding 95\%. This suggests that the $l^k$ method is unsuitable for this application.

The observation method performed poorly, with accuracy below 40\% for the KS statistic at 30 repetitions and no improvement with increased cost. For the Lyapunov exponent, accuracy reached 90\% for the Lorenz attractor but only 44\% for the R\"ossler attractor after 100 repetitions.

Both the RCTs and inferrology methods achieved the highest performance, converging to 98\% accuracy within 60 repetitions. The SCM method, when applied to the full variable space, performed equivalently to do-calculus. A limited study of a partial variable space resulted in either 100\% or 0\% accuracy, depending on the configuration.

In conclusion, RCTs and evaluatology are optimal and statistically indistinguishable for this task. The $2^k$ DoE method remains viable, whereas the $l^k$ method with $l=4$ is not recommended. Accuracy–cost curves for all methods are presented in Figure~\ref{fig:task123}(A).

\begin{figure*}[htbp!]
\centering
\vspace{-15pt}
\includegraphics[width=0.85\textwidth]{figs/task1.png}\\[-5pt]
\textit{(A) Task 1}\\[0pt]

\includegraphics[width=0.85\textwidth]{figs/task2.png}\\[-5pt]
\textit{(B) Task 2}\\[0pt]

\includegraphics[width=0.85\textwidth]{figs/task3.png}\\[-5pt]
\textit{(C) Task 3}

\caption{\textbf{The accuracy–cost curves for various methods in Tasks 1, 2, and 3 are shown above.} Colors represent different methods, shapes correspond to objects, and line styles denote metrics. The red lines indicate the Evaluatology method, which achieves the highest performance.}\label{fig:task123}

\vspace{-10pt}
\end{figure*}

\subsubsection{Rainfall prediction}

This task is defined as inferring the accuracy of the regression models in precipitation prediction problems, using a simplified ab initio dataset~\cite{clausius1850ueber,emanuel1994atmospheric} simulating the real-world annual rainfall condition of a certain city. This task aims at effect-uncovering of the prediction models, whose influence is on the accuracy of predicting rainfall. We infer four classical regression models~\cite{zuur2009glm}: GLM, GAM, RF, and FBM, under time series data computed based on several saturation-specific humidity, vertical airflow intensity, temperature, airflow noise, temperature noise, and observation noise values. The indices taken are MSE, RMSE, and MAE. This experiment aims to infer the accuracy estimation ability of different methods. The detailed illustration of this task is in Section~\cite{framework}.

%Our experiment results of this task is as follows. First, the $2^k$ DoE method achieves the complete 100\% accuracy at the cost of no more than 10 times of evaluations for both the MAE index and the RMSE index, and it achieves the 94\% accuracy for the MSE index. Among all tasks, DoE method at this task shows the best performance. Due to the EC space shape of this task, there is no feasible $l^k$ DoE methods. Second, the observation method's accuracies rapidly descend to 20\% while the times of repetition increase. For the MAE index and the RMSE index, the accuracies at cost of 100 times converge to 20\%, while the accuracy for the MSE index converges to 20\% at the cost of about 60 times of evaluation. This shows that observation totally failed at evaluation. Third, the results for the RCTs method and the evaluatology method achieve the accuracies that converge to 98\% accuracies within the cost of no more than 60 times of evaluation. The SCM method is equivalent to do-calculus, if the entire EC space is evaluated through. As a complement, an experiment of the SCM method of partial space is set. The results report that it all failed in evaluation, at the accuracy of 0\%. The intervention staggering method is inferior than the RCTs and evaluatology method, at the maximum 90\% accuracy. As a conclusion, using $2^k$ DoE method, the RCTs, or the evaluatology method should be considered as the optimal method and taking which one has no significant impact. Figure~\ref{fig:task123}(B) shows the accuracy-cost curve of different evaluation methods in this task.

Our experimental results for this task are summarized as follows. First, the $2^k$ DoE approach achieved 100\% accuracy in both the MAE and RMSE indices with fewer than 10 repetitions, and attained 94\% accuracy for the MSE index. Among all six tasks, the DoE method performed most effectively here. Owing to the structure of the EC space in this task, no feasible $l^k$ DoE configurations could be applied.

Second, the observation method exhibited a rapid decline in accuracy, converging to approximately 20\% as the number of repetitions increased. For the MAE and RMSE indices, accuracy stabilized near 20\% after 100 repetitions, whereas for the MSE index, it reached this level after approximately 60 repetitions, indicating a comprehensive failure.

Third, both the RCTs and evaluatology methods achieved accuracies converging to 98\% within 60 repetitions. The SCM method, equivalent to do-calculus when exhaustively inferring the entire EC space, was also tested on a partial subspace for comparison. Results from this partial analysis showed complete failure, with accuracies consistently at 0\%. The intervention staggering method underperformed relative to both RCTs and evaluatology, reaching a maximum accuracy of only 90\%.

In conclusion, the $2^k$ DoE, RCTs, and evaluatology methods are optimal choices for this task, with no significant performance differences observed among them. Accuracy–cost curves for each method are presented in Figure~\ref{fig:task123}(B).

\subsubsection{Population dynamics simulation}

This task is defined as inferring population dynamic simulation based on cellular automata in a two-dimensional grid, where each cell can represent an individual of a species, and the state of the cell describes whether there is an individual at that location. The expansion and competition of a population can be simulated according to certain rules, such as growth, reproduction, and death. This task aims at effect-uncovering of the initial population states, whose influence is the dynamic population characteristics. We simulate the population size changes of bacteria~\cite{ben2000cooperative}, yeast~\cite{pfeiffer2001cooperation}, and algae~\cite{huisman1999biodiversity}, where the simulation rules for each are slightly different. We adopt the following indices used to infer the stability of the population in the cellular automaton model: average population, standard deviation of the population, half-life, and growth rate~\cite{murray2007mathematical}. We put forward an analysis under different initial distributions, simulation steps, environment capacity, and resource concentration degrees. This experiment aims to infer the accuracy estimation ability of different methods. The detailed illustration of this task is in Section~\cite{framework}.

%Our experiment results of this task is as follows. First, the $2^k$ DoE method achieves over 90\% accuracy at the cost of 30 times of evaluations. This conclusion is the same for the predictions on the growth of bacteria, yeast, and algae. For the average population, standard deviation of the population, and growth rate indexes, the accuracy is up to 95\%. The $2^{k-p}$ method achieves the same accuracy at a lower cost, taking $p=1$ and $p=2$. However, the $l^k$ DoE method taking $l=4$ has no accuracy improvement but a 8-times cost. Therefore, it is not a feasible evaluation method for this task. Second, the observation method achieves unbearable low performance, while taking more evaluation cost by repetition results in even worse performance. For the average population index, the accuracy at cost of 30 times is close to 0\%, and there is no improvement when increasing evaluation cost. The accuracies for the growth rate and half-life indexes converge to about 20\% to 40\%. The accuracy for standard deviation index converges to lower than 20\%, where bacteria's is slightly better than algae's, and then yeast's. Third, the results for the RCTs method and the evaluatology method is generally the same as those of the $2^k$ DoE method. As a conclusion, using $2^k$, $2^{k-p}$, RCTs, or the evaluatology method should be considered as the optimal method and taking which one has no significant impact. Figure~\ref{fig:task123}(C) shows the accuracy-cost curve of different evaluation methods in this task.

Our experimental findings for this task are summarized as follows. First, the $2^k$ full-factorial DoE method consistently achieved accuracy exceeding 90\% across all metrics -- bacteria, yeast, and algae growth -- within 30 repetitions. Specifically, for the average population, population standard deviation, and growth rate indices, accuracy reached up to 95\%. The $2^{k-p}$ factorial design attained comparable performance at reduced cost for $p=1$ and $p=2$. In contrast, an $l^k$ DoE configuration with $l=4$ showed no improvement in accuracy despite an approximately eightfold increase in cost, rendering it impractical for this application.

Second, the observation method exhibited consistently poor performance. Accuracy for the average population fell to nearly 0\% after 30 repetitions, with no improvement upon more repetition. For growth rate and half-life indices, accuracy plateaued between 20\% and 40\%, while the standard deviation metric converged below 20\%—with marginally better results for bacteria than algae or yeast.

Third, both RCTs and evaluatology methods delivered performance equivalent to the $2^k$ DoE approach, achieving high accuracy within a constrained cost concern.

In conclusion, the $2^k$, $2^{k-p}$, RCTs, and evaluatology methods all represent robust and efficient strategies for this task, with no statistically significant differences in outcome. Accuracy–cost curves for each method are presented in Figure~\ref{fig:task123}(C).

\subsubsection{Formation energy prediction}

This task is defined as formation energy prediction~\cite{peterson2021materials} analysis using classical machine learning algorithms. This task includes several different patterns of materials, described by general molecule modules and different radicals made of different chemical elements. This task aims at effect-uncovering of the prediction models, whose influence is on the prediction accuracy. We simulate the formation energies of materials by computation using ab initio methods within a certain range of errors. The objects inferred in this task include several state-of-the-art models. We adopt the following two indices used to infer the accuracy of objects: MAPE and SMAPE. This experiment aims to infer the indexes' accuracy estimation ability of different methods. The detailed illustration of this task is in Section~\cite{framework}.

%Our experiment results of this task is as follows. The RCTs method and the evaluatology method achieve over 80\% accuracy and over 90\% accuracy at the cost of 60 times of evaluations for MAPE and SMAPE indexes respectively. Besides, there are no more improvements when taking more evaluation times cost. The results for the four AI models (i.e., PRB'14~\cite{meredig2014combinatorial}, ICDM'16~\cite{agrawal2016formation}, LR, and SVM) are no significant differences. The evaluatology method is slightly better compared to the RCTs method, due to less evaluation cost at the same accuracy performances. DoE methods are not suitable for this task, because the EC values are not completely independent, and there are some materials that cannot exist. The NRCTs method achieves similar but slightly lower performance to that of RCTs. In details, for the MAPE index, the accuracy is about 60\% at the cost of 30 evaluations, and it ascends to 90\% at the cost of 60. For the SMAPE index, they are about 75\% and 90\% at the cost of 30 and 60 respectively. As a conclusion, using RCTs, or the evaluatology method should be considered as a feasible practice and the evaluatology method is better at cost. The NRCTs method is also an inferior however feasible alternative approach for this task. Figure~\ref{fig:task456}(A) shows the accuracy-cost curve of different evaluation methods in this task.

Our experimental results for this task are summarized as follows. Both the RCTs and the evaluatology method achieved high performance, exceeding 80\% accuracy for the MAPE index and 90\% for the SMAPE index, within 60 repetitions. No further improvements were observed with additional repetitions. Performance across the four AI models -- PRB'14~\cite{meredig2014combinatorial}, ICDM'16~\cite{agrawal2016formation}, logistic regression (LR), and support vector machine (SVM) -- showed no significant differences. The evaluatology method demonstrated a marginal advantage over RCTs in terms of efficiency, achieving comparable accuracy at a lower cost.

DoE methods were deemed unsuitable for this task due to dependencies among EC values and the presence of infeasible material combinations.

The non-randomized controlled trials (NRCTs) method exhibited slightly inferior but acceptable performance relative to RCTs. Specifically, for the MAPE index, NRCTs reached approximately 60\% accuracy at 30 repetitions and improved to 90\% at 60 repetitions. Corresponding values for the SMAPE index were approximately 75\% and 90\%, respectively.

In conclusion, both RCTs and evaluatology represent robust strategies for this task, with evaluatology offering superior cost efficiency. NRCTs may serve as a viable alternative where full randomization is impractical. Accuracy–cost curves for all methods are presented in Figure~\ref{fig:task456}(A).

\begin{figure*}[htbp!]
\centering
\vspace{-12pt}         
\includegraphics[width=0.84\textwidth]{figs/task4.png}\\[-3pt]
(A) Task 4\\[0pt]

\includegraphics[width=0.84\textwidth]{figs/task5.png}\\[-3pt]
(B) Task 5\\[0pt]

\includegraphics[width=0.84\textwidth]{figs/task6.png}\\[-3pt]
(C) Task 6

\vspace{0pt}
\caption{\textbf{The accuracy–cost curves for various methods of effect-uncovering in Tasks 4, 5, and 6 are shown above.} Colors represent different methods, shapes correspond to objects, and line styles denote metrics. The red lines indicate the Evaluatology method, which achieves the highest performance.}\label{fig:task456}
\vspace{-10pt}
\end{figure*}

\subsubsection{Pseudo-random number verification}

This task is defined as inferring the ability of multiple randomness test algorithms of pseudo-random sequences, where the sequences are generated using the logistic map~\cite{may1976simple} with multiple initial values, parameters, and lengths, which is totally decisive rather than random. According to the definition, the less confident a randomness test algorithm is in its randomness, the more correct it achieves. This task aims at effect-uncovering of the verification methods, whose influence is the correctness rate of verification. We adopt four test algorithms from the Dieharder randomness testing tool~\cite{brown2018dieharder}: frequency test, runs test, overlapping permutations test, and birthday spacings test. The indices taken are the true rate that the sequence is randomly generated, and the average p-value measuring the confidence. This experiment aims to infer the index's accuracy estimation ability of different methods. The detailed illustration of this task is in Section~\cite{framework}.

%Our experiment results of this task is as follows. First, the $2^k$ method achieves different convergent accuracies for the four test algorithms and the true rate and the p-value indexes. Specifically, for the p-value index, the maximum accuracies of frequency test, runs test, overlapping permutations test, birthday spacings test are respectively 64\%, 90\%, 88\%, and 100\%; for the true rate index, those accuracies are respectively 78\%, 92\%, 98\%, and 100\%. This means that the $2^k$ method is effective for estimating some indexes and some objects (i.e., algorithms), while is not for some others. The $2^{k-p}$ method achieves a similar but lower accuracy. Taking $p=1$, for the p-value index, the maximum accuracies are respectively 44\%, 76\%, 68\%, and 100\%; for the true rate index, those accuracies are respectively 54\%, 80\%, 82\%, and 98\%. Taking $p=2$, for the p-value index, the maximum accuracies are respectively 26\%, 52\%, 48\%, and 78\%; for the true rate index, those accuracies are respectively 24\%, 54\%, 58\%, and 86\%. Second, as a comparison, the $l^k$ DoE method that $k=4$ achieves all the accuracies converge to more than 95\% at the cost of about 100 times of evaluation. Third, the RCTs method and the evaluatology method achieve over 95\% accuracy at the cost of 30 times of evaluations for the two indexes respectively, if regardless of some minor out-of-distribution data. Besides, there are no more improvements when taking more evaluation times cost. The results for the four algorithms are no significant differences. As a conclusion, using RCTs, or the evaluatology method should be considered as a feasible practice and the evaluatology method is better at cost. The $l^k$ DoE method is forsaken due to the high cost. Figure~\ref{fig:task456}(B) shows the accuracy-cost curve of different evaluation methods in this task.

Our experimental results for this task are summarized as follows. First, the $2^k$ full-factorial design exhibited variable convergence in accuracy across the four test algorithms and the two performance indices: true rate and p-value. For the p-value index, maximum accuracies were 64\% for the frequency test, 90\% for the runs test, 88\% for the overlapping permutations test, and 100\% for the birthday spacings test. Corresponding values for the true rate index were 78\%, 92\%, 98\%, and 100\%, respectively. These results indicate that the $2^k$ method is effective for certain algorithms and metrics, but performance is inconsistent.

The $2^{k-p}$ design showed similar but reduced accuracy. For $p=1$, maximum p-value accuracies were 44\%, 76\%, 68\%, and 100\%, while true rate accuracies reached 54\%, 80\%, 82\%, and 98\%, respectively. For $p=2$, performance further declined to 26\%, 52\%, 48\%, and 78\% for the p-value index, and 24\%, 54\%, 58\%, and 86\% for the true rate index.

Second, in contrast, the $l^k$ DoE method with $l=4$ achieved accuracies exceeding 95\% across all algorithms and indices after approximately 100 repetitions, albeit at substantially higher cost.

Third, both RCTs and evaluatology methods attained over 95\% accuracy within 30 repetitions for both indices, barring minor out-of-distribution deviations. No further improvements were observed with additional repetitions. Performance was consistent across all four algorithms.

In conclusion, RCTs and evaluatology represent efficient and effective strategies for this task, with evaluatology offering marginally better cost efficiency. The $l^k$ DoE method is less desirable due to its high computational expense. Accuracy–cost curves for each method are presented in Figure~\ref{fig:task456}(B).

\subsubsection{Revenue evaluation of game strategies}

This task is defined as revenue evaluation of game strategies, measured by the effectiveness of the company's bidding strategy in multiple rounds of games. Unlike other tasks, this one is carried out in the form of multiple rounds of two-on-two competition. The conditions of the task include multiple values of game time, income matrix, decay factor, and noise, where the discount factor is defined as the ratio of the profit value of the next round to that of the current round, and the noise is defined as the probability of failure in making decisions. We defined the cumulative decay income as the index, fully explained in Section~\cite{framework}. This task aims at effect-uncovering of the initial game strategies, whose influence is the final earnings. We infer the following six bidding strategies derived from the work of Axelrod et al.~\cite{axelrod1981evolution,axelrod1988further}: always defect, always cooperate, tic-for-tac, generous tic-for-tac, mixed strategy, consequence strategy. The indices taken are the total earnings and net earnings on different occasions. This experiment aims to infer the indices' accuracy estimation ability of different methods. The detailed illustration of this task is in Section~\cite{framework}.

%Our experiment results for this task is as follows. First, the $2^k$ DoE method achieves over 95\% accuracy at the cost of 20 times of evaluations. This conclusion is the same for the predictions on every strategy. However, the $l^k$ DoE method taking $l=4$ has no accuracy improvement but a 16-times cost. Therefore, it is not a feasible evaluation method for this task. Second, the results for the RCTs method achieve over 90\% at the cost of 60 times of evaluation. The evaluatology method is slightly better compared to the RCTs method, due to the 5\% better accuracy performances. Third, the NRCTs method achieves bad performance. For the total earning index, the accuracies of six objects converge to 0\%, 96\%, 82\%, 0\%, 0\%, and 0\%. For the net earning index, they are 74\%, 30\%, 68\%, 0\%, 48\%, and 0\%. This means most NRCTs evaluations are impractical. As a conclusion, only the evaluatology method should be considered as the optimal method. Figure~\ref{fig:task456}(C) shows the accuracy-cost curve of different evaluation methods in this task.

Our experimental findings for this task are summarized as follows. First, the $2^k$ full-factorial DoE method achieved an accuracy of over 95\% across all strategic predictions within 20 repetitions. In contrast, an $l^k$ DoE configuration with $l=4$ yielded no improvement in accuracy despite a 16-fold increase in cost, rendering it impractical for this application.

Second, the RCTs method attained accuracies exceeding 90\% after 60 repetitions. The evaluatology method again demonstrated marginally superior performance, outperforming RCTs by approximately 5\% in final accuracy.

Third, non-randomized controlled trials (NRCTs) exhibited inconsistent and largely unsatisfactory results. For the total earning index, accuracies converged to 0\%, 96\%, 82\%, 0\%, 0\%, and 0\% across the six test objects. Corresponding values for the net earning index were 74\%, 30\%, 68\%, 0\%, 48\%, and 0\%, indicating that most NRCTs were ineffective or inapplicable.

In conclusion, the evaluatology method represents the optimal strategy for this task, balancing high accuracy with feasible computational cost. Accuracy–cost curves for each method are presented in Figure~\ref{fig:task456}(C).

\subsection{Overall performances}

Table~\ref{tab:results} summarizes the overall performance of each method, assessed by accuracy, cost, and the Cost at 90\% Accuracy (C@0.9A), defined as the minimum number of repetitions required to achieve at least 90\% accuracy. A lower C@0.9A value indicates superior efficiency. If the method fails to reach 90\% accuracy, C@0.9A is defined as $\infty$.

Across most tasks, the evaluatology method~\cite{wang2024achieving} consistently demonstrated optimal performance, achieving the lowest C@0.9A values. The randomized controlled trials (RCTs) method ranked as the second most effective approach, offering robust accuracy at competitive costs. These results underscore the efficiency and validity of evaluatology for high accuracy under constrained cost concern.

\begin{table}[ht]
\centering
\caption{\textbf{Verification results.} This table presents the performance of various methods in the validity verification experiments.}\label{tab:results}
\begin{tabular}{lrcr}
\toprule
Methods           & Accuracy   & Cost   & C@0.9A  \\
\midrule
Obs               & 0$\sim$0.6 & high   & $\infty$  \\
Doe\_$2^k$        & 0.6$\sim$1 & low    & $\infty$  \\
%Doe\_$2^kr$       & 0.2$\sim$1 & low    & $\infty$  \\
Doe\_$2^{k-p}$    & 0.2$\sim$1 & low    & $\infty$  \\
Doe\_$l^k$        & 0.6$\sim$1 & high   & $\infty$  \\
Rct               & 0.9$\sim$1 & low    & 30  \\
Ci\_do            & 1          & high   & 200+  \\
Ci\_scm           & 0$\sim$1   & high   & 200+  \\
Qe\_nrct          & 0.2$\sim$1 & middle & 60$\sim\infty$  \\
Qe\_stagger       & 0.9        & high   & 30$\sim$200+  \\
Eva               & 0.9$\sim$1 & low    & 30  \\
\bottomrule
\end{tabular}
\end{table}

Our interpretation of the suboptimal performance exhibited by the observation method centers on its inherent reliance on the probability distribution of experimental condition (EC) values. In our simulations, which mirror real-world heterogeneity, certain EC values were assigned significantly higher occurrence probabilities -- differing by orders of magnitude -- than others. Consequently, as the number of observations increases, the aggregated results increasingly reflect the average behavior of high-probability instances rather than the true global average across the entire EC space. This sampling bias systematically skews the outcome and undermines the validity of observational estimates.

The inadequacy of DoE methods arises from two main limitations. First, DoE is fundamentally designed to elucidate the influence of various factors on experimental outcomes, not to isolate or quantify the causal effect of a specific object under study. Second, all DoE approaches rely on a sparse sampling of the configuration space, which is generally insufficient to capture the full complexity and variability of the ground-truth distribution. This insufficient sampling leads to significant estimation bias and weak correlation with results that would be obtained from exhaustive research of the entire EC space.

%In the task of LLM problem solving, we found cyclical fluctuation in performances of the $2^k$ and $2^kr$ DoE methods. Hereby, we provide probable explanation of this phenomenon. Taking the $2^k$ method as example, the EC space of this task is shaped $8\times4\times4\times3\times2\times4=3072$, while that of the size of the $2^k$ DoE method is $2^6=64$. As a result, different configurations are chosen in every 64 times of repetition. If less than 64, there must be some configurations that have some unique characteristics that are not chosen. Therefore, the performances close to the integer times of 64 are better, as they are more close to the global average performance of the entire EC space.

The outcome of the intervention staggering method in Task 2 shows a somewhat yet not critical accuracy decrease compared to the controlled experiments and evaluatology method. This is because the rainfall prediction task in the now-casting scenario is not sensitive to the non-recent data.

There is a leap of accuracy at about 30 repetition times, because the distribution estimating methods are different. Specifically, a Z-score less than 30 and a T-score otherwise.

\section{Discussion}

In this work, we propose validity verification of effect-uncovering as an operational scientific framework. By defining the experimental space, formalizing its representation, and instantiating it through AxiaBench, we enable -- for the first time -- a systematic, quantitative validity analysis of nine representative methods across six canonical scientific domains.

Our analysis demonstrates that no existing method simultaneously delivers accuracy and efficiency across diverse scenarios or illustrates the feasibility of methods in different occasions, except a very recently proposed entire-space stratified sampling method~\cite{wang2024achieving} based on evaluatology~\cite{zhan2024evaluatology} consistently achieves superior performance with substantially lower cost. Together, these results position our methodology not merely as a methodological exercise but as an essential instrument for trustworthy scientific inquiry.

A na\"ive test design on an effect uncovering method is carried out on a certain given, sampled, or generated dataset. The validity of a method is asserted according to whether the conclusions and findings the method made are in accordance with the dataset. Our well-defined verification methodology asserts the validity of a method based on whether the conclusions and findings the method made are in accordance with \textit{most of the dataset instances in the entire configuration space}. In such contexts, a certain dataset of a na\"ive design is only a single point in the entire space. We argue that our methodology yields more reliable insights because it evaluates the method's validity across the entire configuration space, ensuring that conclusions are not biased or overfitted to a specific dataset instance, as is often the case with a na\"ive test design.

Several limitations merit consideration. First, the effectiveness of our approach diminishes in settings where the condition space is intrinsically small, poorly characterized, or inaccessible, as stratification relies on a sufficiently rich configuration space. Second, while our benchmark spans a wide range of domains, some tasks necessarily rely on simulated rather than real-world experimental data, raising questions about external validity. We mitigated this trade-off by carefully balancing realism with tractability, and by cross-checking simulated findings against domains where ground-truth data are available. Nonetheless, extending AxiaBench to more real-world experimental pipelines remains an important direction for future work.

%As a limitation, this work and the evaluatology method does not work in good performance when the EC space is small or inaccessible. Another limitation is that the external effectiveness of this work may be questioned, because part of the evaluation designs are simulation instead of real-world experiments. As an explanation, we spared no effort in the balance between restoring real-world scenarios and experiment cost. We argue that there are no or few influence on the effectiveness of our novel evaluation methodology, either using real-world data achieved or ab initio computed or simulated data.

Despite these limitations, the broader implications of this work are profound. By elevating validity verification to a formal scientific object, we provide both a conceptual foundation and a practical tool set for interrogating the validity of effect uncovering itself -- a question that underpins all empirical disciplines. AxiaBench enables researchers to uncover when and why established methods fail, and offers a pathway toward principled, generalizable alternatives. Beyond methodological refinement, this work contributes to a larger scientific vision: bridging the gap between experimental designs and the complexity of real-world systems, accelerating the deployment of analysis methods that are both rigorous and efficient, and ultimately strengthening the epistemic foundations on which scientific progress depends.

%Through estimating the ground-truth evaluation outcomes of different objects, our meta evaluation methodology and AxiaBench provide a systematic approach for measuring and analyzing the gap between the outputs of AI models and the ground-truth data derived from the real world. In addition, the analysis of evaluation results of different objects in multiple configurations provide causal explanation of the effect of objects. This work lays a solid foundation for bridging the gap between experimental settings and a high dimensional space of the real-world applications, not only accelerating the development and deployment of the novel evaluation methods, but also affirming the usability of existing evaluation methods (e.g., RCTs, and DoE methods in partial tasks) in various fields, which can bring about efficiency improvements that have not been thoroughly studied before this work.

\section{Methodology}

We firstly investigated multiple essential problems in various AI for science fields, including but not limited to physics, chemistry, biology, finance, and selected over ten of the most influential ones. We found that the problems in different fields can be abstracted into the same formulation. We formalize the causes and effects of an effect-uncovering problem. However, besides the common properties, the problems in different fields also have mutually different internal structures. First, for some problems, every part is totally known and definite, while for some problems, it is not. Second, the cause under study and its affected objects are independent, while some are dependent. Third, some problems involve mutual influence and intertwinement of objects, and some do not. Fourth, the fundamental models of the problems are decisive and indecisive, respectively.

We abstracted six classical tasks for various fields: celestial motion prediction (Task 1), rainfall prediction (Task 2), population dynamics simulation (Task 3), formation energy prediction (Task 4), pseudo-random number verification (Task 5), and revenue evaluation of game strategies (Task 6). The existing essential problems with their causes, effects, and fundamental models, and the corresponding simplified tasks are presented in Tables~\ref{tab:problems_part1} and~\ref{tab:problems_part2}. Every part of Tasks 1, 5, and 6 is definite, while Tasks 2, 3, and 4 are not. The causes and affected objects of Tasks 1, 2, 3, and 4 are independent, and those of Tasks 5 and 6 are not. Task 6 includes the mutual influence of objects. The mathematical models of Tasks 1, 3, 5, and 6 are decisive, and the others are not.

The next step is to define the full configuration space of every task and define the test oracle. And then, define the validity and efficiency, decided by the accuracy and cost of every effect-uncovering method on every task.

We followed the basic methodology of testing, which includes the following four steps according to the book of evaluatology~\cite{zhan2025evaluatology}:
\begin{enumerate}
    \item Test oracle definition,
    \item Test case design for a test oracle,
    \item Test case implementation,
    \item Test case execution.
\end{enumerate}

\subsection{Framework}

After defining the verification problem, the simplified tasks, the configuration space, and test oracle of each task, the verification process is carried out with our framework. The framework consists of the method module, the task module, the verification module, and the execution module. The method module includes all adopted effect-uncovering methods. The task module consists of the task set, task definition, and configuration space. The task set is made of the six simplified effect-uncovering tasks in Table~\ref{tab:tasks}. The task definition is the formalized definition of all concerning components: EO, effect, AO, EXO, and SES. The configuration space of a task is an essential part of the AO, consisting of several variables of different values. The verification module includes the accuracy and cost index definition of each task, test oracles, and validity. The execution module is the implementation of AxiaBench on a computer, which allows the experiments to be executed. For all tasks but Task 5, the experiments are simulations. The framework is presented in Figure~\ref{fig:framework}.

\begin{figure}
	\centering
		\includegraphics[scale=.6]{figs/framework.png}
	\caption{\textbf{The framework of AxiaBench, including method module, task module, verification module, and execution module.}}
	\label{fig:framework}
\end{figure}

\subsubsection{Method module}

This module consists of multiple state-of-the-art effect-uncovering methods in respective fields. They are: observation, multiple DoE variants~\cite{jain1990art}, classical RCTs~\cite{chalmers1981method}, do-calculus~\cite{imbens2015causal,powell2018book} and Structural Causal Models (SCM)\cite{imbens2015causal,powell2018book} from causal inference, as well as non-randomized controlled trials (NRCTs)\cite{thyer2012quasi} and intervention-staggering approaches~\cite{thyer2012quasi} from quasi-experimental designs. 

\subsubsection{Task module}

This module consists of a range of computational tasks, including celestial motion prediction~\cite{lichtenberg2013regular,cambel1993applied}, rainfall prediction~\cite{burrows1991objective}, population dynamics simulation~\cite{turing1990chemical,tilman1997spatial}, formation energy prediction~\cite{wicks1963thermodynamic,kohn1965self}, pseudo-random number verification~\cite{may1976simple,rukhin2001statistical}, revenue evaluation of game strategies~\cite{axelrod1981evolution}, CPU performance evaluation~\cite{wang2024achieving}, and large language model (LLM) problem-solving~\cite{yang2023gpt,liu2024finemath}. Each task is assessed using multiple methodologies: observation, DoE designs ($2^k$, $2^{k-p}$, $l^k$)~\cite{jain1990art}, randomized controlled trials (RCTs)~\cite{chalmers1981method}, do-calculus~\cite{imbens2015causal,powell2018book}, structural causal model (SCM)~\cite{imbens2015causal,powell2018book}, non-randomized controlled trials (NRCTs)~\cite{thyer2012quasi}, intervention staggering~\cite{thyer2012quasi}, and the evaluatology method~\cite{wang2024achieving}. Experimental designs, parameter configurations, and the existing methods vary across the six tasks, as detailed in Table~\ref{tab:designs}. The holistic view of six tasks are shown as Figures~\ref{fig:ses1}, ~\ref{fig:ses2}, ~\ref{fig:ses3}, ~\ref{fig:ses4}, ~\ref{fig:ses5}, and ~\ref{fig:ses6}.

\begin{figure}
	\centering
		\includegraphics[scale=.7]{figs/ses1.png}
	\caption{\textbf{This figure shows the holistic view of Task 1.} The parts combine to form the SES.}
	\label{fig:ses1}
\end{figure}

\begin{figure}
	\centering
		\includegraphics[scale=.6]{figs/ses2.png}
	\caption{\textbf{This figure shows the holistic view of Task 2.} The parts combine to the SES.}
	\label{fig:ses2}
\end{figure}

\begin{figure}
	\centering
		\includegraphics[scale=.6]{figs/ses3.png}
	\caption{\textbf{This figure shows the holistic view of Task 3.} The parts combine to the SES.}
	\label{fig:ses3}
\end{figure}

\begin{figure}
	\centering
		\includegraphics[scale=.6]{figs/ses4.png}
	\caption{\textbf{This figure shows the holistic view of Task 4.} The parts combine to the SES.}
	\label{fig:ses4}
\end{figure}

\begin{figure}
	\centering
		\includegraphics[scale=.7]{figs/ses5.png}
	\caption{\textbf{This figure shows the holistic view of Task 5.} The parts combine to the SES.}
	\label{fig:ses5}
\end{figure}

\begin{figure}
	\centering
		\includegraphics[scale=.7]{figs/ses6.png}
	\caption{\textbf{This figure shows the holistic view of Task 6.} The parts combine to the SES.}
	\label{fig:ses6}
\end{figure}

\begin{table}[ht]
\centering
\caption{\textbf{Experimental designs.} This table provides a detailed account of the tested methods for each task, with ``Done'' indicating conducted and analyzed experiments, and ``N/A'' denoting experiment designs that are not applicable. The rationale behind these distinctions is elucidated in Results. The abbreviations of the methods are clarified in preceding sections.}\label{tab:designs}
\begin{tabular}{lcccccc}
\toprule
Methods         & Task 1 & Task 2 & Task 3 & Task 4 & Task 5 & Task 6  \\
\midrule
Obs             & Done   & Done   & Done   & N/A    & N/A    & N/A     \\
Doe\_2\textasciicircum k      & Done   & Done   & Done   & N/A    & Done   & Done    \\
Doe\_2\textasciicircum(k-p)  & N/A    & N/A    & Done   & N/A    & Done   & N/A     \\
Doe\_l\textasciicircum k      & Done   & N/A    & Done   & N/A    & Done   & Done     \\
Rct             & Done   & Done   & Done   & Done   & Done   & Done       \\
Ci\_do          & Done   & Done   & Done   & Done   & Done   & Done       \\
Ci\_scm         & Done   & Done   & N/A    & N/A    & N/A    & N/A     \\
Qe\_nrct        & N/A    & N/A    & N/A    & Done   & N/A    & Done     \\
Qe\_stagger     & N/A    & Done   & Done   & N/A    & N/A    & N/A     \\
Eva             & Done   & Done   & Done   & Done   & Done   & Done    \\
\bottomrule
\end{tabular}
\end{table}

\subsubsection{Verification module}

In this module, we concrete formalization of several basic concepts for every task. We first formalize the verification problem. We delineate the verification problem, the space of concerned experiment objects $O$, and the space of variable configurations $C$. The combination of components $C$ and $O$ forms a holistic system that produces various interim variables and outcomes denoted as $m(C,O)$.

Taking Task 5 as an example, different algorithms infer the randomness of given logistic sequences. According to our notation, $O$ stands for the algorithm space, where $o\in O$ stands for a certain algorithm. $C$ stands for the space of various configurations, i.e., $C=N\times X_0\times r\times|y|$, where $c\in C$ stands for a certain configuration, e.g., $c=(1,0.1,-1.8,10)$. For validity verification methods, as defined, different methods verify the ability of multiple randomness test algorithms. In our notation, $m$ stands for a certain method, e.g., $2^k$ DoE, which inputs a set of configurations $c\in C$ and an object $o$, and outputs the results, with given indexes. In notation, $m(C,O)=i_1,i_2,\cdots$.

We define the total cost of a method as the product of $\Vert C\Vert$.
%and experiment repetition (as $r=3$ in Task 7).
Besides the cost, the accuracy is defined as the distance between the estimated performance and the actual performance. In notation,\\
\centerline{$\mathbf{cost}(m,C,O)==\Vert C\Vert$,}\\
\centerline{$\mathbf{accuracy}(m,C,O,i)==\mathrm{dist(}\mathbf{GT}_i(o),\mathbf{mean}_{m,i}(o))$,}\\
where $\mathrm{dist}(\cdot)$ is a distance function, $\mathbf{GT}$ is the test oracle (ground-truth). We provide its concrete computation for every method in Section~\cite{quantification}.

\subsubsection{Execution module}

In the execution module, the details of four steps of testing, i.e., test oracle definition, test case design for a test oracle, test case implementation, and test case execution, are presented in later sections.

\subsubsection{Test oracle}

%We focus on two fundamental indexes: accuracy and cost.
We define the test oracle, in other words, ground-truth, as the average results of all configurations. In notation,\\
\centerline{$\mathbf{GT}_i(o)=\dfrac{\sum_{C}i(c,o)}{\Vert C\Vert}$,}\\
where $\Vert C\Vert$ is the size of the space (as $1440$ in Task 5), $i\in I$ is any index in the index space $I$ (as the true rate in Task 5).

%Once the accuracy and the cost of a method is defined, it can be computed by experiment. An accuracy-cost curve is defined as the accuracy tendency within the range of cost. The horizontal axis represents the cost and the vertical axis represents the accuracy. When the accuracy is over 90\%, namely the estimated performance of the object covers the ground-truth performance with the 90\% confidence interval, we consider such evaluation feasible. The horizontal line of accuracy=90\% in the result figures and the index C@0.9A in Results is based on this.

Once the precision and cost-effectiveness of a method are determined, they can be quantified through implementation. An accuracy-cost curve illustrates how accuracy varies across different cost levels. The cost is depicted on the horizontal axis and accuracy on the vertical axis. A method is deemed viable when its accuracy surpasses 90\%, indicating that the estimated performance of the object aligns with the ground-truth performance within a 90\% confidence interval. In the results section, the horizontal line representing accuracy at 90\% and the index C@0.9A are derived from this criterion.

%The test oracle is defined as whether an uncovering methodology is valid. Table~\ref{tab:results} shows the C@0.9A level of every method. If a method's C@0.9A index has a low cost (especially lower than 100) on a certain task, we deem it valid on that task.

\subsubsection{Test case}

The verification of a method's validity is not directly computable. We describe a method's general validity by its performance in six defined tasks in AxiaBench, as is already shown in early sections. The theoretical basis of such design is that the six tasks represent essential problems in major scientific fields, as shown in Tables~\ref{tab:problems_part1} and~\ref{tab:problems_part2}.

\subsubsection{Test case implementation}

This part involves detailed test case implementation. We illustrate it in Section~\cite{quantification}.

\subsubsection{Test case execution}

After implementing the six tasks through the concrete configuration, the verification process is executed. This part involves an illustration of how the verification of every method's validity is computed. We present it in Section~\cite{quantification}.

\subsection{Quantification}

We provide explicit descriptions of how results (as shown in Results) are generated for each task, how accuracy and cost are quantified for each method, and whether it is valid in the circumstance. To ensure reproducibility and minimize stochastic variability, all tasks were executed using fixed random seeds (0, 37, and 42), thereby reducing the need for repeated trials.

\subsubsection{Celestial motion prediction}

As defined in Results, this task infers the accuracies and costs of different methods. The methods infer the degree of chaos of the given Lorenz system equations. In a formal definition, a strange attractor of a certain Lorenz system requires three parameters $a,b,c$ and the initial values $x,y,z$. The full form of a system is as follows:\\
\centerline{$\begin{cases}
\dfrac{\text{d}x}{\text{d}t}=f(x,y,z)\\
\dfrac{\text{d}y}{\text{d}t}=g(x,y,z)\\
\dfrac{\text{d}z}{\text{d}t}=h(x,y,z)
\end{cases}$,}\\
where $f,g,h$ are usually linear or pseudo-linear functions. This task simulates equations with precision $p$ for $N$ steps, generating sequences. The Lyapunov exponent~\cite{dingwell2006lyapunov} and Kolmogorov-Smirnov (KS) statistics~\cite{fasano1987multidimensional} (3-dimensional) are used to measure the chaos degree of sequences.

The EC space is defined as $p\times(a,b,c)\times N\times(x,y,z)$ with given values as follows:\\
\[
\begin{aligned}
&p \in \{0.01,\ 0.005,\ 0.001,\ 0.0005,\ 0.0001\} \\
&(a,b,c) \in\left\{ \begin{aligned} 
(0.1, 0.2, 5.0),\ (0.5, 1.2, 12.0),\ (1.0, 0.3, 20.0),\ (1.5, 1.8, 8.0),\\ (0.3, 1.6, 30.0), \ (1.8, 0.6, 18.0),\ (0.7, 1.1, 25.0),\ (1.3, 0.9, 35.0),\\
(0.2, 1.7, 10.0),\ (1.9, 0.4, 40.0)
\end{aligned} \right\} \\
&N \in \{200,\ 400,\ 800,\ 1600\} \\
&(x,y,z) \in\left\{ \begin{aligned} 
(1.0, 5.0, 10.0),\ (-2.0, -8.0, -15.0),\ (3.0, -12.0, 20.0),\\
(-4.0, 16.0, -25.0),\ (5.0, 20.0, 30.0), \ (-6.0, -24.0, -35.0),\\
(7.0, -28.0, 40.0),\ (-8.0, 32.0, -45.0),\ (9.0, 36.0, 50.0),\\
(-10.0, -40.0, -55.0)
\end{aligned} \right\}
\end{aligned}
\]
For a certain group of $(p,(a,b,c),N,(x,y,z))$ values, the sequence is generated by the above definition. In total, there are $5\times10\times4\times10=2000$ number of configurations.

The inferred objects are Lorentz attractor~\cite{lorenz2017deterministic} (\texttt{obj1}), and R\"ossler attractor~\cite{rossler1983chaotic} (\texttt{obj2}), defined as follows:\\
Lorentz attractor: $\begin{cases}
\dfrac{\text{d}x}{\text{d}t}=a(y-x)\\
\dfrac{\text{d}y}{\text{d}t}=x(b-z)-y\\
\dfrac{\text{d}z}{\text{d}t}=xy-cz
\end{cases}$,
R\"ossler attractor: $\begin{cases}
\dfrac{\text{d}x}{\text{d}t}=-y-z\\
\dfrac{\text{d}y}{\text{d}t}=x+ay\\
\dfrac{\text{d}z}{\text{d}t}=b+z(x-c)
\end{cases}$.\\
Each object generates a sequence with the above configuration of parameters. The Lyapunov exponent and the KS statistics are indices computed with the sequence. So far, the verification result of every index and every object is defined and computed.

Seven methods (observation, $2^k$, $l^k$, RCTs, do-calculus, SCM, and evaluatology methods) compute with the result of every index and every object as the input, and the accuracies and costs as the output, as shown in Results. The detailed computing method is explained in this section, right after the explanation of the six tasks.

\subsubsection{Rainfall prediction}

As defined in Results, this task infers the accuracies and costs of different methods. The methods infer the accuracy of the regression models in the prediction of the ab initio precipitation sequences. In formal definition, a sequence $R(t)$ is given by\\
\centerline{$R(t)=\rho\dfrac{q_S(1+\epsilon(t))L\Gamma_mw(1+\sigma_W\eta(t))}{R_\nu (T+\sigma_T\xi(t))^2}+\sigma_R\nu(t)$,}\\
where $t$ is the time in a day referring the corresponding sequence position, $\rho\equiv3.6$ is the air density, $q_S$ is the saturation specific humidity, $\epsilon$ is the vapor noise, $L\equiv2.5\times10^6$ is the latent heat of vaporization, $\Gamma_m\equiv0.006$ is the moist adiabatic lapse rate, $w$ is the updraft airflow speed, $\sigma_W$ is the airflow noise coefficient, $\eta$ is the airflow noise, $R_\nu\equiv461$ is the water gas constant, $T$ is the temperature, $\sigma_T$ is the temperature noise coefficient, $\xi$ is the temperature noise, $\sigma_R$ is the observation noise strength, and $\nu$ is the observation noise. All the noises are in normal distribution. The range of $t$ is within $[0,8760)$ representing the precipitation conditions per hour in an entire year.

The EC space is defined as $q_S\times w\times T,\times\sigma_W\times\sigma_T\times\sigma_R$ with given values as follows:\\
\[
\begin{aligned}
&q_S \in \{0.01,\ 0.02,\ 0.04\} \\
&w \in \{0.01,\ 0.1\} \\
&T \in \{253,\ 261,\ 269,\ 277,\ 285,\ 293,\ 301,\ 309,\ 317,\ 325\} \\
&\sigma_W \in \{0.01,\ 0.03\} \\
&\sigma_T \in \{0.5,\ 1,\ 1.5,\ 2\} \\
&\sigma_R \in \{0.1,\ 0.2,\ 0.3,\ 0.4,\ 0.5\}
\end{aligned}
\]
For a certain group of $(q_S,w,T,\sigma_W,\sigma_T,\sigma_R)$ values, the precipitation sequence is generated by the above definition. In total, there are $3\times2\times10\times2\times4\times5=2400$ number of sequences.

The inferred objects are four classical regression models~\cite{zuur2009glm}: GLM (\texttt{obj1}), GAM (\texttt{obj2}), RF (\texttt{obj3}), and FBM (\texttt{obj4}). The accuracy of each object on each sequence is computed according to the indexes of MSE, RMSE, and MAE. So far, the verification result of the index and objects has been decided and obtained.

Seven methods (observation, $2^k$, RCTs, do-calculus, SCM, intervention staggering, and evaluatology methods) compute with the result of every index and every object as the input, and the accuracies and costs as the output, as shown in Results. The detailed computing method is explained in this section, right after the explanation of the six tasks.

\subsubsection{Population dynamics simulation}

As defined in Results, this task infers the accuracies and costs of different methods. The methods infer population dynamic stability of different species. This task simulates the three species based on cellular automata in a two-dimensional grid, with states of the cells and rules defined above. In a formal definition, a grid is a $a\times a$ sized matrix with positive integer values that stands for the number of species. The initial distribution of species is decided by $A_{init}$, which has six patterns with different centers and non-centers. This task simulates the species' evolution with grids for $n$ steps, with given environment capacity $L$ and resource density $s$. According to the real-world condition, the environmental capacity provides a constraint on the upper bound of species density. The rank of environment capacity sensitivity is: bacteria>yeast>algae (This task sets $24,11,7$ as coefficients, respectively). The resource density influences the evolution speed and sustainability. The rank of resource density demand is: yeast>algae>bacteria (This task sets $0.6,0.5,0.3$ as coefficients respectively).

The EC space is defined as $a\times A_{init}\times n\times L\times s$ with given values as follows:\\
\[
\begin{aligned}
&a \in \{10, 20, 50, 100\} \\
&A_{\text{init}} \in \left\{ \begin{aligned} \text{up centered}, \text{down centered}, \text{left centered},\\
\text{right centered}, \text{centered}, \text{spread}\end{aligned}\right\} \\
&n \in \{5, 10, 25, 100\} \\
&L \in \{4, 5, 6, 7\} \\
&s \in \{1.0, 1.5, 2.0, 2.5\}
\end{aligned}
\]
For a certain group of $(a,A_{init},n,L,s)$ values, the cellular grid is generated by the above definition. In total, there are $4\times5\times4\times4\times4=1536$ number of configurations.

The inferred objects are bacteria~\cite{ben2000cooperative} (\texttt{obj1}), yeast~\cite{pfeiffer2001cooperation} (\texttt{obj2}), and algae~\cite{huisman1999biodiversity} (\texttt{obj3}). The stability of each object on each cellular grid is computed according to four indices as follows: average population, standard deviation of the population, half-life, and growth rate~\cite{murray2007mathematical}. So far, the verification result of the index and objects has been decided and obtained.

Eight methods (observation, $2^k$, $2^{k-p}$, $l^k$, RCTs, do-calculus, intervention staggering, and evaluatology methods) compute with the result of every index and every object as the input, and the accuracies and costs of evaluation as the output, as shown in Results. The detailed computing method is explained in this section, right after the explanation of the six tasks.

\subsubsection{Formation energy prediction}

As defined in Results, this task infers the accuracies and costs of different methods. The methods infer the performance of material formation energy prediction models. A set of materials is given with their structure formulae and formation energy value computed with ab initio methods within a certain range of errors. The materials are defined by four general molecule modules $M$, and atom/ion types $X,Y,Z$. The formation energy $\Delta H_f$ of a material is computed based on $\Delta H_f=\sum_in_iE_{atom,i}-\sum_jk_jE_{bond,j}$, where $n_i$ and $k_j$ are the numbers of the $i$-th kind of atom and the $j$-th kind of chemical bond respectively, while $E_{atom,i}$ and $E_{bond,j}$ are formation energies of the $i$-th kind of elementary substance and $j$-th kind of chemical bond respectively, given as chemical constants in Table~\ref{tab:energy}.

\begin{table}[ht]
\centering
\caption{\textbf{Formation energy (kJ/mol).} The formation energy of substances and chemical bonds.}\label{tab:energy}
\begin{tabular}{cccccccc}
\toprule
\ce{Na+} & 0.0 & \ce{-H} & 218.0 & \ce{-OH} & 4.2 & \ce{-CH3} & 146.7 \\
\ce{-NH2} & 190.0 & \ce{-C#N} & 302.4 & \ce{=O} & 249.2 & \ce{=CH2} & 385.0 \\
\ce{=NH} & 260.0 & \ce{=C=O} & 749.0 & \ce{H-H} & -436.0 & \ce{O-H} & -463.0 \\
\ce{C-H} & -413.0 & \ce{N-H} & -391.0 & \ce{C-C} & -348.0 & \ce{C-O} & -358.0 \\
\ce{C-N} & -305.0 & \ce{O-O} & -142.0 & \ce{O-N} & -200.0 & \ce{N-N} & -163.0 \\
\ce{Na-H} & -188.7 & \ce{Na-O} & -260.0 & \ce{Na-C} & -150.0 & \ce{Na-N} & -243.0 \\
\bottomrule
\end{tabular}
\end{table}

The EC space is defined as $M\times X\times Y\times Z$ with given values as follows:\\
\[
\begin{aligned}
&M \in \{\text{X}_2, \text{XYZ}, \text{XZ}, \text{XY}_2\text{Z}\} \\
&X \in \{\ce{Na+}, \ce{-H}, \ce{-OH}, \ce{-CH3}, \ce{-NH2}, \ce{-}\chemfig{C\equiv N}\} \\
&Y \in \{\ce{-O-}, \ce{-CH2-}, \ce{-NH-}, \ce{-}\chemfig{C(=[2,0.5]O)}\ce{-}\} \\
&Z \in \{\ce{Na+}, \ce{-H}, \ce{-OH}, \ce{-CH3}, \ce{-NH2}, \ce{-}\chemfig{C\equiv N}\}
\end{aligned}
\]
Unlike other tasks, there are $5+6\times5\times4+6\times5+6\times5\times4=275$ number of substances in total, due to the duplication removal and illegal chemical formulae.

The inferred objects are: PRB'14~\cite{meredig2014combinatorial} (\texttt{obj1}), ICDM'16~\cite{agrawal2016formation} (\texttt{obj2}), LR (\texttt{obj3}), and SVM (\texttt{obj4}). Each object inputs the substances and outputs its formation energy predicted, inferred by MAPE and SMAPE. So far, the verification result of every index and every object is defined and computed.

Four methods (RCTs, do-calculus, NRCTs, and evaluatology methods) compute with the result of every index and every object as the input, and the accuracies and costs as the output, as shown in Results. The detailed computing method is explained in this section, right after the explanation of the six tasks.

\subsubsection{Pseudo-random number verification}

As defined in Results, this task infers the accuracies and costs of different methods. The methods infer the ability of multiple randomness test algorithms on certain logistic map~\cite{may1976simple} sequences. In formal definition, a sequence $y_n$ is given by\\
\centerline{$x_{n+1}=rx_n(1-x_n),y_n=\sum_{i=1,\cdots,N}x_{n,i},\text{ for any }n\in|y|$,}\\
where $|y|$ stands for the sequence length, $x_{n,i}$ is the $i$-th logistic map, $r$ is the mapping coefficient, $X_0=x_{1,0},\cdots,x_{n,0}$ are initial values, and $N$ is the number of maps.

The EC space is defined as $N\times X_0\times r\times|y|$ with given values as follows:\\
\[
\begin{aligned}
&N \in \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \\
&X_0 \in \{0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9\} \\
&r \in \{-1.8, -1.5, 3.6, 3.8\} \\
&|y| \in \{10, 20, 50, 100\}
\end{aligned}
\]
For a certain group of $(N,X_0,r,|y|)$ values, the sequence $y_n$ is generated by the above definition. In total, there are $10\times9\times4\times4=1440$ number of sequences.

The inferred objects are defined as frequency test (\texttt{obj1}), runs test (\texttt{obj2}), overlapping permutations test (\texttt{obj3}), birthday spacings test (\texttt{obj4}), derived from the Dieharder tool~\cite{brown2018dieharder}. Each object runs and outputs the confidence that the sequence is randomized, given by the two indices: the true rate, and the average p-value as defined in Results. So far, the verification result of every index and every object is defined and computed.

Five methods ($2^k$, $l^k$, RCTs, do-calculus, and evaluatology methods) compute with the result of every index and every object as the input, and the accuracies and costs as the output, as shown in Results. The detailed computing method is explained in this section, right after the explanation of the six tasks.

\subsubsection{Revenue evaluation of game strategies}

As defined in Results, this task evaluates the accuracy and costs of different methods. The methods evaluate the effectiveness of the company's bidding strategy in multiple rounds of games. In a formal definition, a company has to decide its strategy in $n$ rounds. In each round, there are two choices: make a discount or do not make a discount. There is a probability $\epsilon$ (the noise coefficient defined in Results) for a company to execute a wrong decision. The income matrix for two companies is as follows:\\
\centerline{$\begin{matrix}
                       & \text{Com. B disc.} & \text{Com. B no disc.}\\
\text{Com. A \quad~disc.}    & (b,b)               & (d,a)\\
\text{Com. A no disc.} & (a,d)               & (c,c)\\
\end{matrix}$}\\
In the above matrix, $a,b,c,d$ are coefficients with restriction $a<b<c<d$. The total earning $U$ of a company after $n$ rounds of gaming is computed as: $U=\sum_{i=1}^\infty\delta^{i-1}v(i)$, where $v(i)$ is the earning at the $i$-th round, $\delta>0.5$ is the decay factor defined in Results. The computation process of $U$ is actually a finite summation, due to the finite round number $n$.

The EC space is defined as $n\times(a,b,c,d)\times\delta\times\epsilon$ with given values as follows:\\
\[
\begin{aligned}
&n \in \{2, 5, 10, 100\} \\
&(a,b,c,d) \in 
\left\{ \begin{aligned} 
(2,4,5,6),\ (1,4,5,6),\ (2,4,6,7),\ (1,4,6,7),\ (2,3,5,7), \\
(1,3,5,7),\ (2,3,6,7),\ (1,3,6,7),\ (2,5,6,7),\ (1,5,6,7)
\end{aligned} \right\} \\
&\delta \in \{0.5, 0.58, 0.66, 0.74, 0.82, 0.9, 0.95, 0.975, 0.99\} \\
&\epsilon \in \{0.0, 0.05, 0.1, 0.15, 0.2, 0.25, 0.3, 0.35, 0.4, 0.45, 0.5, 1.0\}
\end{aligned}
\]
For a certain group of $(n,(a,b,c,d),\delta,\epsilon)$ values, the game procedure is decided by the above definition. In total, there are $4\times10\times9\times12=4320$ number of configurations.

The inferred objects are six bidding strategies derived from the work of Axelrod et al.~\cite{axelrod1981evolution,axelrod1988further}: always defect (\texttt{obj1}), always cooperate (\texttt{obj2}), tic-for-tac (\texttt{obj3}), generous tic-for-tac (\texttt{obj4}), mixed strategy (\texttt{obj5}), and consequence strategy (\texttt{obj6}). Each object competes with other objects and outputs its earnings, given by the two indexes: total earnings and net earnings as defined in Results. So far, the verification result of every index and every object is defined and computed.

Five methods ($2^k$, $l^k$, RCTs, do-calculus, and evaluatology methods) compute with the result of every index and every object as the input, and the accuracies and costs as the output, as shown in Results. The detailed computing method is explained in this section, right after the explanation of the six tasks.

\subsubsection{Evaluatology method}

We begin by presenting the evaluatology method.
%which, despite being introduced in this study, stands out as the most straightforward approach in our framework.
According to Wang et al.~\cite{wang2024achieving}, this method is structured into the following four steps.

The cost should be decided first. The minimum and maximum cost are $r$ and $\Vert C\Vert\times r$ respectively. If no repetition is required, treat as $r=1$. For example, the minimum and maximum cost of Task 5 are $1$ and $1440$, and $\mathbf{cost}=60$ is chosen.

Secondly, make $\mathbf{cost}$ random sampling without replacements, e.g., $60$ times. Compute the average and standard deviation results of all sampled configurations. In notation,\\
\centerline{$\mathbf{mean}_i(o)=\dfrac{\sum_{C_{\text{sample}}}i(c,o)}{\Vert C_{\text{sample}}\Vert},$}\\
\centerline{$\mathbf{std}_i(o)=\sqrt{\dfrac{\sum_{C_{\text{sample}}}\big(i(c,o)-\mathbf{mean}_{i}(o)\big)^2}{|C_{\text{sample}}|}}$,}\\
where $\Vert C\Vert$ is the size of the space (as 1440 in Task 5), $i$ is any index (as the true rate in Task 5).

%Thirdly, make the $\epsilon$ confidence interval estimation using the average and standard deviation. For configurations less than 30, use the Z-score, otherwise use the T-score. If the ground-truth result falls in the interval, note as success, otherwise note as failure.

Thirdly, to conduct a $\epsilon$ confidence interval estimation, utilize the average and standard deviation. When dealing with configurations below 30, apply the Z-score; for configurations exceeding 30, opt for the T-score. Identify success if the ground-truth result lies within the interval; otherwise, denote it as failure.

%Finally, repeat the second and third steps $K$ times, compute the success rate as the accuracy of the evaluatology method. The result figures with an accuracy-cost curve in Results is drawn in accordance with this rule. This work takes $\epsilon=95\%$ and $K=50$ in practice.

To determine the accuracy of the evaluatology method, repeat the second and third steps for a total of $K$ iterations. Compute the success rate based on how many times the ground-truth result falls within the confidence interval. The accuracy-cost curve in Results illustrates this relationship. In AxiaBench, we set $\epsilon$ to 95\% and utilize $K=50$ iterations to ensure robust analysis.

For other methods, we introduce their differences in each step if there is a difference. The rest of the procedure is the same.

\subsubsection{Observation method}

%The observation method is mostly same as the evaluatology method. However, there is difference on the second step. After deciding cost, this method does not require a uniform sampling of $\mathbf{cost}$ samples. Instead, we simulate the real-world distribution of all configurations in the EC. In Task 1, 2, and 3, we used power, Gauss, and mixed logarithm distributions respectively according to specific scenarios. This distribution is also used in the SCM method in late parts. The computation of average and standard deviation, and other steps are the same.

The observation method aligns closely with the evaluatology approach, with a key distinction in the second step. Following cost determination, this method does not mandate uniform sampling of $\mathbf{cost}$ samples. Rather, it replicates the real-world distribution of all configurations within the EC. For Tasks 1, 2, and 3, power, Gaussian, and mixed logarithm distributions are respectively employed based on distinct scenarios. This distribution is also integrated into the SCM method in subsequent stages. The computation of average, standard deviation, and other procedural steps remains consistent across both methods.

\subsubsection{DoE methods}

We specify the difference of the second step for DoE methods here.

The $2^k$ method takes $2^k$ configurations from the EC space, where $k$ is the dimension of the EC space. In each dimension, half values are considered as `low' and half as `high'. It is required to sample one `low' and one `high' value in each dimension, resulting in $2^k$ samples. Taking Task 3 as an example, the EC space is defined above, we provide a possible sampling as follows:\\
\[
\begin{aligned}
&a \in \{10, 50\} \\
&A_{\text{init}} \in \{\text{up centered}, \text{centered}\} \\
&n \in \{5, 100\} \\
&L \in \{5, 6\} \\
&s \in \{1.0, 2.0\}
\end{aligned}
\]
Different from the Evaluatology method, the average and standard deviation computation is illustrated by Jain et al.~\cite{jain1990art}.

The $2^{k-p}$ method takes the same sampling space as that of $2^k$. However, it uses a $\dfrac{1}{2^p}$ part of the samples for analysis. For example, suppose $k=5$, then the $2^k$ method takes $2^5=32$ configurations, where all `low' and `high' values in the five dimensions are selected. On the contrary, the $2^ {k-1}$ method takes $2^4=16$ examples, where only one is chosen in the two adjacent configurations (e.g., select one of the following two: (low, low, high, high, \underline{low}), and (low, low, high, high, \underline{high}).

%The $2^kr$ method is similar. The only difference is experiment repetition, the average and standard deviation computation is also illustrated by Jain et al.~\cite{jain1990art}. An equivalent form is as follows:\\
%\centerline{$\mathbf{mean}_{i}(o)=\mathbb{E}_{C}[\mu(\cdots,k_i,\cdots;j,o)]$ for $f(\cdots,k_i,\cdots;j)>0$,}\\
%\centerline{$\mathbf{std}_{i}(o)=\sqrt{\dfrac{\Sigma_{i=1}^{2^k}\Sigma_{j=1}^ry_{ij}^2-2^krq_0^2-\Sigma_{j=1}^{2^k-1}2^kr(\dfrac{1}{2^k}\Sigma_{i=1}^{2^k}S_{ij}\overline{y_i})^2}{2^k(r-1)}}$,}\\
%where $k_i$'s are values at different dimensions, $j$ is the repetition, and other symbols are interim variables computed based on the results. This work used a looser average and standard deviation. In fact, the above results in inferior performance.

As for the $l^k$ DoE method, the difference from $2^k$ is that it separates values in each dimension into $l$ parts instead of two, and takes $l^k$ samples for certain. Again, we provide a possible sampling of Task 3 as follows ($l=4$):\\
\[
\begin{aligned}
&a \in \{10, 20, 50, 100\} \\
&A_{\text{init}} \in \{\text{up centered}, \text{left centered}, \text{centered}, \text{spread}\} \\
&n \in \{5, 10, 25, 100\} \\
&L \in \{4, 5, 6, 7\} \\
&s \in \{1.0, 1.5, 2.0, 2.5\}
\end{aligned}
\]
The average and standard deviation computation is illustrated by Jain et al.~\cite{douglas2001design} for $l=3$, but others are similar. This work used $l=3$ and $l=4$ only.

\subsubsection{RCTs method}

%The RCTs method is mostly same as the evaluatology method. However, there is difference on the second step. After deciding cost, take distinct samples for all the objects. The computation of average and standard deviation is the same. Alternately, the RCTs method can also estimate the average discrepancy between two objects by the difference (subtraction). On the third step, if the confidence interval of discrepancy includes the original point (zero point), we assert that the two objects' performances are indistinguishable.

The RCTs method shares similarities with the evaluatology approach, with the only notable difference in the second step. Following cost determination, unique samples are extracted for all objects. The computation of average and standard deviation remains consistent. Additionally, the RCTs method can compute the average discrepancy between two objects through subtraction. In the third step, if the confidence interval of the discrepancy encompasses the original point (zero point), it indicates that the performances of the two objects are statistically indistinguishable.

\subsubsection{Casual inference methods}

Causal methods include do-calculus and the SCM method. The do-calculus is basically the same as the evaluatology method. However, according to the probability formula of do-calculus, the minimum cost is $\Vert C\Vert\times r$, which means high cost and 100\% accuracy. For further explanation, the probability formula is as follows:\\
\centerline{$\sum_w\mathbb{P}(y|t,w)\mathbb{P}(w|\mathbf{do}(t))=\sum_w\mathbb{P}(y|t,w)\mathbb{P}(w)$.}\\
In order to compute the probability and get the distribution of $y$, all the configurations are needed for computation.

The SCM method aligns with the causal model utilized in do-calculus, rendering it equivalent when the entire EC space undergoes inference. Nonetheless, distinctions arise when certain configurations are absent. Our study encompasses Task 1 and Task 2, employing identical distribution functions for EC configurations as observed in the observation method. Subsequently, we mask a specific dimension of the EC space and infer the distribution based on SCM theory and the causal diagrams pertinent to the two tasks. Further details are delineated below, with the EC spaces previously defined.

\textbf{Task 1:}\\
We carry out two experiments, by masking $(a,b,c)$ and $(x,y,z)$ respectively. The causal diagram of this task is in Figure\ref{fig:scm}(A). According to the SCM theory, the distribution of $(a,b,c)$ is delineated by:\\
\centerline{$\mathbb{P}(i|\mathbf{do}((a,b,c)=(a_0,b_0,c_0)))=\sum\limits_{\substack{(x_0,y_0,z_0)\in(x,y,z)}}[\mathbb{P}(i|(a_0,b_0,c_0))\mathbb{P}((a_0,b_0,c_0))]$,}\\
where $(a_0,b_0,c_0)$ and $(x_0,y_0,z_0)$ are values of the corresponding dimension. For the same reason, the distribution of $(x,y,z)$ is delineated by:\\
\centerline{$\mathbb{P}(i|\mathbf{do}((x,y,z)=(x_0,y_0,z_0)))=\sum\limits_{\substack{(a_0,b_0,c_0)\in(a,b,c)}}[\mathbb{P}(i|(x_0,y_0,z_0))\mathbb{P}((x_0,y_0,z_0))]$.}

\textbf{Task 2:}\\
We carry out three experiments, by masking $\sigma_W$, $\sigma_T$, and $\sigma_R$ respectively. The causal diagram of this task is in Figure\ref{fig:scm}(B). According to the SCM theory, the distribution of $\sigma_W$ is delineated by:\\
\centerline{$\mathbb{P}(i|\mathbf{do}(\sigma_W=\sigma_{W,0}))=\mathbb{P}(i|\sigma_{W,0},\sigma_{T,0},\sigma_{R,0})$,}\\
where $\sigma_{W,0}$, $\sigma_{T,0}$, and $\sigma_{R,0}$ are values of the corresponding dimension. For the same reason, the distribution of $\sigma_T$ and $\sigma_R$ are delineated by:\\
\centerline{$\mathbb{P}(i|\mathbf{do}(\sigma_T=\sigma_{T,0}))=\mathbb{P}(i|\sigma_{W,0},\sigma_{T,0},\sigma_{R,0})$,}\\
and\\
\centerline{$\mathbb{P}(i|\mathbf{do}(\sigma_R=\sigma_{R,0}))=\mathbb{P}(i|\sigma_{W,0},\sigma_{T,0},\sigma_{R,0})$.}

\begin{figure}
    \centering
    \begin{minipage}{0.48\textwidth}
        \centering
        \includegraphics[scale=.5]{figs/task1scm.png}\\
        (A) The casual diagram of Task 1.
    \end{minipage}
    \qquad
    \begin{minipage}{0.48\textwidth}
        \centering
        \includegraphics[scale=.5]{figs/task2scm.png}\\
        (B) The casual diagram of Task 2.
    \end{minipage}
    \caption{\textbf{Casual diagrams.}}
    \label{fig:scm}
\end{figure}

\subsubsection{Quasi-experiment methods}

The quasi-experiment methods encompass NRCTs and intervention staggering.

NRCTs are applied in Task 4 and Task 6, differing from the RCTs method primarily in the partitioning approach during the second step. While RCTs randomly divide the configuration space into $x$ parts corresponding to the number of objects, NRCTs partition the space into $x$ parts that are more contiguous and reflective of real-world conditions. The practical partitioning specifics for Task 4 and Task 6 are outlined below, building upon the previously defined EC spaces.

\textbf{Task 4:}\\
\[
\begin{aligned}
&\texttt{obj1}: \text{one half of } M=\text{XYZ} \\
&\texttt{obj2}: \text{another half of } M=\text{XYZ} \\
&\texttt{obj3}: \text{one half of } M=\text{XY}_2\text{Z} \\
&\texttt{obj4}: \text{another half of } M=\text{XY}_2\text{Z}
\end{aligned}
\]

\textbf{Task 6:}\\
\[
\begin{aligned}
&\texttt{obj1}: n \in \{2, 5\},\ \epsilon \in [0.2, 0.4) \\
&\texttt{obj2}: n \in \{10, 100\},\ \epsilon \in [0.2, 0.4) \\
&\texttt{obj3}: n \in \{10, 100\},\ \epsilon \in [0.0, 0.2) \\
&\texttt{obj4}: n \in \{10, 100\},\ \epsilon \in [0.4, 1.0] \\
&\texttt{obj5}: n \in \{2, 5\},\ \epsilon \in [0.4, 1.0] \\
&\texttt{obj6}: n \in \{2, 5\},\ \epsilon \in [0.0, 0.2)
\end{aligned}
\]

Unlike the RCTs and other types of controlled experiments, the intervention staggering method means different intervention times in time sequences for different objects. If such processing does not influence the accuracy of the final results, it means this method is feasible for a certain task.

We deploy experiments on Task 2 as the generated precipitation sequences are time sequences and can be manipulated within such a requirement. We provide the $t$ ranges for four objects as follows.

\textbf{Task 2:}\\
\[
\begin{aligned}
&\texttt{obj1}: t \in [0, 2190) \\
&\texttt{obj2}: t \in [0, 4380) \\
&\texttt{obj3}: t \in [0, 6570) \\
&\texttt{obj4}: t \in [0, 8760)\ (\text{unchanged})
\end{aligned}
\]

Following the reordering of sequences to align with the respective objects, the algorithms undergo training and testing procedures. The outcomes are regarded as the outputs of these objects. Notably, the ground-truth average and standard deviations remain consistent with the original results, mirroring the approach adopted in the evaluatology method.

\section{Acknowledgments}
Some figures of this paper include free icons from the Internet.
\paragraph*{Funding:}
We acknowledge support from the Innovation Funding of ICT, CAS under Grant No. E461070.
\paragraph*{Author contributions:}
H.L. conceptualized this study, proposed the mathematical foundation and the concrete experiment designs, implemented the experiments of all tasks, analyzed all the results, and wrote the manuscript. C.W. and F.F. discussed the insights of this research. J.Z. put forward the basic problem of this study, provided the basic research methods, and revised the manuscript. W.G. and L.W. directed the project and revised the manuscript. All authors have read and approved the final manuscript.

\bibliographystyle{ieeetr}
\bibliography{ref}

\end{document}
