Deep Divide-and-Reduce in Symbolic Regression

# Deep Divide-and-Reduce in Symbolic Regression
Source: [https://arxiv.org/html/2608.02628](https://arxiv.org/html/2608.02628)
Yusong Deng, Yanjie Li, Weijun Li\* School of Advanced Interdisciplinary Sciences University of Chinese Academy of Sciences Beijing, 101408 dengyusong22@mails\.ucas\.ac\.cn

###### Abstract

Symbolic regression \(SR\) is the task of discovering underlying patterns from data and representing them using mathematical expressions\. Current machine learning approaches to SR often lack a profound understanding of the intrinsic mathematical and physical principles governing these expressions\. While the pioneering AI Feynman method leverages the mathematical properties underlying the data, its expression simplification mechanism suffers from a narrow scope of applicability and is prone to failure on complex equations\. Furthermore, its underlying mechanisms rely heavily on brute\-force searches for sub\-expressions, severely limiting its practical utility\. Through rigorous mathematical deduction and proofs, we propose our method, Deep Divide and Reduce in Symbolic Regression \(DDRSR\)\. DDRSR fundamentally broadens the applicability of expression decomposition and reduction, circumvents the need for brute\-force sub\-structure searches, and ensures both wider versatility and strict theoretical correctness\. Empirical evaluations demonstrate that these theoretical principles yield significant advantages in both expression decomposition and numerical regression tasks\. Finally, we discuss the applicable scenarios and inherent limitations of this paradigm, alongside promising directions for future research\.

## 1Introduction

Symbolic Regression \(SR\) stands as a fundamental challenge at the intersection of machine learning and the natural sciences, aiming to automatically discover interpretable mathematical expressions that precisely characterize the intrinsic dynamics of a system from observational data\. Distinct from conventional deep learning regression methods, SR emphasizes model interpretability alongside high predictive accuracy\. Given its unique capability to uncover the exact analytical forms of physical laws, SR has garnered unprecedented attention within the AI for Science paradigm, particularly in contexts involving underlying mathematical and physical principles\.

Contemporary SR methodologies are broadly classified into two dominant paradigms: search\-based and pre\-training\-based approaches\. Search\-based methods—ranging from classical Genetic Programming \(GP\) to recent reinforcement learning frameworks like Deep Symbolic RegressionPetersenet al\.\([2019](https://arxiv.org/html/2608.02628#bib.bib2)\), along with numerous subsequent variantsSunet al\.\([2022](https://arxiv.org/html/2608.02628#bib.bib1)\); Mundhenket al\.\([2021](https://arxiv.org/html/2608.02628#bib.bib3)\); Liet al\.\([2024](https://arxiv.org/html/2608.02628#bib.bib4)\); Liuet al\.\([2025](https://arxiv.org/html/2608.02628#bib.bib33)\); Dong and Zhong \([2025](https://arxiv.org/html/2608.02628#bib.bib34)\); Al\-Helaliet al\.\([2024](https://arxiv.org/html/2608.02628#bib.bib35)\); Xuet al\.\([2024](https://arxiv.org/html/2608.02628#bib.bib37)\)—formulate the problem as a combinatorial optimization task within a discrete mathematical symbol space, typically conducting exploration via policy gradients\. Conversely, pre\-training\-based methodsBiggioet al\.\([2021](https://arxiv.org/html/2608.02628#bib.bib5)\); Kamiennyet al\.\([2022](https://arxiv.org/html/2608.02628#bib.bib6)\); Vastlet al\.\([2024](https://arxiv.org/html/2608.02628#bib.bib7)\); Meidaniet al\.\([2023](https://arxiv.org/html/2608.02628#bib.bib8)\); Wuet al\.\([2023](https://arxiv.org/html/2608.02628#bib.bib9)\); Liet al\.\([2022](https://arxiv.org/html/2608.02628#bib.bib10)\); Shojaeeet al\.\([2024](https://arxiv.org/html/2608.02628#bib.bib36)\)conceptualize SR as a sequence\-to\-sequence machine translation task\. These models leverage neural networks pre\-trained end\-to\-end on massive synthetic equation datasets to achieve exceptionally fast inference speeds\. Furthermore, recent researchLandajuelaet al\.\([2022](https://arxiv.org/html/2608.02628#bib.bib13)\); Xuet al\.\([2023](https://arxiv.org/html/2608.02628#bib.bib15)\); Holtet al\.\([2023](https://arxiv.org/html/2608.02628#bib.bib14)\); Liuet al\.\([2023](https://arxiv.org/html/2608.02628#bib.bib16)\); Chenet al\.\([2023](https://arxiv.org/html/2608.02628#bib.bib17)\); Liet al\.\([2025](https://arxiv.org/html/2608.02628#bib.bib38)\)has increasingly focused on synergizing these two paradigms, utilizing pre\-trained models to guide or constrain the subsequent search process, which has yielded substantial performance gains\.

However, a fundamental deficiency persists across both search\-based and pre\-training\-based paradigms\. Whether driven by data\-driven pattern matching or probabilistic heuristic search, these approaches essentially remain surface\-level data\-fitting techniques\. They lack a profound comprehension of the intrinsic mathematical and physical principles governing the underlying expressions\. In contrast, AI FeynmanUdrescu and Tegmark \([2020](https://arxiv.org/html/2608.02628#bib.bib11)\); Udrescuet al\.\([2020](https://arxiv.org/html/2608.02628#bib.bib12)\)introduced a distinct paradigm that leverages rigorous mathematical properties—such as translational symmetry—to determine functional separability, recursively decomposing complex high\-dimensional expressions into a hierarchy of lower\-dimensional, simpler sub\-problems\. By synergizing first principles with machine learning, this strategy injects a robust mathematical foundation into SR\.

Although AI Feynman represents a pioneering stride in embedding mathematical principles into SR, its applicability in realistic, complex scenarios is hindered by non\-negligible limitations\. Primarily, its criteria for identifying variable separability and translational symmetry exhibit limited generalizability\. These criteria demand highly idealized separable forms, severely restricting the range of separable structures the model can successfully detect and process\. Consequently, it loses efficacy when confronted with complex expressions characterized by overlapping variables or constant interference\. Furthermore, AI Feynman’s criteria for nested composition necessitate simultaneous data access from both the substructure and the current structure\. This mechanism relies heavily on a brute\-force search over candidate sub\-expressions, precipitating severe computational bottlenecks that largely negate the efficiency gains intended by functional decomposition\. Collectively, these deficiencies drastically constrain its generalization capacity and practical utility in high\-dimensional, complex environments\.

To fundamentally overcome these limitations, we conducted rigorous mathematical proofs and derivations to propose Deep Divide and Reduce in Symbolic Regression \(DDRSR\)\. Our primary innovations and contributions are summarized as follows:

- •Generalization of Translational Symmetry: We extend the principle of translational symmetry, significantly broadening its applicability to uncover generalized translational symmetric relationships between variables\.
- •Enhancement of Variable Separation: We generalize the variable separation mechanism to possess greater universality, enabling the resolution of high\-dimensional complex problems even in the presence of overlapping variables and constant interference\.
- •Top\-Down Nested Composition Framework: We propose a novel top\-down framework for nested compositional separation\. Instead of relying on brute\-force searches for candidate sub\-expressions, this framework determines nested compositions directly from the root node by exploiting strict mathematical properties\.
- •Theoretical Bounds on Decomposition: We provide a formal proof elucidating the intrinsic limitations of decomposition methods that operate independently of variable separability\.
- •Empirical Validation: We extensively demonstrate through experiments the substantial advantages of our proposed mathematical principles in both expression decomposition and the broader task of symbolic regression\.

## 2Related Work

The core philosophy of AI FeynmanUdrescu and Tegmark \([2020](https://arxiv.org/html/2608.02628#bib.bib11)\); Udrescuet al\.\([2020](https://arxiv.org/html/2608.02628#bib.bib12)\)lies in employing a data\-driven divide\-and\-conquer strategy\. By exploiting specific mathematical properties—namely, translational symmetry, variable separability, and nested composition—it recursively decomposes computationally intractable high\-dimensional target expressions into a hierarchy of tractable lower\-dimensional sub\-problems\.

### 2\.1Translational Symmetry

For annn\-dimensional functionf​\(X\)f\(X\), whereX=\(x1,…,xn\)X=\(x\_\{1\},\.\.\.,x\_\{n\}\)denotes thenn\-dimensional variables, ifffsatisfiesf​\(xi,xj,…\)=f​\(xi\+xj,…\)f\(x\_\{i\},x\_\{j\},\.\.\.\)=f\(x\_\{i\}\+x\_\{j\},\.\.\.\)for any givenxi,xjx\_\{i\},x\_\{j\}, thenf​\(X\)f\(X\)is said to possess additive translational symmetry with respect to dimensionsxix\_\{i\}andxjx\_\{j\}\. In the AI Feynman framework, for anyf​\(xi,xj,…\)f\(x\_\{i\},x\_\{j\},\.\.\.\), if the conditionf​\(xi,xj,…\)=f​\(xi\+c,xj−c,…\)f\(x\_\{i\},x\_\{j\},\.\.\.\)=f\(x\_\{i\}\+c,x\_\{j\}\-c,\.\.\.\)holds for an arbitrary constantcc, the functionf​\(X\)f\(X\)is considered to exhibit additive translational symmetry\. Analogous definitions apply to subtraction, multiplication, and division\.

However, the AI Feynman method fails to effectively identify these symmetries in the presence of constant interference, such as in the cases off​\(x1\+2​x2\)f\(x\_\{1\}\+2x\_\{2\}\)orf​\(x1​x22\)f\(x\_\{1\}x\_\{2\}^\{2\}\)\.

### 2\.2Variable Separability

Consider annn\-dimensional functionf​\(S​X1,S​X2\)f\(SX\_\{1\},SX\_\{2\}\), whereS​XSXdenotes a set consisting of an arbitrary number of variablesxix\_\{i\}\(i∈\{1,…,n\}i\\in\\\{1,\.\.\.,n\\\}\), and all such disjoint setsS​XSXspan thenn\-dimensional space\. Ifffsatisfiesf​\(S​X1,S​X2\)=g​\(S​X1\)\+h​\(S​X2\)f\(SX\_\{1\},SX\_\{2\}\)=g\(SX\_\{1\}\)\+h\(SX\_\{2\}\)for anyXX, thenf​\(X\)f\(X\)is termed additively separable with respect toS​X1SX\_\{1\}andS​X2SX\_\{2\}\. In AI Feynman, for anyf​\(S​X1,S​X2\)f\(SX\_\{1\},SX\_\{2\}\)and a constantcc, if the relationf​\(S​X1,S​X2\)=f​\(S​X1,c\)\+f​\(c,S​X2\)−f​\(c,c\)f\(SX\_\{1\},SX\_\{2\}\)=f\(SX\_\{1\},c\)\+f\(c,SX\_\{2\}\)\-f\(c,c\)is satisfied,f​\(X\)f\(X\)is deemed decomposable intof​\(S​X1,S​X2\)=g​\(S​X1\)\+h​\(S​X2\)f\(SX\_\{1\},SX\_\{2\}\)=g\(SX\_\{1\}\)\+h\(SX\_\{2\}\)\. Multiplicative variable separability is determined via a similar criterion\.

Nevertheless, this approach is incapable of effectively identifying separability in scenarios involving overlapping variables or constant interference, such asf​\(X\)=g​\(x1,x2\)\+h​\(x2,x3\)f\(X\)=g\(x\_\{1\},x\_\{2\}\)\+h\(x\_\{2\},x\_\{3\}\)orf​\(X\)=g​\(x1\)​h​\(x2\)\+cf\(X\)=g\(x\_\{1\}\)h\(x\_\{2\}\)\+c\.

### 2\.3Nested Composition

Compositionality: Annn\-dimensional functionf​\(X\)f\(X\)is said to possess compositionality if it can be formulated asf​\(X\)=g​\(h​\(X\)\)f\(X\)=g\(h\(X\)\)\. AI Feynman postulates that if the gradient∇f​\(X\)\\nabla f\(X\)aligns with the direction of∇h​\(X\)\\nabla h\(X\)for allXX, thenf​\(X\)f\(X\)can be represented asf​\(X\)=g​\(h​\(X\)\)f\(X\)=g\(h\(X\)\)\.

Generalized Symmetry: Annn\-dimensional functionf​\(S​X1,S​X2\)f\(SX\_\{1\},SX\_\{2\}\)exhibits generalized symmetry if it takes the formf​\(X\)=g​\(h​\(S​X1\),S​X2\)f\(X\)=g\(h\(SX\_\{1\}\),SX\_\{2\}\)\. In AI Feynman, if the direction of the partial gradient∇S​X1f​\(S​X1,S​X2\)\\nabla\_\{SX\_\{1\}\}f\(SX\_\{1\},SX\_\{2\}\)is independent ofS​X2SX\_\{2\}for allXX,f​\(X\)f\(X\)is assumed to be representable asf​\(X\)=g​\(h​\(S​X1\),S​X2\)f\(X\)=g\(h\(SX\_\{1\}\),SX\_\{2\}\)\.

Generalized Additivity: A 2\-dimensional functionf​\(x1,x2\)f\(x\_\{1\},x\_\{2\}\)is characterized by generalized additivity if it can be expressed asf​\(X\)=F​\(g​\(x1\)\+h​\(x2\)\)f\(X\)=F\(g\(x\_\{1\}\)\+h\(x\_\{2\}\)\)\. AI Feynman determines this property by checking whether the ratio of partial derivatives∂f/∂x1∂f/∂x2\\frac\{\\partial f/\\partial x\_\{1\}\}\{\\partial f/\\partial x\_\{2\}\}is multiplicatively separable with respect tox1x\_\{1\}andx2x\_\{2\}\.

However, several critical issues undermine these criteria\. First, the condition for compositionality requires evaluating∇h​\(X\)\\nabla h\(X\), despite the analytical form ofh​\(X\)h\(X\)being intrinsically unknown\. Second, the criterion for generalized additivity is not exclusively sensitive to additive structures and is prone to false positives when encountering multiplicative structures \(a detailed proof is provided in[3\.2\.7](https://arxiv.org/html/2608.02628#S3.SS2.SSS7)of Methodology and[B\.9](https://arxiv.org/html/2608.02628#A2.SS9)of Appendix\)\. Finally, even if generalized symmetry or generalized additivity is successfully identified, these conditions do not yield the explicit forms of the inner functionsggandhh\. Consequently, AI Feynman is compelled to rely on brute\-force search over candidate sub\-structures, a computationally suboptimal approach that severely limits its applicability\.

## 3Methodology

We significantly extend the mathematical principles of translational symmetry, variable separability, and nested composition\. This section presents the primary definitions, criteria, and theorems utilized in our framework; detailed derivations and formal proofs are comprehensively provided in the Appendix[B](https://arxiv.org/html/2608.02628#A2)\.

In the tree representation of a mathematical expression, leaf nodes denote variables, whereas internal nodes represent mathematical operators\. Within our methodology, the objective of translational symmetry is to consolidate multiple leaf nodes into a single equivalent leaf node, a process we define as bottom\-up variable composition\. Conversely, variable separability and nested composition attempt to partition the expression into two distinct sub\-modules starting from the root node, a process we define as top\-down expression separation\.

In the mathematical derivations and proofs presented in methodology and appendix, we intentionally omit explicit domain constraints for operations such as division and logarithms\. This is because the occurrence of mathematically undefined values intrinsically serves as a valid criterion within identification mechanism\. Consider[3\.1\.1](https://arxiv.org/html/2608.02628#S3.SS1.SSS1)as an example: if the truth expression possesses additive translational symmetry, then even as the sampled∂f​\(X\)∂x2\\frac\{\\partial f\(X\)\}\{\\partial x\_\{2\}\}approaches0,∂f​\(X\)∂x1\\frac\{\\partial f\(X\)\}\{\\partial x\_\{1\}\}will concurrently approach0by a proportional constantcc, thereby preserving the validity of the equation in the limit\. Conversely, the breakdown of the division serves as a direct mathematical proof that such additive translational symmetry does not hold\.

![Refer to caption](https://arxiv.org/html/2608.02628v1/method.png)Figure 1:Examples of graph modularity that DDRSR can auto\-discover\. Lines denote realvalued variables and ovals denote functions\.### 3\.1Translational Symmetry

#### 3\.1\.1Additive Translational Symmetry

Definition: For annn\-dimensional functionf​\(X\)f\(X\), if it can be formulated asf​\(x1,x2,…\)=f​\(c1×x1\+c2×x2,…\)f\(x\_\{1\},x\_\{2\},\.\.\.\)=f\(c\_\{1\}\\times x\_\{1\}\+c\_\{2\}\\times x\_\{2\},\.\.\.\), we state thatf​\(X\)f\(X\)possesses additive translational symmetry with respect to the dimensionsx1x\_\{1\}andx2x\_\{2\}\.

Criterion: If, for any arbitraryXX, the following condition holds:

∂f​\(X\)∂x1÷∂f​\(X\)∂x2=c\\frac\{\\partial f\(X\)\}\{\\partial x\_\{1\}\}\\div\\frac\{\\partial f\(X\)\}\{\\partial x\_\{2\}\}=cwhereccis a constant, thenf​\(X\)f\(X\)is deemed to possess additive translational symmetry across dimensionsx1x\_\{1\}andx2x\_\{2\}, yielding the relationshipc1c2=c\\frac\{c\_\{1\}\}\{c\_\{2\}\}=c\.

#### 3\.1\.2Multiplicative Translational Symmetry

Definition: For annn\-dimensional functionf​\(X\)f\(X\), if it can be formulated asf​\(x1,x2,…\)=f​\(x1c1×x2c2,…\)f\(x\_\{1\},x\_\{2\},\.\.\.\)=f\(x\_\{1\}^\{c\_\{1\}\}\\times x\_\{2\}^\{c\_\{2\}\},\.\.\.\), we state thatf​\(X\)f\(X\)possesses multiplicative translational symmetry with respect to the dimensionsx1x\_\{1\}andx2x\_\{2\}\.

Criterion: If, for any arbitraryXX, the following condition holds:

∂f​\(X\)∂x1÷∂f​\(X\)∂x2×x1x2=c\\frac\{\\partial f\(X\)\}\{\\partial x\_\{1\}\}\\div\\frac\{\\partial f\(X\)\}\{\\partial x\_\{2\}\}\\times\\frac\{x\_\{1\}\}\{x\_\{2\}\}=cwhereccis a constant, thenf​\(X\)f\(X\)is deemed to possess multiplicative translational symmetry across dimensionsx1x\_\{1\}andx2x\_\{2\}, yielding the relationshipc1c2=c\\frac\{c\_\{1\}\}\{c\_\{2\}\}=c\.

### 3\.2Variable Separability

#### 3\.2\.1Additive Separability with Overlapping Variables

Definition: For annn\-dimensional functionf​\(X\)f\(X\), if it exhibits the formf​\(S​X1,S​X2,S​X3\)=g​\(S​X1,S​X2\)\+h​\(S​X2,S​X3\)f\(SX\_\{1\},SX\_\{2\},SX\_\{3\}\)=g\(SX\_\{1\},SX\_\{2\}\)\+h\(SX\_\{2\},SX\_\{3\}\), we state thatf​\(X\)f\(X\)possesses additive separability with overlapping variables, specifically separable with respect toS​X1SX\_\{1\}andS​X3SX\_\{3\}\.

Criterion: Iff​\(X\)f\(X\)satisfies the following structure for anyXX:

f​\(S​X1,S​X2,S​X3\)=f​\(S​X1,S​X2,S​C3\)\+f​\(S​C1,S​X2,S​X3\)−f​\(S​C1,S​X2,S​C3\)f\(SX\_\{1\},SX\_\{2\},SX\_\{3\}\)=f\(SX\_\{1\},SX\_\{2\},SC\_\{3\}\)\+f\(SC\_\{1\},SX\_\{2\},SX\_\{3\}\)\-f\(SC\_\{1\},SX\_\{2\},SC\_\{3\}\)whereS​CiSC\_\{i\}denotes setting the corresponding dimensions ofS​XiSX\_\{i\}to constants, we determine thatf​\(X\)f\(X\)possesses additive separability with respect toS​X1SX\_\{1\}andS​X3SX\_\{3\}\.

Furthermore, we can extract the sub\-functions as follows:

g​\(S​X1,S​X2\)=f​\(S​X1,S​X2,S​C3\)−f​\(S​C1,S​X2,S​C3\)g\(SX\_\{1\},SX\_\{2\}\)=f\(SX\_\{1\},SX\_\{2\},SC\_\{3\}\)\-f\(SC\_\{1\},SX\_\{2\},SC\_\{3\}\)h​\(S​X2,S​X3\)=f​\(S​C1,S​X2,S​X3\)h\(SX\_\{2\},SX\_\{3\}\)=f\(SC\_\{1\},SX\_\{2\},SX\_\{3\}\)

#### 3\.2\.2Multiplicative Separability with Overlapping Variables

Definition: For annn\-dimensional functionf​\(X\)f\(X\), if it exhibits the formf​\(S​X1,S​X2,S​X3\)=g​\(S​X1,S​X2\)×h​\(S​X2,S​X3\)f\(SX\_\{1\},SX\_\{2\},SX\_\{3\}\)=g\(SX\_\{1\},SX\_\{2\}\)\\times h\(SX\_\{2\},SX\_\{3\}\), we state thatf​\(X\)f\(X\)possesses multiplicative separability with overlapping variables, specifically separable with respect toS​X1SX\_\{1\}andS​X3SX\_\{3\}\.

Criterion: Iff​\(X\)f\(X\)satisfies the following structure for anyXX:

f​\(S​X1,S​X2,S​X3\)=f​\(S​X1,S​X2,S​C3\)×f​\(S​C1,S​X2,S​X3\)f​\(S​C1,S​X2,S​C3\)f\(SX\_\{1\},SX\_\{2\},SX\_\{3\}\)=\\frac\{f\(SX\_\{1\},SX\_\{2\},SC\_\{3\}\)\\times f\(SC\_\{1\},SX\_\{2\},SX\_\{3\}\)\}\{f\(SC\_\{1\},SX\_\{2\},SC\_\{3\}\)\}we determine thatf​\(X\)f\(X\)possesses multiplicative separability with respect toS​X1SX\_\{1\}andS​X3SX\_\{3\}\.

Furthermore, we can extract the sub\-functions as follows:

g​\(S​X1,S​X2\)=f​\(S​X1,S​X2,S​C3\)f​\(S​C1,S​X2,S​C3\)g\(SX\_\{1\},SX\_\{2\}\)=\\frac\{f\(SX\_\{1\},SX\_\{2\},SC\_\{3\}\)\}\{f\(SC\_\{1\},SX\_\{2\},SC\_\{3\}\)\}h​\(S​X2,S​X3\)=f​\(S​C1,S​X2,S​X3\)h\(SX\_\{2\},SX\_\{3\}\)=f\(SC\_\{1\},SX\_\{2\},SX\_\{3\}\)

#### 3\.2\.3Compositional Form of Additive Separability

Theorem 1: For annn\-dimensional functionf​\(X\)f\(X\), ifffis additively separable with respect to dimensionsS​X1SX\_\{1\}andS​X3SX\_\{3\}, and concurrently additively separable with respect toS​X2SX\_\{2\}andS​X3SX\_\{3\}, thenffcan be rigorously expressed as:

f​\(S​X1,S​X2,S​X3\)=g​\(S​X1,S​X2\)\+h​\(S​X3\)f\(SX\_\{1\},SX\_\{2\},SX\_\{3\}\)=g\(SX\_\{1\},SX\_\{2\}\)\+h\(SX\_\{3\}\)

#### 3\.2\.4Compositional Form of Multiplicative Separability

Theorem 2: For annn\-dimensional functionf​\(X\)f\(X\), ifffis multiplicatively separable with respect to dimensionsS​X1SX\_\{1\}andS​X3SX\_\{3\}, and concurrently multiplicatively separable with respect toS​X2SX\_\{2\}andS​X3SX\_\{3\}, thenffcan be rigorously expressed as:

f​\(S​X1,S​X2,S​X3\)=g​\(S​X1,S​X2\)×h​\(S​X3\)f\(SX\_\{1\},SX\_\{2\},SX\_\{3\}\)=g\(SX\_\{1\},SX\_\{2\}\)\\times h\(SX\_\{3\}\)

#### 3\.2\.5Multiplicative Separability with Constant Interference

Definition: For annn\-dimensional functionf​\(X\)f\(X\), if it takes the formf​\(X\)=g​\(xi,S​X\)×h​\(xj,S​X\)\+cf\(X\)=g\(x\_\{i\},SX\)\\times h\(x\_\{j\},SX\)\+c, whereS​XSXdenotes the remaining dimensions excludingxix\_\{i\}andxjx\_\{j\}, we state thatf​\(X\)f\(X\)possesses multiplicative separability with a constant term across dimensionsxix\_\{i\}andxjx\_\{j\}\.

Criterion: Letui​\(X\)=∂2f​\(X\)∂xi​∂xj÷∂f​\(X\)∂xiu\_\{i\}\(X\)=\\frac\{\\partial^\{2\}f\(X\)\}\{\\partial x\_\{i\}\\partial x\_\{j\}\}\\div\\frac\{\\partial f\(X\)\}\{\\partial x\_\{i\}\}anduj​\(X\)=∂2f​\(X\)∂xi​∂xj÷∂f​\(X\)∂xju\_\{j\}\(X\)=\\frac\{\\partial^\{2\}f\(X\)\}\{\\partial x\_\{i\}\\partial x\_\{j\}\}\\div\\frac\{\\partial f\(X\)\}\{\\partial x\_\{j\}\}\. If∂ui​\(X\)∂xi=0\\frac\{\\partial u\_\{i\}\(X\)\}\{\\partial x\_\{i\}\}=0and∂uj​\(X\)∂xj=0\\frac\{\\partial u\_\{j\}\(X\)\}\{\\partial x\_\{j\}\}=0hold for any arbitraryXX, we definitively determine thatf​\(X\)f\(X\)possesses multiplicative separability with a constant term acrossxix\_\{i\}andxjx\_\{j\}\.

#### 3\.2\.6Separation Mechanism for Multiplicative Forms with Constant Interference

When the functionf​\(X\)f\(X\)is identified as possessing multiplicative separability under constant interference, further exact separation can be executed\.

Theorem 3: Given thatf​\(X\)f\(X\)exhibits the formf​\(X\)=g​\(xi,S​X\)×h​\(xj,S​X\)\+cf\(X\)=g\(x\_\{i\},SX\)\\times h\(x\_\{j\},SX\)\+c, whereS​XSXdenotes the remaining dimensions excludingxix\_\{i\}andxjx\_\{j\}\. If two sample points are obtained for dimensionsxix\_\{i\}andxjx\_\{j\}respectively, denoted as\(xi​1,xi​2\)\(x\_\{i1\},x\_\{i2\}\)and\(xj​1,xj​2\)\(x\_\{j1\},x\_\{j2\}\), the constant termcccan be analytically formulated as:

c=f​\(xi​2,xj​2,S​X\)−\(f​\(xi​2,xj​2,S​X\)−f​\(xi​2,xj​1,S​X\)\)×\(f​\(xi​2,xj​2,S​X\)−f​\(xi​1,xj​2,S​X\)\)f​\(xi​2,xj​2,S​X\)−f​\(xi​1,xj​2,S​X\)−f​\(xi​2,xj​1,S​X\)\+f​\(xi​1,xj​1,S​X\)c=f\(x\_\{i2\},x\_\{j2\},SX\)\-\\frac\{\(f\(x\_\{i2\},x\_\{j2\},SX\)\-f\(x\_\{i2\},x\_\{j1\},SX\)\)\\times\(f\(x\_\{i2\},x\_\{j2\},SX\)\-f\(x\_\{i1\},x\_\{j2\},SX\)\)\}\{f\(x\_\{i2\},x\_\{j2\},SX\)\-f\(x\_\{i1\},x\_\{j2\},SX\)\-f\(x\_\{i2\},x\_\{j1\},SX\)\+f\(x\_\{i1\},x\_\{j1\},SX\)\}

#### 3\.2\.7Generalized Separable Forms

Definition: For annn\-dimensional functionf​\(X\)f\(X\), if it takes the formf​\(X\)=U​\(g​\(xi,S​X\)\+h​\(xj,S​X\)\)f\(X\)=U\(g\(x\_\{i\},SX\)\+h\(x\_\{j\},SX\)\)orf​\(X\)=U​\(g​\(xi,S​X\)×h​\(xj,S​X\)\)f\(X\)=U\(g\(x\_\{i\},SX\)\\times h\(x\_\{j\},SX\)\), whereUUrepresents an arbitrary operator andS​XSXdenotes the remaining dimensions excludingxix\_\{i\}andxjx\_\{j\}, we state thatf​\(X\)f\(X\)possesses a generalized separable form\.

Criterion: If the ratio∂f/∂x1∂f/∂x2\\frac\{\\partial f/\\partial x\_\{1\}\}\{\\partial f/\\partial x\_\{2\}\}exhibits multiplicative separability with respect tox1x\_\{1\}andx2x\_\{2\}for any arbitraryXX, we determine thatf​\(X\)f\(X\)possesses a generalized separable form across dimensionsxix\_\{i\}andxjx\_\{j\}\.

However, under this criterion, it cannot be directly ascertained whether the intrinsic structure is additive or multiplicative\.

### 3\.3Nested Composition

#### 3\.3\.1Nested Separation Form of Standard Logarithmic Functions

Definition: For annn\-dimensional functionf​\(X\)f\(X\), if it exhibits the formf​\(X\)=log⁡\(g​\(X\)\)f\(X\)=\\log\(g\(X\)\), whereg​\(X\)g\(X\)can undergo further separation, we state thatf​\(X\)f\(X\)possesses a nested separation form of standard logarithmic functions\.

Criterion: Letfn​e​w​\(X\)=exp⁡\(f​\(X\)\)f\_\{new\}\(X\)=\\exp\(f\(X\)\)\. Iffn​e​w​\(X\)f\_\{new\}\(X\)exhibits separability, we verify thatf​\(X\)f\(X\)possesses a nested separation form of standard logarithmic functions\.

Consequently, we reduce the problem to the separation of the non\-nested functionfn​e​w​\(X\)f\_\{new\}\(X\)\. Notably, functions formulated asf​\(X\)=log⁡\(g​\(xi\)×h​\(xj\)\)f\(X\)=\\log\(g\(x\_\{i\}\)\\times h\(x\_\{j\}\)\)can be analytically rewritten asf​\(X\)=log⁡\(g​\(xi\)\)\+log⁡\(h​\(xj\)\)f\(X\)=\\log\(g\(x\_\{i\}\)\)\+\\log\(h\(x\_\{j\}\)\)\. This structure intrinsically exhibits additive separability and will thus be correctly processed during the additive separation phase\.

#### 3\.3\.2Nested Separation Form of Standard Exponential Functions

Definition: For annn\-dimensional functionf​\(X\)f\(X\), if it exhibits the formf​\(X\)=exp⁡\(g​\(X\)\)f\(X\)=\\exp\(g\(X\)\), whereg​\(X\)g\(X\)can undergo further separation, we state thatf​\(X\)f\(X\)possesses a nested separation form of standard exponential functions\.

Criterion: Letfn​e​w​\(X\)=log⁡\(f​\(X\)\)f\_\{new\}\(X\)=\\log\(f\(X\)\)\. Iffn​e​w​\(X\)f\_\{new\}\(X\)exhibits separability, we verify thatf​\(X\)f\(X\)possesses a nested separation form of standard exponential functions\.

Consequently, we reduce the problem to the separation of the non\-nested functionfn​e​w​\(X\)f\_\{new\}\(X\)\. Notably, functions formulated asf​\(X\)=exp⁡\(g​\(xi\)\+h​\(xj\)\)f\(X\)=\\exp\(g\(x\_\{i\}\)\+h\(x\_\{j\}\)\)can be analytically rewritten asf​\(X\)=exp⁡\(g​\(xi\)\)×exp⁡\(h​\(xj\)\)f\(X\)=\\exp\(g\(x\_\{i\}\)\)\\times\\exp\(h\(x\_\{j\}\)\)\. This structure intrinsically exhibits multiplicative separability and will thus be correctly processed during the multiplicative separation phase\.

#### 3\.3\.3Nested Separation of Logarithmic Functions with Constant Terms

Functions exhibiting the formf​\(X\)=c1×log⁡\(g​\(xi,S​X\)×h​\(xj,S​X\)\)\+c2f\(X\)=c\_\{1\}\\times\\log\(g\(x\_\{i\},SX\)\\times h\(x\_\{j\},SX\)\)\+c\_\{2\}can be mathematically rewritten asf​\(X\)=c1×log⁡\(g​\(xi,S​X\)\)\+c1×log⁡\(h​\(xj,S​X\)\)\+c2f\(X\)=c\_\{1\}\\times\\log\(g\(x\_\{i\},SX\)\)\+c\_\{1\}\\times\\log\(h\(x\_\{j\},SX\)\)\+c\_\{2\}, allowing them to be systematically reduced to additive separation\.

For functions exhibiting the formf​\(X\)=log⁡\(g​\(xi,S​X\)\+h​\(xj,S​X\)\)\+cf\(X\)=\\log\(g\(x\_\{i\},SX\)\+h\(x\_\{j\},SX\)\)\+c, applying an inverse function mapping yieldsexp⁡\(f​\(X\)\)=exp⁡\(c\)×g​\(xi,S​X\)\+exp⁡\(c\)×h​\(xj,S​X\)\\exp\(f\(X\)\)=\\exp\(c\)\\times g\(x\_\{i\},SX\)\+\\exp\(c\)\\times h\(x\_\{j\},SX\)\. Therefore, the original function can be reduced to the standard logarithmic function \(log\\log\) separation\.

Functions exhibiting the formf​\(X\)=c×log⁡\(g​\(xi,S​X\)\+h​\(xj,S​X\)\)f\(X\)=c\\times\\log\(g\(x\_\{i\},SX\)\+h\(x\_\{j\},SX\)\)are not subjected to standalone separation criteria\.

#### 3\.3\.4Nested Separation of Exponential Functions with Constant Terms

Functions exhibiting the formf​\(X\)=c1×exp⁡\(g​\(xi,S​X\)\+h​\(xj,S​X\)\)\+c2f\(X\)=c\_\{1\}\\times\\exp\(g\(x\_\{i\},SX\)\+h\(x\_\{j\},SX\)\)\+c\_\{2\}can be mathematically rewritten asf​\(X\)=c1×exp⁡\(g​\(xi,S​X\)\)×exp⁡\(h​\(xj,S​X\)\)\+c2f\(X\)=c\_\{1\}\\times\\exp\(g\(x\_\{i\},SX\)\)\\times\\exp\(h\(x\_\{j\},SX\)\)\+c\_\{2\}, allowing them to be reduced to multiplicative separation containing a constant term\.

For functions exhibiting the formf​\(X\)=c×exp⁡\(g​\(xi,S​X\)×h​\(xj,S​X\)\)f\(X\)=c\\times\\exp\(g\(x\_\{i\},SX\)\\times h\(x\_\{j\},SX\)\), applying an inverse function mapping yieldslog⁡\(f​\(X\)\)=log⁡\(c\)\+g​\(xi,S​X\)×h​\(xj,S​X\)\\log\(f\(X\)\)=\\log\(c\)\+g\(x\_\{i\},SX\)\\times h\(x\_\{j\},SX\)\. Therefore, the original function can be reduced to the standard exponential function \(exp\\exp\) separation\.

Functions exhibiting the formf​\(X\)=exp⁡\(g​\(xi,S​X\)×h​\(xj,S​X\)\)\+cf\(X\)=\\exp\(g\(x\_\{i\},SX\)\\times h\(x\_\{j\},SX\)\)\+care not subjected to standalone separation criteria\.

#### 3\.3\.5Nested Separation Form of Standard Trigonometric Functions

Definition: For annn\-dimensional functionf​\(X\)f\(X\), if it takes the formf​\(X\)=sin⁡\(or​cos\)​\(g​\(xi,S​X\)\+h​\(xj,S​X\)\)f\(X\)=\\sin\(\\text\{or \}\\cos\)\(g\(x\_\{i\},SX\)\+h\(x\_\{j\},SX\)\)orf​\(X\)=sin⁡\(or​cos\)​\(g​\(xi,S​X\)×h​\(xj,S​X\)\)f\(X\)=\\sin\(\\text\{or \}\\cos\)\(g\(x\_\{i\},SX\)\\times h\(x\_\{j\},SX\)\), whereS​XSXdenotes the remaining dimensions excludingxix\_\{i\}andxjx\_\{j\}, we state thatf​\(X\)f\(X\)possesses a nested separation form of standard trigonometric functions\.

Criterion: LetT​\(X\)=∂f​\(X\)∂xi×∂f​\(X\)∂xj÷\(1−f2​\(X\)\)T\(X\)=\\frac\{\\partial f\(X\)\}\{\\partial x\_\{i\}\}\\times\\frac\{\\partial f\(X\)\}\{\\partial x\_\{j\}\}\\div\(1\-f^\{2\}\(X\)\)\. If, for any arbitraryXX,T​\(X\)T\(X\)exhibits multiplicative separability with respect to dimensionsxix\_\{i\}andxjx\_\{j\}, we confirm thatf​\(X\)f\(X\)possesses a standard trigonometric nested separation form\.

#### 3\.3\.6Irrationality of Decomposition Without Variable Separability

During the process of expression separation, if an expression is partitioned into two components where the variables involved in the second component are merely a subset of those in the first component—meaning no distinct variable separation has occurred—such a separation strategy is fundamentally irrational\.

Theorem 4: For annn\-dimensional functionf​\(X\)f\(X\), iff​\(X\)f\(X\)can be formulated asf​\(S​X1,S​X2\)=g​\(S​X1,S​X2\)​\+⃝​h​\(S​X2\)f\(SX\_\{1\},SX\_\{2\}\)=g\(SX\_\{1\},SX\_\{2\}\)\+⃝h\(SX\_\{2\}\), where\+⃝\+⃝represents an arbitrary operator\. Provided that\+⃝\+⃝possesses a corresponding inverse operator\-⃝\-⃝and is independent of the computational path, there exist infinitely many valid functional configurations forggandhh\.

Consequently, it is analytically intractable to achieve deterministic expression decomposition under similar circumstances where variables are not strictly separated\.

## 4Experiments

![Refer to caption](https://arxiv.org/html/2608.02628v1/Comprehensive_Regression_R2.png)\(a\)R2R^\{2\}of Comprehensive Regression
![Refer to caption](https://arxiv.org/html/2608.02628v1/Comprehensive_Regression_Rec.png)\(b\)Recovery Rate of Comprehensive Regression
![Refer to caption](https://arxiv.org/html/2608.02628v1/Comprehensive_Separation.png)\(c\)Score of Comprehensive Separation
![Refer to caption](https://arxiv.org/html/2608.02628v1/Separation_Separation.png)\(d\)Score of Decomposition
![Refer to caption](https://arxiv.org/html/2608.02628v1/Separation_Regression.png)\(e\)R2R^\{2\}of Decomposition
![Refer to caption](https://arxiv.org/html/2608.02628v1/Translation_Symmetry_Regression.png)\(f\)R2R^\{2\}of Translation Symmetry

Figure 2:Comprehensive evaluation of the proposed method against baselines\. \(a\)\(b\) and \(c\) show the performance in comprehensive experiments\. \(d\) and \(e\) demonstrate the result in the ablation experiments of decomposition\. \(f\) provides the performance in the ablation experiments of Translation Symmetry\.To demonstrate the correctness and effectiveness of our proposed method, we utilize the separation principles introduced in our framework to decompose expressions\. Subsequently, we apply various symbolic regression algorithms \(E2EKamiennyet al\.\([2022](https://arxiv.org/html/2608.02628#bib.bib6)\), GP\-GOMEAVirgolinet al\.\([2021](https://arxiv.org/html/2608.02628#bib.bib40)\), NGGPMundhenket al\.\([2021](https://arxiv.org/html/2608.02628#bib.bib3)\), PySRCranmer \([2023](https://arxiv.org/html/2608.02628#bib.bib39)\), and MetaSymNetLiet al\.\([2023](https://arxiv.org/html/2608.02628#bib.bib31)\)\) to perform regression on the decomposed sub\-expressions\. We benchmark our approach against the original methods without decomposition, as well as against the decomposition principles utilized in AI Feynman\.

In the ablation studies, we useR2R^\{2\}as the evaluation metric, whereas in the comprehensive experiments, we employ bothR2R^\{2\}and the recovery rate\. Each experiment is repeated 7 times; after discarding the highest and lowest values, the results of the remaining 5 runs are averaged\. Furthermore, we introduce a Relative Separation Score to quantify the relative degree of expression decomposition\. The score is defined as follows:0if no decomposition is achievable;0\.50\.5if decomposition is possible but less thorough than other baseline methods; and11if the decomposition is more thorough than, or equal to, the best baseline\.

Notably, to focus strictly on the theoretical correctness of our principles and to avoid interference from numerical errors and noise, both numerical values and gradients involved in the computations for DDRSR and AI Feynman are directly sampled from the ground\-truth functions\. We omit the brute\-force search components of the AI Feynman method, focusing exclusively on expression decomposition and variable composition\. The expression simplification process incurs minimal computational overhead, completing in less than 1 second\. Furthermore, while the execution times of the respective symbolic regression methods vary, they all consistently remain within tractable limits\. Because our study does not aim to conduct cross\-methodological comparisons of computational efficiency, we omit the reporting of execution times in our evaluation metrics\.

The specific formulas in the datasets, the hyperparameter configurations for the symbolic regression methods, and the complete, detailed experimental results are provided in the Appendix\. All experiments were conducted on a server running Linux with four NVIDIA GV100\(TITAN V, 12GB memory\)\.

### 4\.1Comprehensive Symbolic Regression Experiments

We evaluate the complete method—incorporating variable separation, nested composition, and translational symmetry—on public datasets as well as several custom\-constructed test expressions\. We selected the three best\-performing baseline methods \(E2E, GP\-GOMEA, and PySR\) for these experiments, with the results illustrated in Figure[2\(a\)](https://arxiv.org/html/2608.02628#S4.F2.sf1)–[2\(c\)](https://arxiv.org/html/2608.02628#S4.F2.sf3)\.

Figure[2\(c\)](https://arxiv.org/html/2608.02628#S4.F2.sf3)presents the Relative Separation Scores of DDRSR and AI Feynman\. It can be observed that DDRSR consistently outperforms AI Feynman across the board, demonstrating a stronger capability for expression simplification\. This is theoretically sound, as DDRSR is designed to identify and mathematically parse a broader range of structural cases\.

Figure[2\(a\)](https://arxiv.org/html/2608.02628#S4.F2.sf1)compares theR2R^\{2\}performance of the original methods against those utilizing DDRSR and AI Feynman decompositions\. Evidently, the results following decomposition by either DDRSR or AI Feynman generally surpass those of the raw baseline methods, verifying the efficacy of expression simplification\. A more detailed statistical analysis grouped by variable dimensionality reveals that this performance gain becomes increasingly pronounced as the dimensionality and complexity of the expressions increase\.

Figure[2\(b\)](https://arxiv.org/html/2608.02628#S4.F2.sf2)reports the recovery rates of the symbolic regression methods following decomposition by DDRSR and AI Feynman\. Taken together with the aforementionedR2R^\{2\}results, DDRSR exhibits comprehensive superiority, indicating that it can indeed identify and simplify more complex scenarios than AI Feynman, thereby achieving better overall performance\.

### 4\.2Ablation Studies

#### 4\.2\.1Top\-down Expression Decomposition

We independently evaluate the top\-down expression decomposition—which involves variable separation and nested composition—on all datasets except complex\-test\. The results are shown in Figure[2\(d\)](https://arxiv.org/html/2608.02628#S4.F2.sf4)and[2\(e\)](https://arxiv.org/html/2608.02628#S4.F2.sf5)\.

Figure[2\(d\)](https://arxiv.org/html/2608.02628#S4.F2.sf4)presents the Relative Separation Scores for DDRSR and AI Feynman\. DDRSR achieves consistently higher decomposition scores than AI Feynman, indicating its capability to identify a wider variety of mathematical structures and its more robust expression separation ability\.

Figure[2\(e\)](https://arxiv.org/html/2608.02628#S4.F2.sf5)compares theR2R^\{2\}performance among DDRSR, AI Feynman, and the original methods\. The decomposed variants \(using either DDRSR or AI Feynman\) outperform the original baseline methods, demonstrating that expression separation alone effectively simplifies the target problem\. Furthermore, because DDRSR recognizes and separates a wider array of structures, it yields superior experimental results compared to AI Feynman\.

#### 4\.2\.2Bottom\-up Variable Composition

To highlight the fundamental algorithmic differences, we conduct experiments on our custom complex\-test dataset\. The results are presented in Figure[2\(f\)](https://arxiv.org/html/2608.02628#S4.F2.sf6)\.

Figure[2\(f\)](https://arxiv.org/html/2608.02628#S4.F2.sf6)illustrates theR2R^\{2\}performance comparison, where DDRSR achieves the optimal results\. Furthermore, in the isolated variable composition tests, DDRSR yields a Relative Separation Score of0\.77770\.7777, compared to a mere0\.05550\.0555for AI Feynman\. These confirm that the variable composition module in DDRSR can recognize and compose a significantly wider range of mathematical scenarios, demonstrating substantially enhanced capabilities\.

## 5Discussion

The proposed DDRSR framework exhibits remarkable versatility by effectively transforming complex problems into simpler sub\-tasks\. As demonstrated in experiments, the "Divide\-and\-Reduce" paradigm is broadly compatible to the downstream regressors, allowing seamless integration with any existing symbolic regression algorithm after decomposition\.

Nevertheless, methods like DDRSR come with inherent limitations, details are shown in Appendix[A](https://arxiv.org/html/2608.02628#A1)\.

## 6Conclusion

To overcome the narrow applicability of the principles in AI Feynman, we adopt a "Divide\-and\-Reduce" philosophy to systematically extend the principles of variable separation, nested composition, and translational symmetry, culminating in the proposed DDRSR framework\. Extensive experiments demonstrate that mathematical expression simplification effectively reduces problem complexity and significantly boosts the performance of downstream symbolic regression\. By successfully identifying and mathematically decomposing a significantly broader range of mathematical structures, DDRSR consistently achieves superior predictive performance and recovery accuracy\.

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## Appendix ALimitation

This class of methods like DDRSR and AI Feynman comes with inherent limitations\. In practice, we typically only have access to an initial batch of randomly sampled points\. However, rigorous mathematical deductions often require querying information at specific, critical coordinates\. This discrepancy restricts the application scenarios of such algorithms\. AI Feynman mitigates this by employing neural networks to approximate the underlying true mappings; however, this approach necessitates a massive number of initial samples to capture sufficiently fine\-grained details of the target spatial domain\. Alternatively, if an active sampling environment is available, exact numerical values and gradients can be efficiently obtained\.

Consequently, this methodology is most effective under conditions where we possess abundant observational data or comprehensive control of the target domain\. Conversely, under suboptimal conditions—such as sparse sampling distributions, severe noise interference, or measurement errors—the evaluation of strict mathematical properties becomes highly susceptible to degradation\.

## Appendix BMathematical Derivations and Proofs

### B\.1Multiplicative Translational Symmetry

Derivation: For annn\-dimensional functionf​\(X\)f\(X\), iff​\(X\)f\(X\)possesses additive translational symmetry with respect to dimensionsxix\_\{i\}andxjx\_\{j\}, thenf​\(X\)f\(X\)can be expressed as:

f​\(xi,xj,S​X\)=f​\(c1×xi\+c2×xj,S​X\)f\(x\_\{i\},x\_\{j\},SX\)=f\(c\_\{1\}\\times x\_\{i\}\+c\_\{2\}\\times x\_\{j\},SX\)Here,S​XSXdenotes the remaining dimensions excludingxix\_\{i\}andxjx\_\{j\}\. By introducing a new variablexi​j=c1×xi\+c2×xjx\_\{ij\}=c\_\{1\}\\times x\_\{i\}\+c\_\{2\}\\times x\_\{j\}, we obtain:

∂f​\(X\)∂xi=∂f​\(X\)∂xi​j×∂xi​j∂xi=c1×∂f​\(X\)∂xi​j\\frac\{\\partial f\(X\)\}\{\\partial x\_\{i\}\}=\\frac\{\\partial f\(X\)\}\{\\partial x\_\{ij\}\}\\times\\frac\{\\partial x\_\{ij\}\}\{\\partial x\_\{i\}\}=c\_\{1\}\\times\\frac\{\\partial f\(X\)\}\{\\partial x\_\{ij\}\}∂f​\(X\)∂xj=∂f​\(X\)∂xi​j×∂xi​j∂xj=c2×∂f​\(X\)∂xi​j\\frac\{\\partial f\(X\)\}\{\\partial x\_\{j\}\}=\\frac\{\\partial f\(X\)\}\{\\partial x\_\{ij\}\}\\times\\frac\{\\partial x\_\{ij\}\}\{\\partial x\_\{j\}\}=c\_\{2\}\\times\\frac\{\\partial f\(X\)\}\{\\partial x\_\{ij\}\}Consequently, the ratio of these partial derivatives is given by:

∂f​\(X\)∂xi÷∂f​\(X\)∂xj=c1c2=constant\\frac\{\\partial f\(X\)\}\{\\partial x\_\{i\}\}\\div\\frac\{\\partial f\(X\)\}\{\\partial x\_\{j\}\}=\\frac\{c\_\{1\}\}\{c\_\{2\}\}=\\text\{constant\}If the above equation holds for any arbitraryxix\_\{i\}andxjx\_\{j\}, we determine thatf​\(X\)f\(X\)possesses additive translational symmetry along thexix\_\{i\}andxjx\_\{j\}dimensions, prompting the introduction ofxn​e​w=xi\+c2c1×xjx\_\{new\}=x\_\{i\}\+\\frac\{c\_\{2\}\}\{c\_\{1\}\}\\times x\_\{j\}\.

### B\.2Multiplicative Translational Symmetry

Derivation: For annn\-dimensional functionf​\(X\)f\(X\), iff​\(X\)f\(X\)possesses multiplicative translational symmetry with respect to dimensionsxix\_\{i\}andxjx\_\{j\}, thenf​\(X\)f\(X\)can be formulated as:

f​\(xi,xj,S​X\)=f​\(xic1×xjc2,S​X\)f\(x\_\{i\},x\_\{j\},SX\)=f\(x\_\{i\}^\{c\_\{1\}\}\\times x\_\{j\}^\{c\_\{2\}\},SX\)Here,S​XSXdenotes the remaining dimensions excludingxix\_\{i\}andxjx\_\{j\}\. By introducing a new variablexi​j=xic1×xjc2x\_\{ij\}=x\_\{i\}^\{c\_\{1\}\}\\times x\_\{j\}^\{c\_\{2\}\}, we obtain:

∂f​\(X\)∂xi=∂f​\(X\)∂xi​j×∂xi​j∂xi=c1×∂f​\(X\)∂xi​j×xic1−1×xjc2\\frac\{\\partial f\(X\)\}\{\\partial x\_\{i\}\}=\\frac\{\\partial f\(X\)\}\{\\partial x\_\{ij\}\}\\times\\frac\{\\partial x\_\{ij\}\}\{\\partial x\_\{i\}\}=c\_\{1\}\\times\\frac\{\\partial f\(X\)\}\{\\partial x\_\{ij\}\}\\times x\_\{i\}^\{c\_\{1\}\-1\}\\times x\_\{j\}^\{c\_\{2\}\}∂f​\(X\)∂xj=∂f​\(X\)∂xi​j×∂xi​j∂xj=c2×∂f​\(X\)∂xi​j×xic1×xjc2−1\\frac\{\\partial f\(X\)\}\{\\partial x\_\{j\}\}=\\frac\{\\partial f\(X\)\}\{\\partial x\_\{ij\}\}\\times\\frac\{\\partial x\_\{ij\}\}\{\\partial x\_\{j\}\}=c\_\{2\}\\times\\frac\{\\partial f\(X\)\}\{\\partial x\_\{ij\}\}\\times x\_\{i\}^\{c\_\{1\}\}\\times x\_\{j\}^\{c\_\{2\}\-1\}Consequently, dividing the two equations yields:

∂f​\(X\)∂xi÷∂f​\(X\)∂xj=c1c2×xjxi\\frac\{\\partial f\(X\)\}\{\\partial x\_\{i\}\}\\div\\frac\{\\partial f\(X\)\}\{\\partial x\_\{j\}\}=\\frac\{c\_\{1\}\}\{c\_\{2\}\}\\times\\frac\{x\_\{j\}\}\{x\_\{i\}\}∂f​\(X\)∂xi÷∂f​\(X\)∂xj×xixj=c1c2=constant\\frac\{\\partial f\(X\)\}\{\\partial x\_\{i\}\}\\div\\frac\{\\partial f\(X\)\}\{\\partial x\_\{j\}\}\\times\\frac\{x\_\{i\}\}\{x\_\{j\}\}=\\frac\{c\_\{1\}\}\{c\_\{2\}\}=\\text\{constant\}If the above equation holds for any arbitraryxix\_\{i\}andxjx\_\{j\}, we determine thatf​\(X\)f\(X\)possesses multiplicative translational symmetry along thexix\_\{i\}andxjx\_\{j\}dimensions, prompting the introduction ofXn​e​w=xic1×xjc2X\_\{new\}=x\_\{i\}^\{c\_\{1\}\}\\times x\_\{j\}^\{c\_\{2\}\}\.

### B\.3Additive Separability with Overlapping Variables

Derivation: For annn\-dimensional functionf​\(X\)f\(X\), iff​\(X\)f\(X\)exhibits additive separability with overlapping variables with respect to dimensionsS​X1SX\_\{1\}andS​X3SX\_\{3\}, thenf​\(X\)f\(X\)can be expressed as:

f​\(S​X1,S​X2,S​X3\)=g​\(S​X1,S​X2\)\+h​\(S​X2,S​X3\)f\(SX\_\{1\},SX\_\{2\},SX\_\{3\}\)=g\(SX\_\{1\},SX\_\{2\}\)\+h\(SX\_\{2\},SX\_\{3\}\)By manipulating the terms, we derive:

f​\(S​X1,S​X2,S​X3\)=\[g​\(S​X1,S​X2\)\+h​\(S​X2,S​C3\)\]\+\[g​\(S​C1,S​X2\)\+h​\(S​X2,S​X3\)\]−\[g​\(S​C1,S​X2\)\+h​\(S​X2,S​C3\)\]f\(SX\_\{1\},SX\_\{2\},SX\_\{3\}\)=\[g\(SX\_\{1\},SX\_\{2\}\)\+h\(SX\_\{2\},SC\_\{3\}\)\]\+\[g\(SC\_\{1\},SX\_\{2\}\)\+h\(SX\_\{2\},SX\_\{3\}\)\]\-\[g\(SC\_\{1\},SX\_\{2\}\)\+h\(SX\_\{2\},SC\_\{3\}\)\]=f​\(S​X1,S​X2,S​C3\)\+f​\(S​C1,S​X2,S​X3\)−f​\(S​C1,S​X2,S​C3\)=f\(SX\_\{1\},SX\_\{2\},SC\_\{3\}\)\+f\(SC\_\{1\},SX\_\{2\},SX\_\{3\}\)\-f\(SC\_\{1\},SX\_\{2\},SC\_\{3\}\)If the above relationship holds for anyS​X1SX\_\{1\}andS​X3SX\_\{3\}alongside any fixed constantsS​CSC, we conclude thatf​\(X\)f\(X\)is additively separable across dimensionsS​X1SX\_\{1\}andS​X3SX\_\{3\}\. To obtain the explicit function mappings for the sub\-functionsggandhhto facilitate further separation, we define:

g​\(S​X1,S​X2\)=f​\(S​X1,S​X2,S​C3\)−f​\(S​C1,S​X2,S​C3\)g\(SX\_\{1\},SX\_\{2\}\)=f\(SX\_\{1\},SX\_\{2\},SC\_\{3\}\)\-f\(SC\_\{1\},SX\_\{2\},SC\_\{3\}\)h​\(S​X2,S​X3\)=f​\(S​C1,S​X2,S​X3\)h\(SX\_\{2\},SX\_\{3\}\)=f\(SC\_\{1\},SX\_\{2\},SX\_\{3\}\)

### B\.4Multiplicative Separability with Overlapping Variables

Derivation: For annn\-dimensional functionf​\(X\)f\(X\), iff​\(X\)f\(X\)exhibits multiplicative separability with overlapping variables with respect to dimensionsS​X1SX\_\{1\}andS​X3SX\_\{3\}, thenf​\(X\)f\(X\)can be expressed as:

f​\(S​X1,S​X2,S​X3\)=g​\(S​X1,S​X2\)×h​\(S​X2,S​X3\)f\(SX\_\{1\},SX\_\{2\},SX\_\{3\}\)=g\(SX\_\{1\},SX\_\{2\}\)\\times h\(SX\_\{2\},SX\_\{3\}\)Through algebraic substitution, we establish:

f​\(S​X1,S​X2,S​X3\)=\[g​\(S​X1,S​X2\)×h​\(S​X2,S​C3\)\]×\[g​\(S​C1,S​X2\)×h​\(S​X2,S​X3\)\]\[g​\(S​C1,S​X2\)×h​\(S​X2,S​C3\)\]f\(SX\_\{1\},SX\_\{2\},SX\_\{3\}\)=\\frac\{\[g\(SX\_\{1\},SX\_\{2\}\)\\times h\(SX\_\{2\},SC\_\{3\}\)\]\\times\[g\(SC\_\{1\},SX\_\{2\}\)\\times h\(SX\_\{2\},SX\_\{3\}\)\]\}\{\[g\(SC\_\{1\},SX\_\{2\}\)\\times h\(SX\_\{2\},SC\_\{3\}\)\]\}=f​\(S​X1,S​X2,S​C3\)×f​\(S​C1,S​X2,S​X3\)f​\(S​C1,S​X2,S​C3\)=\\frac\{f\(SX\_\{1\},SX\_\{2\},SC\_\{3\}\)\\times f\(SC\_\{1\},SX\_\{2\},SX\_\{3\}\)\}\{f\(SC\_\{1\},SX\_\{2\},SC\_\{3\}\)\}If the above relationship holds for anyS​X1SX\_\{1\}andS​X3SX\_\{3\}alongside any fixed constantsS​CSC, we conclude thatf​\(X\)f\(X\)is multiplicatively separable across dimensionsS​X1SX\_\{1\}andS​X3SX\_\{3\}\. To obtain the function mappings for sub\-functionsggandhhto enable further decomposition, we define:

g​\(S​X1,S​X2\)=f​\(S​X1,S​X2,S​C3\)f​\(S​C1,S​X2,S​C3\)g\(SX\_\{1\},SX\_\{2\}\)=\\frac\{f\(SX\_\{1\},SX\_\{2\},SC\_\{3\}\)\}\{f\(SC\_\{1\},SX\_\{2\},SC\_\{3\}\)\}h​\(S​X2,S​X3\)=f​\(S​C1,S​X2,S​X3\)h\(SX\_\{2\},SX\_\{3\}\)=f\(SC\_\{1\},SX\_\{2\},SX\_\{3\}\)

### B\.5Compositional Form of Additive Separability

Proof: Assume the functionffis additively separable with respect to dimensionsS​X1SX\_\{1\}andS​X3SX\_\{3\}, and concurrently additively separable with respect toS​X2SX\_\{2\}andS​X3SX\_\{3\}\. The functionffcan therefore be represented as:

f​\(S​X1,S​X2,S​X3\)=g1​\(S​X1,S​X2\)\+h1​\(S​X2,S​X3\)=g2​\(S​X1,S​X2\)\+h2​\(S​X1,S​X3\)f\(SX\_\{1\},SX\_\{2\},SX\_\{3\}\)=g\_\{1\}\(SX\_\{1\},SX\_\{2\}\)\+h\_\{1\}\(SX\_\{2\},SX\_\{3\}\)=g\_\{2\}\(SX\_\{1\},SX\_\{2\}\)\+h\_\{2\}\(SX\_\{1\},SX\_\{3\}\)Taking the partial derivative with respect toS​X3SX\_\{3\}yields:

∂f​\(S​X1,S​X2,S​X3\)∂S​X3=∂h1​\(S​X2,S​X3\)∂S​X3=∂h2​\(S​X1,S​X3\)∂S​X3\\frac\{\\partial f\(SX\_\{1\},SX\_\{2\},SX\_\{3\}\)\}\{\\partial SX\_\{3\}\}=\\frac\{\\partial h\_\{1\}\(SX\_\{2\},SX\_\{3\}\)\}\{\\partial SX\_\{3\}\}=\\frac\{\\partial h\_\{2\}\(SX\_\{1\},SX\_\{3\}\)\}\{\\partial SX\_\{3\}\}LetT​\(X\)=h1​\(S​X2,S​X3\)−h2​\(S​X1,S​X3\)T\(X\)=h\_\{1\}\(SX\_\{2\},SX\_\{3\}\)\-h\_\{2\}\(SX\_\{1\},SX\_\{3\}\)\. It follows that:

∂T​\(X\)∂S​X3=∂h1​\(S​X2,S​X3\)∂S​X3−∂h2​\(S​X1,S​X3\)∂S​X3=0\\frac\{\\partial T\(X\)\}\{\\partial SX\_\{3\}\}=\\frac\{\\partial h\_\{1\}\(SX\_\{2\},SX\_\{3\}\)\}\{\\partial SX\_\{3\}\}\-\\frac\{\\partial h\_\{2\}\(SX\_\{1\},SX\_\{3\}\)\}\{\\partial SX\_\{3\}\}=0This indicates thatT​\(X\)T\(X\)is independent ofS​X3SX\_\{3\}and can be strictly expressed in the formT​\(X\)=T​\(S​X1,S​X2\)T\(X\)=T\(SX\_\{1\},SX\_\{2\}\)\. Simultaneously, we have:

∂∂T​\(S​X1,S​X2\)∂S​X1∂S​X2=∂∂h2​\(S​X1,S​X3\)∂S​X1∂S​X2=0,∂∂T​\(S​X1,S​X2\)∂S​X2∂S​X1=∂∂h1​\(S​X2,S​X3\)∂S​X2∂S​X1=0\\frac\{\\partial\\frac\{\\partial T\(SX\_\{1\},SX\_\{2\}\)\}\{\\partial SX\_\{1\}\}\}\{\\partial SX\_\{2\}\}=\\frac\{\\partial\\frac\{\\partial h\_\{2\}\(SX\_\{1\},SX\_\{3\}\)\}\{\\partial SX\_\{1\}\}\}\{\\partial SX\_\{2\}\}=0,\\quad\\frac\{\\partial\\frac\{\\partial T\(SX\_\{1\},SX\_\{2\}\)\}\{\\partial SX\_\{2\}\}\}\{\\partial SX\_\{1\}\}=\\frac\{\\partial\\frac\{\\partial h\_\{1\}\(SX\_\{2\},SX\_\{3\}\)\}\{\\partial SX\_\{2\}\}\}\{\\partial SX\_\{1\}\}=0Thus,T​\(X\)T\(X\)can be further decomposed into the formT​\(X\)=A​\(S​X1\)\+B​\(S​X2\)T\(X\)=A\(SX\_\{1\}\)\+B\(SX\_\{2\}\)\. Furthermore:

h1​\(S​X2,S​X3\)−B​\(S​X2\)=h2​\(S​X1,S​X3\)\+A​\(S​X1\)h\_\{1\}\(SX\_\{2\},SX\_\{3\}\)\-B\(SX\_\{2\}\)=h\_\{2\}\(SX\_\{1\},SX\_\{3\}\)\+A\(SX\_\{1\}\)Given that the left side is a function exclusively ofS​X2SX\_\{2\}andS​X3SX\_\{3\}, and the right side is a function exclusively ofS​X1SX\_\{1\}andS​X3SX\_\{3\}, both sides must equal a function solely dependent onS​X3SX\_\{3\}\. We therefore deduce:

h1​\(S​X2,S​X3\)=C​\(S​X3\)\+B​\(S​X2\),h2​\(S​X1,S​X3\)=C​\(S​X3\)−A​\(S​X1\)h\_\{1\}\(SX\_\{2\},SX\_\{3\}\)=C\(SX\_\{3\}\)\+B\(SX\_\{2\}\),\\quad h\_\{2\}\(SX\_\{1\},SX\_\{3\}\)=C\(SX\_\{3\}\)\-A\(SX\_\{1\}\)Substituting this back yields:

f​\(S​X1,S​X2,S​X3\)=g1​\(S​X1,S​X2\)\+h1​\(S​X2,S​X3\)=\[g1​\(S​X1,S​X2\)\+B​\(S​X2\)\]\+C​\(S​X3\)f\(SX\_\{1\},SX\_\{2\},SX\_\{3\}\)=g\_\{1\}\(SX\_\{1\},SX\_\{2\}\)\+h\_\{1\}\(SX\_\{2\},SX\_\{3\}\)=\[g\_\{1\}\(SX\_\{1\},SX\_\{2\}\)\+B\(SX\_\{2\}\)\]\+C\(SX\_\{3\}\)This rigorously proves thatffcan be uniquely expressed in the formf​\(S​X1,S​X2,S​X3\)=g​\(S​X1,S​X2\)\+h​\(S​X3\)f\(SX\_\{1\},SX\_\{2\},SX\_\{3\}\)=g\(SX\_\{1\},SX\_\{2\}\)\+h\(SX\_\{3\}\)\.

### B\.6Compositional Form of Multiplicative Separability

Proof: Assume the functionffis multiplicatively separable with respect to dimensionsS​X1SX\_\{1\}andS​X3SX\_\{3\}, and concurrently multiplicatively separable with respect toS​X2SX\_\{2\}andS​X3SX\_\{3\}\. The functionffcan therefore be represented as:

f​\(S​X1,S​X2,S​X3\)=g1​\(S​X1,S​X2\)×h1​\(S​X2,S​X3\)=g2​\(S​X1,S​X2\)×h2​\(S​X1,S​X3\)f\(SX\_\{1\},SX\_\{2\},SX\_\{3\}\)=g\_\{1\}\(SX\_\{1\},SX\_\{2\}\)\\times h\_\{1\}\(SX\_\{2\},SX\_\{3\}\)=g\_\{2\}\(SX\_\{1\},SX\_\{2\}\)\\times h\_\{2\}\(SX\_\{1\},SX\_\{3\}\)Taking the logarithm of both sides provides:

log⁡\(f​\(S​X1,S​X2,S​X3\)\)=log⁡\(g1​\(S​X1,S​X2\)\)\+log⁡\(h1​\(S​X2,S​X3\)\)=log⁡\(g2​\(S​X1,S​X2\)\)\+log⁡\(h2​\(S​X1,S​X3\)\)\\log\(f\(SX\_\{1\},SX\_\{2\},SX\_\{3\}\)\)=\\log\(g\_\{1\}\(SX\_\{1\},SX\_\{2\}\)\)\+\\log\(h\_\{1\}\(SX\_\{2\},SX\_\{3\}\)\)=\\log\(g\_\{2\}\(SX\_\{1\},SX\_\{2\}\)\)\+\\log\(h\_\{2\}\(SX\_\{1\},SX\_\{3\}\)\)Invoking the conclusions derived in[B\.5](https://arxiv.org/html/2608.02628#A2.SS5), it is evident thatlog⁡\(f\)\\log\(f\)strictly adheres to the additive form:

log⁡\(f​\(S​X1,S​X2,S​X3\)\)=g​\(S​X1,S​X2\)\+h​\(S​X3\)\\log\(f\(SX\_\{1\},SX\_\{2\},SX\_\{3\}\)\)=g\(SX\_\{1\},SX\_\{2\}\)\+h\(SX\_\{3\}\)Consequently, taking the exponential yields:

f​\(S​X1,S​X2,S​X3\)=exp⁡\(g​\(S​X1,S​X2\)\+h​\(S​X3\)\)=exp⁡\(g​\(S​X1,S​X2\)\)×exp⁡\(h​\(S​X3\)\)f\(SX\_\{1\},SX\_\{2\},SX\_\{3\}\)=\\exp\(g\(SX\_\{1\},SX\_\{2\}\)\+h\(SX\_\{3\}\)\)=\\exp\(g\(SX\_\{1\},SX\_\{2\}\)\)\\times\\exp\(h\(SX\_\{3\}\)\)This rigorously proves thatffcan be expressed in the formf​\(S​X1,S​X2,S​X3\)=g​\(S​X1,S​X2\)×h​\(S​X3\)f\(SX\_\{1\},SX\_\{2\},SX\_\{3\}\)=g\(SX\_\{1\},SX\_\{2\}\)\\times h\(SX\_\{3\}\)\.

### B\.7Multiplicative Separability with Constant Interference

Derivation: For annn\-dimensional functionf​\(X\)f\(X\), iff​\(X\)f\(X\)exhibits multiplicative separability containing a constant term across dimensionsS​X1SX\_\{1\}andS​X3SX\_\{3\}, it can be expressed as:

f​\(X\)=g​\(xi,S​X\)×h​\(xj,S​X\)\+cf\(X\)=g\(x\_\{i\},SX\)\\times h\(x\_\{j\},SX\)\+cHere,S​XSXdenotes the remaining dimensions excludingxix\_\{i\}andxjx\_\{j\}\. The partial derivatives are:

∂f​\(X\)∂xi=h​\(xj,S​X\)×∂g​\(xi,S​X\)∂xi,∂f​\(X\)∂xj=g​\(xi,S​X\)×∂h​\(xj,S​X\)∂xj\\frac\{\\partial f\(X\)\}\{\\partial x\_\{i\}\}=h\(x\_\{j\},SX\)\\times\\frac\{\\partial g\(x\_\{i\},SX\)\}\{\\partial x\_\{i\}\},\\quad\\frac\{\\partial f\(X\)\}\{\\partial x\_\{j\}\}=g\(x\_\{i\},SX\)\\times\\frac\{\\partial h\(x\_\{j\},SX\)\}\{\\partial x\_\{j\}\}∂2f​\(X\)∂xi​∂xj=∂g​\(xi,S​X\)∂xi×∂h​\(xj,S​X\)∂xj\\frac\{\\partial^\{2\}f\(X\)\}\{\\partial x\_\{i\}\\partial x\_\{j\}\}=\\frac\{\\partial g\(x\_\{i\},SX\)\}\{\\partial x\_\{i\}\}\\times\\frac\{\\partial h\(x\_\{j\},SX\)\}\{\\partial x\_\{j\}\}By computing the ratios, we obtain:

∂2f​\(X\)∂xi​∂xj÷∂f​\(X\)∂xi=∂h​\(xj,S​X\)∂xj÷h​\(xj,S​X\),∂2f​\(X\)∂xi​∂xj÷∂f​\(X\)∂xj=∂g​\(xi,S​X\)∂xi÷g​\(xi,S​X\)\\frac\{\\partial^\{2\}f\(X\)\}\{\\partial x\_\{i\}\\partial x\_\{j\}\}\\div\\frac\{\\partial f\(X\)\}\{\\partial x\_\{i\}\}=\\frac\{\\partial h\(x\_\{j\},SX\)\}\{\\partial x\_\{j\}\}\\div h\(x\_\{j\},SX\),\\quad\\frac\{\\partial^\{2\}f\(X\)\}\{\\partial x\_\{i\}\\partial x\_\{j\}\}\\div\\frac\{\\partial f\(X\)\}\{\\partial x\_\{j\}\}=\\frac\{\\partial g\(x\_\{i\},SX\)\}\{\\partial x\_\{i\}\}\\div g\(x\_\{i\},SX\)Letui​\(X\)=∂2f​\(X\)∂xi​∂xj÷∂f​\(X\)∂xiu\_\{i\}\(X\)=\\frac\{\\partial^\{2\}f\(X\)\}\{\\partial x\_\{i\}\\partial x\_\{j\}\}\\div\\frac\{\\partial f\(X\)\}\{\\partial x\_\{i\}\}anduj​\(X\)=∂2f​\(X\)∂xi​∂xj÷∂f​\(X\)∂xju\_\{j\}\(X\)=\\frac\{\\partial^\{2\}f\(X\)\}\{\\partial x\_\{i\}\\partial x\_\{j\}\}\\div\\frac\{\\partial f\(X\)\}\{\\partial x\_\{j\}\}\. Iff​\(X\)f\(X\)exhibits multiplicative separability with a constant term,ui​\(X\)u\_\{i\}\(X\)is independent ofxix\_\{i\}, anduj​\(X\)u\_\{j\}\(X\)is independent ofxjx\_\{j\}\. Thus,∂ui​\(X\)∂xi=0\\frac\{\\partial u\_\{i\}\(X\)\}\{\\partial x\_\{i\}\}=0and∂uj​\(X\)∂xj=0\\frac\{\\partial u\_\{j\}\(X\)\}\{\\partial x\_\{j\}\}=0\.

Therefore, for any functionf​\(X\)f\(X\)of unknown form, we constructu​\(X\)u\(X\)strictly according to the above formulation\. If there exists a pair\(xi,xj\)\(x\_\{i\},x\_\{j\}\)such that the aforementioned conditions hold true across any 2\-dimensional subspace with fixed remaining dimensions and any values ofXiX\_\{i\}andXjX\_\{j\}, we determine thatf​\(X\)f\(X\)possesses multiplicative separability with a constant term across dimensionsxix\_\{i\}andxjx\_\{j\}\.

### B\.8Separation Mechanism for Multiplicative Forms with Constant Interference

Whenf​\(X\)f\(X\)is determined to possess a multiplicatively separable form under constant interference, precise structural separation can be executed\.

Derivation: We knowf​\(X\)f\(X\)has the structural formf​\(X\)=g​\(xi,S​X\)×h​\(xj,S​X\)\+cf\(X\)=g\(x\_\{i\},SX\)\\times h\(x\_\{j\},SX\)\+c\. First, we fix all dimensions excludingxix\_\{i\}andxjx\_\{j\}to valid random values, establishing a 2\-dimensional subspace spanned byxix\_\{i\}andxjx\_\{j\}\. We sample two valid points along thexix\_\{i\}andxjx\_\{j\}dimensions, denoted as\(xi​1,xi​2\)\(x\_\{i1\},x\_\{i2\}\)and\(xj​1,xj​2\)\(x\_\{j1\},x\_\{j2\}\)respectively, forming a 2\-dimensional bounding rectangle\. Within this subspace \(whereS​XSXacts as a constant, and is thus omitted from the subsequent notation for brevity\), we evaluate the integrals:

f​\(xi​2,xj​2\)=f​\(xi​1,xj​2\)\+∫xi​1xi​2∂f​\(X\)∂xi\|xj=xj​2=f​\(xi​1,xj​2\)\+h​\(xj​2\)×∫xi​1xi​2∂g​\(xi\)∂xif\(x\_\{i2\},x\_\{j2\}\)=f\(x\_\{i1\},x\_\{j2\}\)\+\\int\_\{x\_\{i1\}\}^\{x\_\{i2\}\}\\frac\{\\partial f\(X\)\}\{\\partial x\_\{i\}\}\\bigg\|\_\{x\_\{j\}=x\_\{j2\}\}=f\(x\_\{i1\},x\_\{j2\}\)\+h\(x\_\{j2\}\)\\times\\int\_\{x\_\{i1\}\}^\{x\_\{i2\}\}\\frac\{\\partial g\(x\_\{i\}\)\}\{\\partial x\_\{i\}\}f​\(xi​2,xj​1\)=f​\(xi​1,xj​1\)\+h​\(xj​1\)×∫xi​1xi​2∂g​\(xi\)∂xif\(x\_\{i2\},x\_\{j1\}\)=f\(x\_\{i1\},x\_\{j1\}\)\+h\(x\_\{j1\}\)\\times\\int\_\{x\_\{i1\}\}^\{x\_\{i2\}\}\\frac\{\\partial g\(x\_\{i\}\)\}\{\\partial x\_\{i\}\}Therefore, by computing the ratio, we derive:

h​\(xj​2\)h​\(xj​1\)=f​\(xi​2,xj​2\)−f​\(xi​1,xj​2\)f​\(xi​2,xj​1\)−f​\(xi​1,xj​1\)\\frac\{h\(x\_\{j2\}\)\}\{h\(x\_\{j1\}\)\}=\\frac\{f\(x\_\{i2\},x\_\{j2\}\)\-f\(x\_\{i1\},x\_\{j2\}\)\}\{f\(x\_\{i2\},x\_\{j1\}\)\-f\(x\_\{i1\},x\_\{j1\}\)\}By identical logic, we deduce:

g​\(xi​2\)g​\(xi​1\)=f​\(xi​2,xj​2\)−f​\(xi​2,xj​1\)f​\(xi​1,xj​2\)−f​\(xi​1,xj​1\)\\frac\{g\(x\_\{i2\}\)\}\{g\(x\_\{i1\}\)\}=\\frac\{f\(x\_\{i2\},x\_\{j2\}\)\-f\(x\_\{i2\},x\_\{j1\}\)\}\{f\(x\_\{i1\},x\_\{j2\}\)\-f\(x\_\{i1\},x\_\{j1\}\)\}Furthermore, we establish the difference:

f​\(xi​2,xj​2\)−f​\(xi​2,xj​1\)=\(g​\(xi​2\)×h​\(xj​2\)\+c\)−\(g​\(xi​2\)×h​\(xj​1\)\+c\)f\(x\_\{i2\},x\_\{j2\}\)\-f\(x\_\{i2\},x\_\{j1\}\)=\(g\(x\_\{i2\}\)\\times h\(x\_\{j2\}\)\+c\)\-\(g\(x\_\{i2\}\)\\times h\(x\_\{j1\}\)\+c\)Which yields:

h​\(xj​2\)−h​\(xj​1\)=f​\(xi​2,xj​2\)−f​\(xi​2,xj​1\)g​\(xi​2\)h\(x\_\{j2\}\)\-h\(x\_\{j1\}\)=\\frac\{f\(x\_\{i2\},x\_\{j2\}\)\-f\(x\_\{i2\},x\_\{j1\}\)\}\{g\(x\_\{i2\}\)\}Similarly:

g​\(xi​2\)−g​\(xi​1\)=f​\(xi​2,xj​2\)−f​\(xi​1,xj​2\)h​\(xj​2\)g\(x\_\{i2\}\)\-g\(x\_\{i1\}\)=\\frac\{f\(x\_\{i2\},x\_\{j2\}\)\-f\(x\_\{i1\},x\_\{j2\}\)\}\{h\(x\_\{j2\}\)\}By solving this system of simultaneous equations, we can analytically extract:

g​\(xi​2\)×h​\(xj​2\)=\(f​\(xi​2,xj​2\)−f​\(xi​2,xj​1\)\)×\(f​\(xi​2,xj​2\)−f​\(xi​1,xj​2\)\)f​\(xi​2,xj​2\)−f​\(xi​1,xj​2\)−f​\(xi​2,xj​1\)\+f​\(xi​1,xj​1\)g\(x\_\{i2\}\)\\times h\(x\_\{j2\}\)=\\frac\{\(f\(x\_\{i2\},x\_\{j2\}\)\-f\(x\_\{i2\},x\_\{j1\}\)\)\\times\(f\(x\_\{i2\},x\_\{j2\}\)\-f\(x\_\{i1\},x\_\{j2\}\)\)\}\{f\(x\_\{i2\},x\_\{j2\}\)\-f\(x\_\{i1\},x\_\{j2\}\)\-f\(x\_\{i2\},x\_\{j1\}\)\+f\(x\_\{i1\},x\_\{j1\}\)\}Substituting this back into the original expression forf​\(xi​2,xj​2\)f\(x\_\{i2\},x\_\{j2\}\), the precise value of the constantccis computed as:

c=f​\(xi​2,xj​2\)−\(f​\(xi​2,xj​2\)−f​\(xi​2,xj​1\)\)×\(f​\(xi​2,xj​2\)−f​\(xi​1,xj​2\)\)f​\(xi​2,xj​2\)−f​\(xi​1,xj​2\)−f​\(xi​2,xj​1\)\+f​\(xi​1,xj​1\)c=f\(x\_\{i2\},x\_\{j2\}\)\-\\frac\{\(f\(x\_\{i2\},x\_\{j2\}\)\-f\(x\_\{i2\},x\_\{j1\}\)\)\\times\(f\(x\_\{i2\},x\_\{j2\}\)\-f\(x\_\{i1\},x\_\{j2\}\)\)\}\{f\(x\_\{i2\},x\_\{j2\}\)\-f\(x\_\{i1\},x\_\{j2\}\)\-f\(x\_\{i2\},x\_\{j1\}\)\+f\(x\_\{i1\},x\_\{j1\}\)\}Finally, by settingfn​e​w​\(X\)=f​\(X\)−cf\_\{new\}\(X\)=f\(X\)\-c, the problem is mathematically reduced to standard multiplicative separability, thereby enabling the execution of subsequent variable separation procedures\.

### B\.9Generalized Separable Forms

Derivation: For annn\-dimensional functionf​\(X\)f\(X\), iff​\(X\)f\(X\)exhibits a generalized separable form with respect to dimensionsS​X1SX\_\{1\}andS​X3SX\_\{3\}, it can be expressed as:

f​\(X\)=U​\(g​\(xi,S​X\)\+h​\(xj,S​X\)\)orf​\(X\)=U​\(g​\(xi,S​X\)×h​\(xj,S​X\)\)f\(X\)=U\(g\(x\_\{i\},SX\)\+h\(x\_\{j\},SX\)\)\\quad\\text\{or\}\\quad f\(X\)=U\(g\(x\_\{i\},SX\)\\times h\(x\_\{j\},SX\)\)Here,S​XSXdenotes the remaining dimensions excludingxix\_\{i\}andxjx\_\{j\}\. Iff​\(X\)f\(X\)assumes the additive internal formf​\(X\)=U​\(g​\(xi,S​X\)\+h​\(xj,S​X\)\)f\(X\)=U\(g\(x\_\{i\},SX\)\+h\(x\_\{j\},SX\)\), applying the chain rule yields:

∂f​\(X\)∂xi=∂f​\(X\)∂\(g​\(xi,S​X\)\+h​\(xj,S​X\)\)×∂g​\(xi,S​X\)∂xi\\frac\{\\partial f\(X\)\}\{\\partial x\_\{i\}\}=\\frac\{\\partial f\(X\)\}\{\\partial\(g\(x\_\{i\},SX\)\+h\(x\_\{j\},SX\)\)\}\\times\\frac\{\\partial g\(x\_\{i\},SX\)\}\{\\partial x\_\{i\}\}∂f​\(X\)∂xj=∂f​\(X\)∂\(g​\(xi,S​X\)\+h​\(xj,S​X\)\)×∂h​\(xj,S​X\)∂xj\\frac\{\\partial f\(X\)\}\{\\partial x\_\{j\}\}=\\frac\{\\partial f\(X\)\}\{\\partial\(g\(x\_\{i\},SX\)\+h\(x\_\{j\},SX\)\)\}\\times\\frac\{\\partial h\(x\_\{j\},SX\)\}\{\\partial x\_\{j\}\}∂f​\(X\)∂xi÷∂f​\(X\)∂xj=∂g​\(xi,S​X\)∂xi÷∂h​\(xj,S​X\)∂xj\\frac\{\\partial f\(X\)\}\{\\partial x\_\{i\}\}\\div\\frac\{\\partial f\(X\)\}\{\\partial x\_\{j\}\}=\\frac\{\\partial g\(x\_\{i\},SX\)\}\{\\partial x\_\{i\}\}\\div\\frac\{\\partial h\(x\_\{j\},SX\)\}\{\\partial x\_\{j\}\}Letv​\(X\)=∂f​\(X\)∂xi÷∂f​\(X\)∂xjv\(X\)=\\frac\{\\partial f\(X\)\}\{\\partial x\_\{i\}\}\\div\\frac\{\\partial f\(X\)\}\{\\partial x\_\{j\}\}\. It is evident thatv​\(X\)v\(X\)is multiplicatively separable across dimensionsxix\_\{i\}andxjx\_\{j\}\. Similarly, iff​\(X\)f\(X\)assumes the multiplicative internal formf​\(X\)=U​\(g​\(xi,S​X\)×h​\(xj,S​X\)\)f\(X\)=U\(g\(x\_\{i\},SX\)\\times h\(x\_\{j\},SX\)\), we have:

v​\(X\)=∂f​\(X\)∂xi÷∂f​\(X\)∂xj=∂g​\(xi,S​X\)∂xi×h​\(xj,S​X\)÷\(∂h​\(xj,S​X\)∂xj×g​\(xi,S​X\)\)v\(X\)=\\frac\{\\partial f\(X\)\}\{\\partial x\_\{i\}\}\\div\\frac\{\\partial f\(X\)\}\{\\partial x\_\{j\}\}=\\frac\{\\partial g\(x\_\{i\},SX\)\}\{\\partial x\_\{i\}\}\\times h\(x\_\{j\},SX\)\\div\\left\(\\frac\{\\partial h\(x\_\{j\},SX\)\}\{\\partial x\_\{j\}\}\\times g\(x\_\{i\},SX\)\\right\)In this scenario,v​\(X\)v\(X\)is equally multiplicatively separable across dimensionsxix\_\{i\}andxjx\_\{j\}\.

Therefore, for any unknown functionf​\(X\)f\(X\), we construct the metricv​\(X\)v\(X\)according to the definitions above\. Ifv​\(X\)v\(X\)satisfies multiplicative separability across dimensionsxix\_\{i\}andxjx\_\{j\}for anyXX, we definitively classifyf​\(X\)f\(X\)as possessing a generalized separable form\.

### B\.10Nested Separation of Standard Logarithmic and Exponential Functions

Derivation: For annn\-dimensional functionf​\(X\)f\(X\), iff​\(X\)f\(X\)possesses a nested separation form of a standard logarithmic function, it can be expressed asf​\(X\)=log⁡\(g​\(X\)\)f\(X\)=\\log\(g\(X\)\), whereg​\(X\)g\(X\)can be subjected to further separation\.

Letfn​e​w​\(X\)=exp⁡\(f​\(X\)\)=g​\(X\)f\_\{new\}\(X\)=\\exp\(f\(X\)\)=g\(X\)\. Under this mapping,fn​e​wf\_\{new\}inherently maintains the capacity for further separation\. Therefore, for any unknown target mappingf​\(X\)f\(X\), iffn​e​w​\(X\)f\_\{new\}\(X\)is proven to be separable, we confirm that the original functionf​\(X\)f\(X\)possesses a nested separation form of a standard logarithmic function\. The derivation and logic for exponential functions \(exp\\exp\) follow an identical paradigm\.

It is worth noting that when evaluating the internal separability during this nested decomposition, we typically prohibit subsequent nested separation to prevent infinite recursive calls\. Theoretically, however, guided by specific priors, the maximum nesting depth can be established as a tunable hyperparameter, thereby permitting a predefined number of nested hierarchical layers\.

### B\.11Nested Separation Form of Standard Trigonometric Functions

Derivation: For annn\-dimensional functionf​\(X\)f\(X\), iff​\(X\)f\(X\)possesses a nested separation form of standard trigonometric functions, it can be expressed as:

f​\(X\)=sin⁡\(or​cos\)​\(g​\(xi,S​X\)\+h​\(xj,S​X\)\)orf​\(X\)=sin⁡\(or​cos\)​\(g​\(xi,S​X\)×h​\(xj,S​X\)\)f\(X\)=\\sin\(\\text\{or \}\\cos\)\(g\(x\_\{i\},SX\)\+h\(x\_\{j\},SX\)\)\\quad\\text\{or\}\\quad f\(X\)=\\sin\(\\text\{or \}\\cos\)\(g\(x\_\{i\},SX\)\\times h\(x\_\{j\},SX\)\)Here,S​XSXdenotes the remaining dimensions excludingxix\_\{i\}andxjx\_\{j\}\. Iff​\(X\)f\(X\)assumes the internal additive structuref​\(X\)=sin⁡\(g​\(xi,S​X\)\+h​\(xj,S​X\)\)f\(X\)=\\sin\(g\(x\_\{i\},SX\)\+h\(x\_\{j\},SX\)\), computing the partial derivatives yields:

∂f​\(X\)∂xi=∂f​\(X\)∂g​\(xi,S​X\)×∂g​\(xi,S​X\)∂xi=cos⁡\(g​\(xi,S​X\)\+h​\(xj,S​X\)\)×∂g​\(xi,S​X\)∂xi\\frac\{\\partial f\(X\)\}\{\\partial x\_\{i\}\}=\\frac\{\\partial f\(X\)\}\{\\partial g\(x\_\{i\},SX\)\}\\times\\frac\{\\partial g\(x\_\{i\},SX\)\}\{\\partial x\_\{i\}\}=\\cos\(g\(x\_\{i\},SX\)\+h\(x\_\{j\},SX\)\)\\times\\frac\{\\partial g\(x\_\{i\},SX\)\}\{\\partial x\_\{i\}\}∂f​\(X\)∂xj=cos⁡\(g​\(xi,S​X\)\+h​\(xj,S​X\)\)×∂h​\(xj,S​X\)∂xj\\frac\{\\partial f\(X\)\}\{\\partial x\_\{j\}\}=\\cos\(g\(x\_\{i\},SX\)\+h\(x\_\{j\},SX\)\)\\times\\frac\{\\partial h\(x\_\{j\},SX\)\}\{\\partial x\_\{j\}\}Further multiplying these terms produces:

∂f​\(X\)∂xi×∂f​\(X\)∂xj=cos2⁡\(g​\(xi,S​X\)\+h​\(xj,S​X\)\)×∂g​\(xi,S​X\)∂xi×∂h​\(xj,S​X\)∂xj\\frac\{\\partial f\(X\)\}\{\\partial x\_\{i\}\}\\times\\frac\{\\partial f\(X\)\}\{\\partial x\_\{j\}\}=\\cos^\{2\}\(g\(x\_\{i\},SX\)\+h\(x\_\{j\},SX\)\)\\times\\frac\{\\partial g\(x\_\{i\},SX\)\}\{\\partial x\_\{i\}\}\\times\\frac\{\\partial h\(x\_\{j\},SX\)\}\{\\partial x\_\{j\}\}=\(1−f2​\(X\)\)×∂g​\(xi,S​X\)∂xi×∂h​\(xj,S​X\)∂xj=\(1\-f^\{2\}\(X\)\)\\times\\frac\{\\partial g\(x\_\{i\},SX\)\}\{\\partial x\_\{i\}\}\\times\\frac\{\\partial h\(x\_\{j\},SX\)\}\{\\partial x\_\{j\}\}LetT​\(X\)=∂f​\(X\)∂xi×∂f​\(X\)∂xj÷\(1−f2​\(X\)\)T\(X\)=\\frac\{\\partial f\(X\)\}\{\\partial x\_\{i\}\}\\times\\frac\{\\partial f\(X\)\}\{\\partial x\_\{j\}\}\\div\(1\-f^\{2\}\(X\)\)\. Iff​\(X\)f\(X\)conforms to the specified structure,T​\(X\)=∂g​\(xi,S​X\)∂xi×∂h​\(xj,S​X\)∂xjT\(X\)=\\frac\{\\partial g\(x\_\{i\},SX\)\}\{\\partial x\_\{i\}\}\\times\\frac\{\\partial h\(x\_\{j\},SX\)\}\{\\partial x\_\{j\}\}is demonstrably multiplicatively separable with respect to dimensionsxix\_\{i\}andxjx\_\{j\}\. Analogously, assuming the functionfftakes the internal multiplicative formf​\(X\)=sin⁡\(g​\(xi,S​X\)×h​\(xj,S​X\)\)f\(X\)=\\sin\(g\(x\_\{i\},SX\)\\times h\(x\_\{j\},SX\)\), we derive:

∂f​\(X\)∂xi=∂f​\(X\)∂g​\(xi,S​X\)×∂g​\(xi,S​X\)∂xi=cos⁡\(g​\(xi,S​X\)×h​\(xj,S​X\)\)×h​\(xj,S​X\)×∂g​\(xi,S​X\)∂xi\\frac\{\\partial f\(X\)\}\{\\partial x\_\{i\}\}=\\frac\{\\partial f\(X\)\}\{\\partial g\(x\_\{i\},SX\)\}\\times\\frac\{\\partial g\(x\_\{i\},SX\)\}\{\\partial x\_\{i\}\}=\\cos\(g\(x\_\{i\},SX\)\\times h\(x\_\{j\},SX\)\)\\times h\(x\_\{j\},SX\)\\times\\frac\{\\partial g\(x\_\{i\},SX\)\}\{\\partial x\_\{i\}\}∂f​\(X\)∂xj=cos⁡\(g​\(xi,S​X\)×h​\(xj,S​X\)\)×g​\(xi,S​X\)×∂h​\(xj,S​X\)∂xj\\frac\{\\partial f\(X\)\}\{\\partial x\_\{j\}\}=\\cos\(g\(x\_\{i\},SX\)\\times h\(x\_\{j\},SX\)\)\\times g\(x\_\{i\},SX\)\\times\\frac\{\\partial h\(x\_\{j\},SX\)\}\{\\partial x\_\{j\}\}Further multiplying these terms produces:

∂f​\(X\)∂xi×∂f​\(X\)∂xj=cos2⁡\(g​\(xi,S​X\)×h​\(xj,S​X\)\)×\[g​\(xi,S​X\)×∂g​\(xi,S​X\)∂xi\]×\[h​\(xj,S​X\)×∂h​\(xj,S​X\)∂xj\]\\frac\{\\partial f\(X\)\}\{\\partial x\_\{i\}\}\\times\\frac\{\\partial f\(X\)\}\{\\partial x\_\{j\}\}=\\cos^\{2\}\(g\(x\_\{i\},SX\)\\times h\(x\_\{j\},SX\)\)\\times\\left\[g\(x\_\{i\},SX\)\\times\\frac\{\\partial g\(x\_\{i\},SX\)\}\{\\partial x\_\{i\}\}\\right\]\\times\\left\[h\(x\_\{j\},SX\)\\times\\frac\{\\partial h\(x\_\{j\},SX\)\}\{\\partial x\_\{j\}\}\\right\]=\(1−f2​\(X\)\)×\[g​\(xi,S​X\)×∂g​\(xi,S​X\)∂xi\]×\[h​\(xj,S​X\)×∂h​\(xj,S​X\)∂xj\]=\(1\-f^\{2\}\(X\)\)\\times\\left\[g\(x\_\{i\},SX\)\\times\\frac\{\\partial g\(x\_\{i\},SX\)\}\{\\partial x\_\{i\}\}\\right\]\\times\\left\[h\(x\_\{j\},SX\)\\times\\frac\{\\partial h\(x\_\{j\},SX\)\}\{\\partial x\_\{j\}\}\\right\]LetT​\(X\)=∂f​\(X\)∂xi×∂f​\(X\)∂xj÷\(1−f2​\(X\)\)T\(X\)=\\frac\{\\partial f\(X\)\}\{\\partial x\_\{i\}\}\\times\\frac\{\\partial f\(X\)\}\{\\partial x\_\{j\}\}\\div\(1\-f^\{2\}\(X\)\)\. Iff​\(X\)f\(X\)conforms to the specified structure,T​\(X\)=\[g​\(xi,S​X\)×∂g​\(xi,S​X\)∂xi\]×\[h​\(xj,S​X\)×∂h​\(xj,S​X\)∂xj\]T\(X\)=\\left\[g\(x\_\{i\},SX\)\\times\\frac\{\\partial g\(x\_\{i\},SX\)\}\{\\partial x\_\{i\}\}\\right\]\\times\\left\[h\(x\_\{j\},SX\)\\times\\frac\{\\partial h\(x\_\{j\},SX\)\}\{\\partial x\_\{j\}\}\\right\]is multiplicatively separable across dimensionsxix\_\{i\}andxjx\_\{j\}\.

The derivation for the cosine function \(cos\\cos\) follows an identical mathematical progression\. Therefore, for an unknown functionf​\(X\)f\(X\), we construct the metricT​\(X\)T\(X\)\. WhenT​\(X\)T\(X\)is verified to be multiplicatively separable across any two arbitrary dimensions, the functionf​\(X\)f\(X\)is classified as exhibiting a standard trigonometric nested separation form\.

It is critical to note that, due to the inherent periodicity of trigonometric operators and the uncertainty regarding the internal sub\-functions, acquiring explicit deterministic mappings for the sub\-functions is analytically intractable\. Consequently, upon identifying this special nested form forsin\\sin\(orcos\\cos\), the algorithm does not proceed with further variable separation along these nodes\.

### B\.12Irrationality of Decomposition Without Variable Separability

Proof: Assume a given functionffstructurally satisfies the following form:

f​\(S​X1,S​X2\)=g​\(S​X1,S​X2\)​\+⃝​h​\(S​X2\)f\(SX\_\{1\},SX\_\{2\}\)=g\(SX\_\{1\},SX\_\{2\}\)\+⃝h\(SX\_\{2\}\)If the operator\+⃝\+⃝possesses a corresponding inverse operator\-⃝\-⃝and its evaluation is independent of the order of operations, then for any arbitrary functiont​\(S​X2\)t\(SX\_\{2\}\)defined over the subspaceS​X2SX\_\{2\}, the following identity holds:

f​\(S​X1,S​X2\)=g​\(S​X1,S​X2\)​\+⃝​h​\(S​X2\)​\+⃝​t​\(S​X2\)​\-⃝​t​\(S​X2\)f\(SX\_\{1\},SX\_\{2\}\)=g\(SX\_\{1\},SX\_\{2\}\)\+⃝h\(SX\_\{2\}\)\+⃝t\(SX\_\{2\}\)\-⃝t\(SX\_\{2\}\)=\(g​\(S​X1,S​X2\)​\-⃝​t​\(S​X2\)\)​\+⃝​\(h​\(S​X2\)​\+⃝​t​\(S​X2\)\)=gt​\(S​X1,S​X2\)​\+⃝​ht​\(S​X2\)=\(g\(SX\_\{1\},SX\_\{2\}\)\-⃝t\(SX\_\{2\}\)\)\+⃝\(h\(SX\_\{2\}\)\+⃝t\(SX\_\{2\}\)\)=g\_\{t\}\(SX\_\{1\},SX\_\{2\}\)\+⃝h\_\{t\}\(SX\_\{2\}\)This rigorously demonstrates that an infinite number of valid functional configurations exist forg​\(S​X1,S​X2\)g\(SX\_\{1\},SX\_\{2\}\)andh​\(S​X2\)h\(SX\_\{2\}\), rendering deterministic decomposition fundamentally impossible under these unconstrained conditions\.

## Appendix CHyperparameter configurations of symbolic regression methods\.

Table 1:Hyperparameter configurations of symbolic regression methods\.MethodHyperparameterValueGeneralDDR sample variable number20DDR sample constant number10SR sample number40SR test number40variable sample range\(0,1\)OthersdefaultMetaSymNetFunction Set \(SS\)\+\+,∗\*,//,sin\\sin,cos\\cos,exp\\exp,log\\log,⋅\\sqrt\{\\cdot\}Outer Iterations \(ITE\)5OthersdefaultPySRpopulations6population\_size40ncycles\_per\_iteration400niterations10,000timeout\_in\_seconds300maxsize50maxdepth15binary\_operators∗\*,\+\+,−\-,//unary\_operatorssquare, cube, exp, log, sin, cosearly\_stop\_conditionloss<1​e−4<1e\-4&& complexity<30<30constraints//: \(\-1, 20\), square: 9, cube: 9, exp: 9nested\_constraints\(square/cube\)limits nested trig/exp operationscomplexity\_of\_operators//: 2, exp: 3complexity\_of\_constants2turboTrueOthersdefaultNGGP \(DSO\)task\.function\_setadd, sub, mul, div, sin, cos, exp, log, consttask\.metricinv\_nrmsemetric\_params\[1\.0\]task\.threshold1e\-6task\.protectedTruetraining\.n\_samples20,000training\.batch\_size500training\.epsilon0\.05training\.alpha0\.5training\.n\_cores\_batch1controller\.learning\_rate0\.0005controller\.entropy\_weight0\.005controller\.entropy\_gamma0\.7controller\.celllstmprior\.lengthmin: 1, max: 50prior\.repeattokens: \[sin, cos, exp, log\], max: 1prior\.inverse / trig / const / no\_inputson: Trueprior\.soft\_lengthloc: 10, scale: 5, on: Truegp\_meld\.run\_gp\_meldTruegp\_meld\.population\_size100gp\_meld\.generations20gp\_meld\.p\_crossover0\.8gp\_meld\.p\_mutate0\.05gp\_meld\.tourney\_size3OthersdefaultGP\-GOMEAgomeaTruefunctions\+\+,−\-,∗\*,p/p/,⋅\\sqrt\{\\cdot\},p​l​o​gplog,exp\\expcoeffmut0\.5\_0\.5\_0\.5\_10time120generations/evaluations\-1 \(No limit\)initmaxtreeheight5ims\(Interleaved multistart\)5\_1popsize64batchsize256parallel4linearscalingFalseOthersdefaultE2Emax\_input\_points200n\_trees\_to\_refine10batch\_size\_eval1rescaleTrueeval\_onlyTruereplace\_opsadd→\\to\+, mul→\\to\*, sub→\\to\-, pow→\\to\*\*, inv→\\to1/Othersdefault
## Appendix DExpressions of Datasets

Table 2:Expressions of DatasetsDatasetVariablesFormulaKeijzer\-102pow\(x1,x2\)Keijzer\-112x1\*x2\+sin\(\(x1\-1\)\*\(x2\-1\)\)Keijzer\-122pow\(x1,4\)\-pow\(x1,3\)\+div\(pow\(x2,2\),2\)\-x2Keijzer\-1326\*sin\(x1\)\*cos\(x2\)Keijzer\-142div\(8,2\+pow\(x1,2\)\+pow\(x2,2\)\)Keijzer\-152div\(pow\(x1,3\),5\)\+div\(pow\(x2,3\),2\)\-x2\-x1Nguyen\-92sin\(x1\)\+sin\(pow\(x2,2\)\)Nguyen\-1022\*sin\(x1\)\*cos\(x2\)Nguyen\-112pow\(x1,x2\)Nguyen\-122pow\(x1,4\)\-pow\(x1,3\)\+div\(pow\(x2,2\),2\)\-x2Nguyen\-10c2sin\(1\.5\*x1\)\*cos\(0\.5\*x2\)Constant\-32sin\(1\.5\*x1\)\*cos\(0\.5\*x2\)Constant\-422\.7\*pow\(x1,x2\); 2\.7exp\(x2\*ln\(x1\)\)Constant\-722\*sin\(1\.3\*x1\)\*cos\(x2\)Livermore\-52pow\(x1,4\)\-pow\(x1,3\)\+pow\(x2,2\)\-x2Livermore\-1026\*sin\(x1\)\*cos\(x2\)Livermore\-112div\(pow\(x1,2\)\*pow\(x2,2\),\(x1\+x2\)\)Livermore\-122div\(pow\(x1,5\),pow\(x2,3\)\)Livermore\-142pow\(x1,3\)\+pow\(x1,2\)\+x1\+sin\(x1\)\+sin\(pow\(x2,2\)\)Livermore\-1724\*sin\(x1\)\*cos\(x2\)Jin\-122\.5\*pow\(x1,4\)\-1\.3\*pow\(x1,3\)\+0\.5\*pow\(x2,2\)\-1\.7\*x2Jin\-228\.0\*pow\(x1,2\)\+8\.0\*pow\(x2,3\)\-15\.0Jin\-320\.2\*pow\(x1,3\)\+0\.5\*pow\(x2,3\)\-1\.2\*x2 \-0\.5\*x1Jin\-421\.5\*exp\(x1\)\+5\.0\*cos\(x2\)Jin\-526\.0\*sin\(x1\)\*cos\(x2\)Jin\-621\.35\*x1\*x2 \+5\.5\*sin\(\(x1\-1\.0\)\*\(x2\-1\.0\)\)bacterial respiration 1220\-x1\-\(x1\*x2/\(1\+0\.5\*pow\(x1,2\)\)\)bacterial respiration 3210\-\(x1\*x2/\(1\+0\.5\*pow\(x1,2\)\)\)bar magnets 120\.5\*sin\(x1\-x2\)\-sin\(x1\)bar magnets 220\.5\*sin\(x2\-x1\)\-sin\(x2\)glider 12\-0\.05\*pow\(x1,2\)\-sin\(x2\)glider 22x1\-cos\(x2\)/x1Lotka\-Volterra 123\*x1\-2\*x1\*x2\-pow\(x1,2\)Lotka\-Volterra 222\*x2\-x1\*x2\-pow\(x2,2\)predator\-prey 12x1\*\(4\-x1\-x2/\(1\+x1\)\)predator\-prey 22x2\*\(x1/\(1\+x1\)\-0\.075\*x2\)shear flow 12cot\(x2\)\*cos\(x1\)shear flow 22\(pow\(cos\(x2\),2\)\+0\.1\*pow\(sin\(x2\),2\)\)\*sin\(x1\)van der Pol 1210\*\(x2\-\(1/3\*\(pow\(x1,3\)\-x1\)\)\)van der Pol 22\-1/10\*x1Korns\-1222 \-2\.1 \*cos\(9\.8\*x1\*\*3\)\*sin\(1\.3\*x2\)Korns\-230\.23 \+14\.2 \*\(x1\+x3\)/\(3\*x2\)Korns\-334\.9 \*\(x2\-x1\+\(x1/x3\)\)/\(3\*x3\)\-5\.41Korns\-836\.87 \+11\*sqrt\(abs\(7\.23\*x1\*x2\*x3\)\)Korns\-9312\*sqrt\(abs\(4\.2\*x1\*x2\*x3\)\)Korns\-1040\.81 \+24\.3 \*\(2\*x1 \+3\*x2\*\*2\)/\(4\*x3\*\*3 \+5\*x4\*\*4\)Korns\-13432\.0 \-3\.0 \*\(tan\(x1\)\+tan\(x3\)\)/\(tan\(x2\)\+tan\(x4\)\)Korns\-14422\.0 \-\(4\.2\*cos\(x1\)\-tan\(x2\)\)\*tan\(x3\)/sin\(x4\)Korns\-15412\.0 \-6\.0 \*tan\(x1\)\*\(log\(x3\)\-tan\(x4\)\)/exp\(x2\)Neat\-522 \*sin\(x1\)\*cos\(x2\)Neat\-722 \-2\.1 \*cos\(9\.8\*x1\)\*sin\(1\.3\*x2\)Neat\-82exp\(\-x1\*\*2\)/\(1\.2 \+\(x2\-2\.5\)\*\*2\)Neat\-921\.0/\(1\+x1\*\*\(\-4\)\)\+1\.0/\(1\+x2\*\*\(\-4\)\)Vladislavleva\-12exp\(\-\(x1\-1\)\*\*2\)/\(1\.2\+\(x2\-2\.5\)\*\*2\)Vladislavleva\-32x1\*\*3 \*exp\(\-x1\)\*cos\(x1\)\*sin\(x1\)\*\(sin\(x1\)\*\*2 \*cos\(x1\)\-1\)\*\(x2\-5\)Vladislavleva\-626 \*sin\(x1\)\*cos\(x2\)Vladislavleva\-722 \-2\.1\*cos\(9\.8\*x1\)\*sin\(1\.3\*x2\)Vladislavleva\-82exp\(\-x1\*\*2\)/\(1\.2 \+\(x2\-2\.5\)\*\*2\)Vladislavleva\-5330\.0 \*\(x1\-1\)\*\(x3\-1\)\*x2\*\*2 /\(x1\-10\)Vladislavleva\-4510\.0/\(5 \+\(x1\-3\)\*\*2 \+\(x2\-3\)\*\*2 \+\(x3\-3\)\*\*2 \+\(x4\-3\)\*\*2 \+\(x5\-3\)\*\*2\)Pagie\-121\.0/\( 1\+x1\*\*\(\-4\)\+1\.0/\(1 \+x2\*\*\(\-4\)\)\)Meier\-32x1\*\*2 \*x2\*\*2 /\(x1 \+x2\)Meier\-42x1\*\*5 /\(x2\*\*3\)Poly\-109x1\*x2 \+x3\*x4 \+x5\*x6 \+x1\*x7\*x8 \+x3\*x6\*x9Feynman\-I\.6\.22exp\(\-\(x2/x1\)\*\*2/2\)/\(sqrt\(2\*pi\)\*x1\)Feynman\-I\.12\.12x1\*x2Feynman\-I\.12\.52x1\*x2Feynman\-I\.14\.421/2\*x1\*pow\(x2,2\)Feynman\-I\.25\.132x1/x2Feynman\-I\.29\.42x1/x2Feynman\-I\.34\.272\(x2/\(2\*pi\)\)\*x1Feynman\-I\.39\.123/2\*x1\*x2Feynman\-II\.8\.312x1\*pow\(x2,2\)/2Feynman\-II\.11\.2821\+x1\*x2/\(1\-\(x1\*x2/3\)\)Feynman\-II\.27\.182x1\*pow\(x2,2\)Feynman\-II\.38\.142x1/\(2\*\(1\+x2\)\)Feynman\-III\.12\.432x1\*\(x2/\(2\*pi\)\)Feynman\-I\.6\.20b3exp\(\-\(x1\-x2\)\*\*2/\(2\*x3\*\*2\)\)/sqrt\(2\*pi\*x3\*\*2\)Feynman\-I\.12\.43x1/\(4\*pi\*x2\*x3\*\*2\)Feynman\-I\.14\.33x1\*x2\*x3Feynman\-I\.16\.63\(x1\+x2\)/\(1\+x1\*x2/\(x3\*\*2\)\)Feynman\-I\.18\.123x1\*x2\*sin\(x3\)Feynman\-I\.27\.631\.0/\(1\.0/x1 \+x3/x2\)Feynman\-I\.30\.33x1 \*\(sin\(x2\*x3/2\.0\)\)\*\*2 /sin\(x3/2\.0\)Feynman\-I\.34\.103x1 /\(1\-x2/x3\)Feynman\-I\.34\.143\(1\+x1/x2\)\*x3 /sqrt\(1\-x1\*\*2/x2\*\*2\)Feynman\-I\.37\.43x1 \+x2 \+2\*sqrt\(x1\*x2\)\*cos\(x3\)Feynman\-I\.39\.113x1\*x2/\(1\-x3\)Feynman\-I\.43\.313x1\*x2\*x3Feynman\-I\.48\.203x1\*x2\*\*2/\(sqrt\(1\-\(x3/x2\)\*\*2\)\)Feynman\-II\.4\.233x1 /\(4\*pi\*x2\*x3\)Feynman\-II\.8\.730\.6 \*x1\*\*2 /\(4\*pi\*x2\*x3\)Feynman\-II\.10\.93x1 /\(x2 \*\(1\+x3\)\)Feynman\-II\.15\.43\-x1\*x2\*cos\(x3\)Feynman\-II\.15\.53\-x1\*x2\*cos\(x3\)Feynman\-II\.24\.173sqrt\(\(x1/x2\)\*\*2 \-\(pi/x3\)\*\*2\)Feynman\-II\.27\.163x1\*x2\*x3\*\*2Feynman\-II\.34\.2a3x1\*x2/\(2\*pi\*x3\)Feynman\-II\.34\.23x1\*x2\*x3/2Feynman\-II\.34\.113x1\*x2/\(2\*x3\)Feynman\-II\.34\.29a3x1\*x2/\(4\*pi\*x3\)Feynman\-II\.37\.13x1\*\(1\+x2\)\*x3Feynman\-III\.8\.543sin\(\(x1\*x2/x3\)\*\*2\)Feynman\-III\.15\.1232\*x1\*\(1\-cos\(x2\*x3\)\)Feynman\-III\.15\.143x1\*\*2 /\(2\*x2\*x3\*\*2\)Feynman\-III\.15\.2732\*pi\*x1/\(x2\*x3\)Feynman\-III\.17\.373x1\*\(1\+x2\*cos\(x3\)\)Feynman\-I\.8\.144sqrt\(\(x2\-x1\)\*\*2 \+\(x4\-x3\)\*\*2\)Feynman\-I\.12\.24x1\*x2 /\(4\*pi\*x3\*x4\*\*2\)Feynman\-I\.13\.440\.5 \*x1 \*\(x2\*\*2 \+x3\*\*2 \+x4\*\*2\)Feynman\-I\.15\.3x4\(x1 \-x2\*x3\)/sqrt\(1 \-\(x2/x4\)\*\*2\)Feynman\-I\.15\.3t4\(x1 \-x2\*x3/\(x4\*\*2\)\)/sqrt\(1 \-\(x2/x4\)\*\*2\)Feynman\-I\.18\.44\(x1\*x3 \+x2\*x4\)/\(x1\+x2\)Feynman\-I\.18\.164x1 \*x2 \*x3 \*sin\(x4\)Feynman\-I\.24\.640\.25 \*x1 \*\(x2\*\*2 \+x3\*\*2\)\*x4\*\*2Feynman\-I\.29\.164sqrt\(x1\*\*2 \+x2\*\*2 \-1\*x1\*x2\*cos\(x3\-x4\)\)Feynman\-I\.32\.54x1\*\*2 \*x2\*\*2 /\(6\*pi\*x3\*x4\*\*3\)Feynman\-I\.34\.84x1\*x2\*x3/x4Feynman\-I\.38\.1244\*pi\*x1\*x2\*\*2/\(x3\*x4\*\*2\)Feynman\-I\.39\.224x1\*x2\*x3/x4Feynman\-I\.43\.164x1\*x2\*x3/x4Feynman\-I\.43\.434x1\*x2/\(\(x3\-1\)\*x4\)Feynman\-I\.47\.234sqrt\(x1\*x2\*x3/x4\)Feynman\-I\.50\.264x1\*abs\(cos\(x3\*x4\)\+x2\*\(cos\(x3\*x4\)\*\*2\)\)Feynman\-II\.6\.114x1 \*cos\(x2\)/\(4\*pi\*x3\*x4\*\*2\)Feynman\-II\.6\.15b43 \*x1 \*cos\(x2\)\*sin\(x2\)/\(4\*pi\*x3\*x4\*\*3\)Feynman\-II\.11\.274x1\*x2\*x3\*x4 /\(1 \-x1\*x2/3\)Feynman\-II\.13\.174\(2\*x1\)/\(4\*pi\*x2\*x3\*\*2\*x4\)Feynman\-II\.13\.234x1\*x2 /sqrt\(1 \-\(x3/x4\)\*\*2\)Feynman\-II\.13\.344x1\*x2\*x3 /sqrt\(1 \-\(x3/x4\)\*\*2\)Feynman\-II\.34\.29b4x1\*x2\*x3/x4Feynman\-II\.38\.34x1\*x2\*x3/x4Feynman\-III\.4\.3241\.0/\(exp\(x1\*x2/\(x3\*x4\)\)\-1\)Feynman\-III\.4\.334x1\*x2/\(exp\(x1\*x2/\(x3\*x4\)\)\-1\)Feynman\-III\.7\.3842\*x1\*x2\*x3/x4Feynman\-III\.10\.194x1\*sqrt\(x2\*\*2\+x3\*\*2\+x4\*\*2\)Feynman\-III\.13\.1842\*x1\*x2\*\*2\*x3/x4Feynman\-I\.12\.115x1\*\(x2\+x3\*x4\*sin\(x5\)\)Feynman\-I\.13\.125x1 \*x2 \*x3 \*\(1\.0/x5 \-1\.0/x4\)Feynman\-I\.41\.165x1\*x2\*\*3/\(\(pi\)\*\*2\*x3\*\*2\*\(exp\(x1\*x2/\(x4\*x5\)\)\-1\)\)Feynman\-I\.44\.45x1\*x2\*x3\*log\(x5/x4\)Feynman\-II\.2\.425x1\*\(x3\-x2\)\*x4 /x5Feynman\-II\.11\.35x1 \*x2 /\(x3\*\(x4\*\*2\-x5\*\*2\)\)Feynman\-II\.11\.205x1\*x2\*\*2\*x3 /\(3\*x4\*x5\)Feynman\-II\.21\.325x1 /\(4\*pi\*x2\*x3\*\(1\-x4/x5\)\)Feynman\-II\.35\.185x1 /\(exp\(x2\*x3/\(x4\*x5\)\)\+exp\(\-x2\*x3/\(x4\*x5\)\)\)Feynman\-II\.35\.215x1\*x2\*tanh\(x2\*x3/\(x4\*x5\)\)Feynman\-III\.14\.145x1\*\(exp\(x2\*x3/\(x4\*x5\)\)\-1\)Feynman\-III\.19\.515x1\*x2\*\*4 /\(2 \*\(4\*pi\*x3\)\*\*2 \*x4\*\*2 \*x5\*\*2\)Feynman\-III\.21\.205\-x1\*x2\*x3\*x4/x5Feynman\-I\.11\.196x1\*x4 \+x2\*x5 \+x3\*x6Feynman\-I\.32\.176\(0\.5\*x1\*x2\*x3\*\*2\)\*\(8\*pi\*x4\*\*4/3\.0\)\*\(x5\*\*4/\(x5\*\*2\-x6\*\*2\)\*\*2\)Feynman\-I\.40\.16x1 \*exp\(x2\*x3\*x4/\(x5\*x6\)\)Feynman\-II\.6\.15a63 \*x1 \*x2 \*sqrt\(x3\*\*2 \+x4\*\*2\)/\(4\*pi\*x5\*x6\*\*5\)Feynman\-II\.11\.76x1 \*\(1 \+\(x2\*x3\*cos\(x4\)/\(x5\*x6\)\)\)Feynman\-III\.9\.526x1\*x2\*x3\*sin\(\(\(x4\-x5\)\*x3/2\)\*\*2\)/\(x6\*\(\(x4\-x5\)\*x3/2\)\*\*2\)Feynman\-II\.36\.288x1\*x2/\(x3\*x4\)\+x1\*x5\*x6/\(x7\*x8\*\*2\*x3\*x4\)Feynman\-I\.9\.189x9\*x1\*x2 /\(\(x3\-x4\)\*\*2 \+\(x5\-x6\)\*\*2 \+\(x7\-x8\)\*\*2\)Feynman\-I\.10\.79x9\*x1\*x2 /\(\(x3\-x4\)\*\*2 \+\(x5\-x6\)\*\*2 \+\(x7\-x8\)\*\*2\)simple\_test\-13x1\*x2\+x2\*x3simple\_test\-23sin\(x1\)\*x2\+x2\*cos\(x3\)simple\_test\-33sin\(x1\)\*exp\(x2\)\+x2\*cos\(x3\)simple\_test\-43\(x1\+x2\)\*\(x2\+x3\)simple\_test\-53\(sin\(x1\)\+x2\)\*\(x2\+cos\(x3\)\)simple\_test\-63\(sin\(x1\)\+exp\(x2\)\)\*\(x2\+cos\(x3\)\)simple\_test\-73exp\(x1\*x2\*x3\)simple\_test\-83log\(x1\+x2\+x3\)simple\_test\-93sin\(x1\*x2\*x3\)simple\_test\-103x1\*x2\*x3\+1\.2846simple\_test\-115x1\*x2\*x3\+x3\*x4\*x5\+x1\*x3\*x5complex\-test15log\(x1\*x2\*x3\+x3\*x4\*x5\+x1\*x3\*x5\)complex\-test25exp\(x1\*x2\*x3\+x3\*x4\*x5\+x1\*x3\*x5\)complex\-test35x1\*x2\*\*2\*\(0\.5\*x3\+1\)/\(2\*\(x4\+1\)\*x5\)\+1\.234complex\-test45\(x1\+1\.2\*x2\+1\.4\*x3\)\*\(x3\+1\.2\*x4\+1\.4\*x5\)complex\-test55sin\(x1\*\(x2\+1\)\*\(x3\-1\)\+x4\*x5\*\*2\)complex\-test66sqrt\(x1\*x2\*\*2\+\(x3\+1\.5\*x4\)\)/\(x5\*\*2\*x6\*\*3\)\+2\.134complex\-test78\(x1\+0\.5\)\*\(x2\-0\.5\)/\(x3\+2\*x4\)\+\(x1\-0\.5\)\*\(x5\-0\.5\*x6\)/\(x7\*x8\*\*2\)complex\-test89\(x9\+1\.5\)\*\(x1\-1\.2\)\*\(x2\-1\.4\)/\(\(x3\-2\*x4\)\*\*2 \+\(x5\-3\*x6\)\*\*2 \+\(x7\-4\*x8\)\*\*2\)complex\-test99x1\*x2\*x3\*x4\*x5\+x1\*x3\*x5\*x7\*x9\+x5\*x6\*x7\*x8\*x9
## Appendix EStatistical Experimental Results

Table 3:Separation Score of DDRSR and AI Feynman in comprehensive experiments\.Table 4:Performance \(R2R^\{2\}and Recovery Rate\) across different number of variables in comprehensive experiments\.Table 5:Performance \(R2R^\{2\}and Recovery Rate\) across different datasets in comprehensive experiments\.Table 6:Separation Score of DDRSR and AI Feynman in Ablation Experiments of Expression Decomposition\.Table 7:Performance \(R2R^\{2\}\) across different number of variables in Ablation Experiments of Expression Decomposition\.Table 8:Performance \(R2R^\{2\}\) across different datasets in Ablation Experiments of Expression Decomposition\.
## Appendix FDetailed Experimental Results

Table 9:Detailed Separation of DDRSR and AI Feynman in comprehensive experiments\.DatasetDDRSR ScoreDDRSR FormAIF ScoreAIF FormKeijzer\-101\.0exp\(f1\(x1\)\*f2\(x2\)\)0\.0f1\(x1,x2\)Keijzer\-110\.0f1\(x1,x2\)0\.0f1\(x1,x2\)Keijzer\-121\.0f1\(x1\)\+f2\(x2\)1\.0f1\(x1\)\+f2\(x2\)Keijzer\-131\.0\(f1\(x1\)\)\*\(f2\(x2\)\)1\.0\(f1\(x1\)\)\*\(f2\(x2\)\)Keijzer\-140\.0f1\(x1,x2\)0\.0f1\(x1,x2\)Keijzer\-151\.0f1\(x1\)\+f2\(x2\)1\.0f1\(x1\)\+f2\(x2\)Nguyen\-91\.0f1\(x1\)\+f2\(x2\)1\.0f1\(x1\)\+f2\(x2\)Nguyen\-101\.0\(f1\(x1\)\)\*\(f2\(x2\)\)1\.0\(f1\(x1\)\)\*\(f2\(x2\)\)Nguyen\-111\.0exp\(f1\(x1\)\*f2\(x2\)\)0\.0f1\(x1,x2\)Nguyen\-121\.0f1\(x1\)\+f2\(x2\)1\.0f1\(x1\)\+f2\(x2\)Nguyen\-10c1\.0\(f1\(x1\)\)\*\(f2\(x2\)\)1\.0\(f1\(x1\)\)\*\(f2\(x2\)\)Constant\-31\.0\(f1\(x1\)\)\*\(f2\(x2\)\)1\.0\(f1\(x1\)\)\*\(f2\(x2\)\)Constant\-41\.0exp\(f1\(x1,x2\)\+0\.9933\)0\.0f1\(x1,x2\)Constant\-71\.0\(f1\(x1\)\)\*\(f2\(x2\)\)1\.0\(f1\(x1\)\)\*\(f2\(x2\)\)Livermore\-51\.0f1\(x1\)\+f2\(x2\)1\.0f1\(x1\)\+f2\(x2\)Livermore\-101\.0\(f1\(x1\)\)\*\(f2\(x2\)\)1\.0\(f1\(x1\)\)\*\(f2\(x2\)\)Livermore\-110\.0f1\(x1,x2\)0\.0f1\(x1,x2\)Livermore\-121\.0f1\(\(x1^\(5\)\*x2^\(\-3\)\)\)0\.0f1\(x1,x2\)Livermore\-141\.0f1\(x1\)\+f2\(x2\)1\.0f1\(x1\)\+f2\(x2\)Livermore\-171\.0\(f1\(x1\)\)\*\(f2\(x2\)\)1\.0\(f1\(x1\)\)\*\(f2\(x2\)\)Jin\-11\.0f1\(x1\)\+f2\(x2\)1\.0f1\(x1\)\+f2\(x2\)Jin\-21\.0f1\(x1\)\+f2\(x2\)1\.0f1\(x1\)\+f2\(x2\)Jin\-31\.0f1\(x1\)\+f2\(x2\)1\.0f1\(x1\)\+f2\(x2\)Jin\-41\.0f1\(x1\)\+f2\(x2\)1\.0f1\(x1\)\+f2\(x2\)Jin\-51\.0\(f1\(x1\)\)\*\(f2\(x2\)\)1\.0\(f1\(x1\)\)\*\(f2\(x2\)\)Jin\-60\.0f1\(x1,x2\)0\.0f1\(x1,x2\)bacterial respiration 10\.0f1\(x1,x2\)0\.0f1\(x1,x2\)bacterial respiration 31\.0\(f1\(x1\)\)\*\(f2\(x2\)\)\+10\.00\.0f1\(x1,x2\)bar magnets 10\.0f1\(x1,x2\)0\.0f1\(x1,x2\)bar magnets 20\.0f1\(x1,x2\)0\.0f1\(x1,x2\)glider 11\.0f1\(x1\)\+f2\(x2\)1\.0f1\(x1\)\+f2\(x2\)glider 20\.0f1\(x1,x2\)0\.0f1\(x1,x2\)Lotka\-Volterra 10\.0f1\(x1,x2\)0\.0f1\(x1,x2\)Lotka\-Volterra 20\.0f1\(x1,x2\)0\.0f1\(x1,x2\)predator\-prey 10\.0f1\(x1,x2\)0\.0f1\(x1,x2\)predator\-prey 20\.0f1\(x1,x2\)0\.0f1\(x1,x2\)shear flow 11\.0\(f1\(x1\)\)\*\(f2\(x2\)\)1\.0\(f1\(x1\)\)\*\(f2\(x2\)\)shear flow 21\.0\(f1\(x1\)\)\*\(f2\(x2\)\)1\.0\(f1\(x1\)\)\*\(f2\(x2\)\)van der Pol 11\.0f1\(x1\)\+f2\(x2\)1\.0f1\(x1\)\+f2\(x2\)van der Pol 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I\.14\.31\.0f1\(\(x1^\(1\)\*x2^\(1\)\*x3^\(1\)\)\)1\.0f1\(\(x1\*x2\*x3\)\)Feynman\-I\.16\.60\.0f1\(x1,x2,x3\)0\.0f1\(x1,x2,x3\)Feynman\-I\.18\.121\.0\(f1\(\(x1^\(1\)\*x2^\(1\)\)\)\)\*\(f2\(x3\)\)1\.0\(f1\(\(x1\*x2\)\)\)\*\(f2\(x3\)\)Feynman\-I\.27\.61\.0f1\(x1,\(x2^\(1\)\*x3^\(\-1\)\)\)1\.0f1\(x1,\(x2/x3\)\)Feynman\-I\.30\.31\.0\(f1\(x1\)\)\*\(f2\(x2,x3\)\)0\.0f1\(x1,x2,x3\)Feynman\-I\.34\.101\.0\(f1\(x1\)\)\*\(f2\(\(x2^\(1\)\*x3^\(\-1\)\)\)\)1\.0\(f1\(x1\)\)\*\(f2\(\(x2/x3\)\)\)Feynman\-I\.34\.141\.0\(f1\(\(x1^\(1\)\*x2^\(\-1\)\)\)\)\*\(f2\(x3\)\)0\.5\(f1\(x3\)\)\*\(f2\(x1,x2\)\)Feynman\-I\.37\.40\.0f1\(x1,x2,x3\)0\.0f1\(x1,x2,x3\)Feynman\-I\.39\.111\.0\(f1\(\(x1^\(1\)\*x2^\(1\)\)\)\)\*\(f2\(x3\)\)1\.0\(f1\(\(x1\*x2\)\)\)\*\(f2\(x3\)\)Feynman\-I\.43\.311\.0f1\(\(x1^\(1\)\*x2^\(1\)\*x3^\(1\)\)\)1\.0f1\(\(x1\*x2\*x3\)\)Feynman\-I\.48\.201\.0\(f1\(x1\)\)\*\(f2\(x2,x3\)\)1\.0\(f1\(x1\)\)\*\(f2\(x2,x3\)\)Feynman\-II\.4\.231\.0f1\(\(x1^\(1\)\*x2^\(\-1\)\*x3^\(\-1\)\)\)1\.0f1\(\(x1/x2/x3\)\)Feynman\-II\.8\.71\.0f1\(\(x1^\(2\)\*x2^\(\-1\)\*x3^\(\-1\)\)\)0\.5\(f1\(x1\)\)\*\(f2\(\(x2\*x3\)\)\)Feynman\-II\.10\.91\.0\(f1\(\(x1^\(1\)\*x2^\(\-1\)\)\)\)\*\(f2\(x3\)\)1\.0\(f1\(\(x1/x2\)\)\)\*\(f2\(x3\)\)Feynman\-II\.15\.41\.0\(f1\(\(x1^\(1\)\*x2^\(1\)\)\)\)\*\(f2\(x3\)\)1\.0\(f1\(\(x1\*x2\)\)\)\*\(f2\(x3\)\)Feynman\-II\.15\.51\.0\(f1\(\(x1^\(1\)\*x2^\(1\)\)\)\)\*\(f2\(x3\)\)1\.0\(f1\(\(x1\*x2\)\)\)\*\(f2\(x3\)\)Feynman\-II\.24\.171\.0f1\(\(x1^\(1\)\*x2^\(\-1\)\),x3\)0\.0f1\(x1,x2,x3\)Feynman\-II\.27\.161\.0f1\(\(x1^\(1\)\*x2^\(1\)\*x3^\(2\)\)\)0\.5\(f1\(\(x1\*x2\)\)\)\*\(f2\(x3\)\)Feynman\-II\.34\.2a1\.0f1\(\(x1^\(1\)\*x2^\(1\)\*x3^\(\-1\)\)\)1\.0f1\(\(x1\*x2/x3\)\)Feynman\-II\.34\.21\.0f1\(\(x1^\(1\)\*x2^\(1\)\*x3^\(1\)\)\)1\.0f1\(\(x1\*x2\*x3\)\)Feynman\-II\.34\.111\.0f1\(\(x1^\(1\)\*x2^\(1\)\*x3^\(\-1\)\)\)1\.0f1\(\(x1\*x2/x3\)\)Feynman\-II\.34\.29a1\.0f1\(\(x1^\(1\)\*x2^\(1\)\*x3^\(\-1\)\)\)1\.0f1\(\(x1\*x2/x3\)\)Feynman\-II\.37\.11\.0\(f1\(\(x1^\(1\)\*x3^\(1\)\)\)\)\*\(f2\(x2\)\)1\.0\(f1\(\(x1\*x3\)\)\)\*\(f2\(x2\)\)Feynman\-III\.8\.541\.0f1\(\(x1^\(1\)\*x2^\(1\)\*x3^\(\-1\)\)\)1\.0f1\(\(\(x1\*x2\)/\(x3\)\)\)Feynman\-III\.15\.121\.0\(f1\(x1\)\)\*\(f2\(\(x2^\(1\)\*x3^\(1\)\)\)\)1\.0\(f1\(\(x2\*x3\)\)\)\*\(f2\(x1\)\)Feynman\-III\.15\.141\.0f1\(\(x1^\(2\)\*x2^\(\-1\)\*x3^\(\-2\)\)\)0\.5\(f1\(\(x1/x3\)\)\)\*\(f2\(x2\)\)Feynman\-III\.15\.271\.0f1\(\(x1^\(1\)\*x2^\(\-1\)\*x3^\(\-1\)\)\)1\.0f1\(\(x1/x2/x3\)\)Feynman\-III\.17\.371\.0\(f1\(x1\)\)\*\(\(f2\(x2\)\)\*\(f3\(x3\)\)\+1\.0000\)0\.5\(f1\(x1\)\)\*\(f2\(x2,x3\)\)Feynman\-I\.8\.141\.0f1\(\(x1\+\(\-1\.0000\)\*x2\),\(x3\+\(\-1\.0000\)\*x4\)\)1\.0f1\(\(x1\-x2\),\(x3\-x4\)\)Feynman\-I\.12\.21\.0f1\(\(x1^\(1\)\*x2^\(1\)\*x3^\(\-1\)\*x4^\(\-2\)\)\)0\.5\(f1\(\(x1\*x2/x3\)\)\)\*\(f2\(x4\)\)Feynman\-I\.13\.41\.0f1\(\(x2^\(2\)\*x1^\(1\)\)\)\+f2\(\(x4^\(2\)\*x1^\(1\)\)\)\+f3\(\(x3^\(2\)\*x1^\(1\)\)\)0\.5\(f1\(x1\)\)\*\(f2\(x2\)\+f3\(x3\)\+f4\(x4\)\)Feynman\-I\.15\.3x1\.0\(f1\(x1\)\)\*\(f2\(\(x2^\(1\)\*x4^\(\-1\)\)\)\)\+\(f3\(x3\)\)\*\(f4\(x2,x4\)\)0\.0f1\(x1,x2,x3,x4\)Feynman\-I\.15\.3t1\.0\(f1\(x1\)\)\*\(f2\(\(x2^\(1\)\*x4^\(\-1\)\)\)\)\+\(f3\(x3\)\)\*\(f4\(x2,x4\)\)0\.0f1\(x1,x2,x3,x4\)Feynman\-I\.18\.41\.0\(f1\(x3\)\)\*\(f2\(\(x1^\(1\)\*x2^\(\-1\)\)\)\)\+\(f3\(x4\)\)\*\(f4\(\(x1^\(1\)\*x2^\(\-1\)\)\)\)0\.5f1\(\(x1/x2\),x3,x4\)Feynman\-I\.18\.161\.0\(f1\(\(x1^\(1\)\*x2^\(1\)\*x3^\(1\)\)\)\)\*\(f2\(x4\)\)0\.5\(f1\(\(x1\*x2\*x3\)\)\)\*\(f2\(x4\)\)Feynman\-I\.24\.61\.0\(f1\(x2\)\)\*\(f2\(\(x1^\(1\)\*x4^\(2\)\)\)\)\+\(f3\(x3\)\)\*\(f4\(\(x1^\(1\)\*x4^\(2\)\)\)\)0\.5\(f1\(x1\)\)\*\(\(f2\(x4\)\)\*\(f3\(x2\)\+f4\(x3\)\)\)Feynman\-I\.29\.161\.0f1\(x1,x2,\(x3\+\(\-1\.0000\)\*x4\)\)1\.0f1\(x1,x2,\(x3\-x4\)\)Feynman\-I\.32\.51\.0f1\(\(x1^\(2\)\*x2^\(2\)\*x3^\(\-1\)\*x4^\(\-3\)\)\)0\.5\(f1\(\(x1\*x2\)\)\)\*\(\(f2\(x3\)\)\*\(f3\(x4\)\)\)Feynman\-I\.34\.81\.0f1\(\(x1^\(1\)\*x2^\(1\)\*x3^\(1\)\*x4^\(\-1\)\)\)1\.0f1\(\(x1\*x2\*x3/x4\)\)Feynman\-I\.38\.121\.0f1\(\(x1^\(1\)\*x2^\(2\)\*x3^\(\-1\)\*x4^\(\-2\)\)\)0\.5\(f1\(\(x1/x3\)\)\)\*\(f2\(\(x2/x4\)\)\)Feynman\-I\.39\.221\.0f1\(\(x1^\(1\)\*x2^\(1\)\*x3^\(1\)\*x4^\(\-1\)\)\)1\.0f1\(\(x1\*x2\*x3/x4\)\)Feynman\-I\.43\.161\.0f1\(\(x1^\(1\)\*x2^\(1\)\*x3^\(1\)\*x4^\(\-1\)\)\)1\.0f1\(\(x1\*x2\*x3/x4\)\)Feynman\-I\.43\.431\.0\(f1\(\(x1^\(1\)\*x2^\(1\)\*x4^\(\-1\)\)\)\)\*\(f2\(x3\)\)1\.0\(f1\(\(x1\*x2/x4\)\)\)\*\(f2\(x3\)\)Feynman\-I\.47\.231\.0f1\(\(x1^\(1\)\*x2^\(1\)\*x3^\(1\)\*x4^\(\-1\)\)\)1\.0f1\(\(x1\*x2\*x3/x4\)\)Feynman\-I\.50\.261\.0\(f1\(x1\)\)\*\(f2\(x2,\(x3^\(1\)\*x4^\(1\)\)\)\)1\.0\(f1\(x1\)\)\*\(f2\(x2,\(x3\*x4\)\)\)Feynman\-II\.6\.111\.0\(f1\(\(x1^\(1\)\*x3^\(\-1\)\*x4^\(\-2\)\)\)\)\*\(f2\(x2\)\)0\.5\(f1\(\(x1/x3\)\)\)\*\(\(f2\(x2\)\)\*\(f3\(x4\)\)\)Feynman\-II\.6\.15b1\.0\(f1\(\(x1^\(1\)\*x3^\(\-1\)\*x4^\(\-3\)\)\)\)\*\(f2\(x2\)\)0\.5\(f1\(\(x1/x3\)\)\)\*\(\(f2\(x2\)\)\*\(f3\(x4\)\)\)Feynman\-II\.11\.271\.0\(f1\(\(x1^\(1\)\*x2^\(1\)\)\)\)\*\(f2\(\(x3^\(1\)\*x4^\(1\)\)\)\)1\.0\(f1\(\(x1\*x2\)\)\)\*\(f2\(\(x3\*x4\)\)\)Feynman\-II\.13\.171\.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I\.12\.111\.0f1\(\(x2^\(1\)\*x1^\(1\)\)\)\+\(f2\(\(x3^\(1\)\*x4^\(1\)\)\)\)\*\(f3\(x1\)\)\*\(f4\(x5\)\)1\.0\(f1\(x1\)\)\*\(f2\(x2\)\+\(f3\(\(x3\*x4\)\)\)\*\(f4\(x5\)\)\)Feynman I\.13\.121\.0\(f1\(x4\)\)\*\(f2\(\(x1^\(1\)\*x2^\(1\)\*x3^\(1\)\)\)\)\+\(f3\(x5\)\)\*\(f4\(\(x1^\(1\)\*x2^\(1\)\*x3^\(1\)\)\)\)1\.0\(f1\(\(x1\*x2\*x3\)\)\)\*\(f2\(x4\)\+f3\(x5\)\)Feynman I\.41\.161\.0\(f1\(x1,x2,\(x4^\(1\)\*x5^\(1\)\)\)\)\*\(f2\(x3\)\)1\.0\(f1\(x3\)\)\*\(f2\(x1,x2,\(x4\*x5\)\)\)Feynman I\.44\.41\.0\(f1\(\(x1^\(1\)\*x2^\(1\)\*x3^\(1\)\)\)\)\*\(f2\(\(x4^\(1\)\*x5^\(\-1\)\)\)\)1\.0\(f1\(\(x1\*x2\*x3\)\)\)\*\(f2\(\(x4/x5\)\)\)Feynman II\.2\.421\.0f1\(\(x1^\(1\)\*\(x2\+\(\-1\.0000\)\*x3\)^\(1\)\*x4^\(1\)\*x5^\(\-1\)\)\)1\.0f1\(\(x1\*\(x2\-x3\)\*x4/x5\)\)Feynman II\.11\.31\.0\(f1\(\(x1^\(1\)\*x2^\(1\)\*x3^\(\-1\)\)\)\)\*\(f2\(x4,x5\)\)1\.0\(f1\(\(x1\*x2/x3\)\)\)\*\(f2\(x4,x5\)\)Feynman II\.11\.201\.0f1\(\(x1^\(1\)\*x2^\(2\)\*x3^\(1\)\*x4^\(\-1\)\*x5^\(\-1\)\)\)0\.5\(f1\(\(x1\*x3/x4/x5\)\)\)\*\(f2\(x2\)\)Feynman II\.21\.321\.0\(f1\(\(x1^\(1\)\*x2^\(\-1\)\*x3^\(\-1\)\)\)\)\*\(f2\(\(x4^\(1\)\*x5^\(\-1\)\)\)\)1\.0\(f1\(\(x1/x2/x3\)\)\)\*\(f2\(\(x4/x5\)\)\)Feynman II\.35\.181\.0\(f1\(x1\)\)\*\(f2\(\(x2^\(1\)\*x3^\(1\)\*x4^\(\-1\)\*x5^\(\-1\)\)\)\)1\.0\(f1\(x1\)\)\*\(f2\(\(x2\*x3/x4/x5\)\)\)Feynman II\.35\.211\.0\(f1\(x1\)\)\*\(f2\(x2,\(x3^\(1\)\*x4^\(\-1\)\*x5^\(\-1\)\)\)\)1\.0\(f1\(x1\)\)\*\(f2\(x2,\(x3/x4/x5\)\)\)Feynman III\.14\.141\.0\(f1\(x1\)\+f2\(\(x2^\(1\)\*x3^\(1\)\)\)\)\*\(f3\(x4,x5,\(x2^\(1\)\*x3^\(1\)\)\)\)0\.5\(f1\(x1\)\)\*\(f2\(x2,x3,x4,x5\)\)Feynman III\.19\.511\.0f1\(\(x1^\(1\)\*x2^\(4\)\*x3^\(\-2\)\*x4^\(\-2\)\*x5^\(\-2\)\)\)0\.5f1\(x1,x2,\(x3\*x4\*x5\)\)Feynman III\.21\.201\.0f1\(\(x1^\(1\)\*x2^\(1\)\*x3^\(1\)\*x4^\(1\)\*x5^\(\-1\)\)\)1\.0f1\(\(x1\*x2\*x3\*x4/x5\)\)Feynman\-I\.11\.191\.0f1\(\(x1^\(1\)\*x4^\(1\)\)\)\+f2\(\(x3^\(1\)\*x6^\(1\)\)\)\+f3\(\(x2^\(1\)\*x5^\(1\)\)\)1\.0f1\(\(\(x1\*x4\)\+\(x2\*x5\)\+\(x3\*x6\)\)\)Feynman\-I\.32\.171\.0\(f1\(\(x1^\(1\)\*x2^\(1\)\*x3^\(2\)\*x4^\(4\)\)\)\)\*\(f2\(\(x5^\(1\)\*x6^\(\-1\)\)\)\)0\.5f1\(\(x1\*x2\),x3,x4,\(x5/x6\)\)Feynman\-I\.40\.11\.0\(f1\(x1\)\+f2\(\(x2^\(1\)\*x3^\(1\)\*x4^\(1\)\*x5^\(\-1\)\)\)\)\*\(f3\(\(x6^\(1\)\),\(x2^\(1\)\*x3^\(1\)\*x4^\(1\)\*x5^\(\-1\)\)\)\)0\.5\(f1\(x1\)\)\*\(f2\(x2,x3,x4,x5,x6\)\)Feynman\-II\.6\.15a1\.0\(f1\(\(x1^\(1\)\*x2^\(1\)\*x5^\(\-1\)\*x6^\(\-5\)\)\)\)\*\(f2\(x3,x4\)\)1\.0\(f1\(\(x1\*x2/x5\)\)\)\*\(\(f2\(x6\)\)\*\(f3\(x3,x4\)\)\)Feynman\-II\.11\.71\.0\(f1\(x1\)\)\*\(\(f2\(\(x2^\(1\)\*x3^\(1\)\*x5^\(\-1\)\*x6^\(\-1\)\)\)\)\*\(f3\(x4\)\)\+1\.0000\)0\.5\(f1\(x1\)\)\*\(f2\(\(x2\*x3/x5/x6\),x4\)\)Feynman\-III\.9\.521\.0\(f1\(\(x1^\(1\)\*x2^\(1\)\*x6^\(\-1\)\)\)\)\*\(f2\(x3,x5\)\)\+f3\(x4\)\+f4\(x5\)\+f5\(x3\)1\.0\(f1\(\(x1\*x2/x6\)\)\)\*\(f2\(x3,\(x4\-x5\)\)\)Feynman\-II\.36\.281\.0\(f1\(x2\)\)\*\(f2\(\(x1^\(1\)\*x3^\(\-1\)\*x4^\(\-1\)\)\)\)\+\(f3\(\(x5^\(1\)\*x6^\(1\)\*x7^\(\-1\)\*x8^\(\-2\)\)\)\)\*\(f4\(\(x1^\(1\)\*x3^\(\-1\)\*x4^\(\-1\)\)\)\)0\.5\(f1\(\(x1/x3/x4\)\)\)\*\(f2\(x2\)\+f3\(\(x5\*x6/x7\),x8\)\)Feynman\-I\.9\.181\.0\(f1\(\(x1^\(1\)\*x2^\(1\)\*x9^\(1\)\)\)\)\*\(f2\(\(x3\+\(\-1\.0000\)\*x4\),\(x5\+\(\-1\.0000\)\*x6\),\(x7\+\(\-1\.0000\)\*x8\)\)\)1\.0\(f1\(\(x1\*x2\*x9\)\)\)\*\(f2\(\(x3\-x4\),\(x5\-x6\),\(x7\-x8\)\)\)Feynman\-I\.10\.71\.0\(f1\(\(x1^\(1\)\*x2^\(1\)\*x9^\(1\)\)\)\)\*\(f2\(\(x3\+\(\-1\.0000\)\*x4\),\(x5\+\(\-1\.0000\)\*x6\),\(x7\+\(\-1\.0000\)\*x8\)\)\)1\.0\(f1\(\(x1\*x2\*x9\)\)\)\*\(f2\(\(x3\-x4\),\(x5\-x6\),\(x7\-x8\)\)\)simple\_test\-11\.0f1\(\(\(x1\+\(1\.0000\)\*x3\)^\(1\)\*x2^\(1\)\)\)1\.0f1\(\(\(x1\+x3\)\*x2\)\)simple\_test\-21\.0\(f1\(x1\)\)\*\(f2\(x2\)\)\+\(f3\(x3\)\)\*\(f4\(x2\)\)1\.0\(f1\(x2\)\)\*\(f2\(x1\)\+f3\(x3\)\)simple\_test\-31\.0\(f1\(x1\)\)\*\(f2\(x2\)\)\+\(f3\(x3\)\)\*\(f4\(x2\)\)0\.0f1\(x1,x2,x3\)simple\_test\-41\.0\(\(f1\(x1\)\)\*\(f2\(x2\)\)\+1\.0000\)\*\(f3\(x3,x2\)\)0\.0f1\(x1,x2,x3\)simple\_test\-51\.0\(f1\(x1,x2\)\+1\.0000\)\*\(f2\(x3,x2\)\)0\.0f1\(x1,x2,x3\)simple\_test\-61\.0\(\(f1\(x1\)\)\*\(f2\(x2\)\)\+1\.0000\)\*\(f3\(x3,x2\)\)0\.0f1\(x1,x2,x3\)simple\_test\-71\.0f1\(\(x1^\(1\)\*x2^\(1\)\*x3^\(1\)\)\)1\.0f1\(\(x1\*x2\*x3\)\)simple\_test\-81\.0f1\(\(x1\+\(1\.0000\)\*x2\+\(1\.0000\)\*x3\)\)1\.0f1\(\(x1\+x2\+x3\)\)simple\_test\-91\.0f1\(\(x1^\(1\)\*x2^\(1\)\*x3^\(1\)\)\)1\.0f1\(\(x1\*x2\*x3\)\)simple\_test\-101\.0f1\(\(x1^\(1\)\*x2^\(1\)\*x3^\(1\)\)\)1\.0f1\(\(x1\*x2\*x3\)\)simple\_test\-111\.0f1\(\(x2^\(1\)\*x1^\(1\)\*x3^\(1\)\)\)\+f2\(\(x5^\(1\)\*x1^\(1\)\*x3^\(1\)\)\)\+f3\(\(x4^\(1\)\*x3^\(1\)\*x5^\(1\)\)\)0\.5\(f1\(x3\)\)\*\(f2\(x1,x2,x4,x5\)\)complex\-test11\.0log\(\(f1\(x2\)\)\*\(f2\(x1\)\)\+\(f3\(x5\)\)\*\(f4\(x1\)\)\+f5\(\(x4^\(1\)\*x5^\(1\)\)\)\)\+f6\(x3\)0\.5f1\(x3\)\+f2\(x1,x2,x4,x5\)complex\-test21\.0\(exp\(f1\(x2\)\*f2\(\(x1^\(1\)\*x3^\(1\)\)\)\)\)\*\(exp\(f3\(x5\)\*f4\(x3\)\*f5\(x1\)\)\)\*\(exp\(\(f6\(x4\)\)\*\(f7\(x5\)\)\+0\.3333\*f8\(x3\)\)\)0\.0f1\(x1,x2,x3,x4,x5\)complex\-test31\.0\(f1\(\(x1^\(1\)\*x2^\(2\)\*x5^\(\-1\)\)\)\)\*\(f2\(x4\)\)\*\(f3\(x3\)\)\+1\.23390\.5f1\(\(x1/x5\),x2,x3,x4\)complex\-test41\.0\(f1\(\(x1\+\(1\.2000\)\*x2\),x3\)\+1\.0001\)\*\(f2\(\(x4\+\(1\.1667\)\*x5\),x3\)\)0\.0f1\(x1,x2,x3,x4,x5\)complex\-test51\.0sin\(f1\(\(x1\+\(2\.0000\)\*x2\+\(3\.0000\)\*x3\),\(x4^\(1\)\*x5^\(2\)\)\)\)0\.0f1\(x1,x2,x3,x4,x5\)complex\-test61\.0f1\(\(x1^\(1\)\*x2^\(2\)\),\(x3\+\(1\.5000\)\*x4\),\(x5^\(2\)\*x6^\(3\)\)\)0\.0f1\(x1,x2,x3,x4,x5,x6\)complex\-test71\.0\(\(f1\(x2\)\)\*\(f2\(\(x3\+\(2\.0000\)\*x4\)\)\)\+0\.3847605586051941\)\*\(f3\(x1\)\)\+f4\(\(\(x5\+\(\-0\.5000\)\*x6\)^\(1\)\*x7^\(\-1\)\*x8^\(\-2\)\),x1\)0\.0f1\(x1,x2,x3,x4,x5,x6,x7,x8\)complex\-test81\.0\(f1\(x1\)\)\*\(f2\(x9\)\)\*\(f3\(\(x3\+\(\-2\.0000\)\*x4\),\(x5\+\(\-3\.0000\)\*x6\),\(x7\+\(\-4\.0000\)\*x8\)\)\)\*\(f4\(x2\)\)0\.5\(f1\(x1\)\)\*\(\(f2\(x2\)\)\*\(\(f3\(x9\)\)\*\(f4\(x3,x4,x5,x6,x7,x8\)\)\)\)complex\-test91\.0\(f1\(\(x2^\(1\)\*x4^\(2\)\)\)\)\*\(f2\(\(x1^\(1\)\*x3^\(1\)\*x5^\(1\)\)\)\)\+f3\(\(x7^\(2\)\*x9^\(1\)\*x1^\(1\)\*x3^\(2\)\*x5^\(1\)\)\)\+\(f4\(\(x6^\(2\)\*x8^\(1\)\)\)\)\*\(f5\(\(x5^\(1\)\*x7^\(1\)\*x9^\(1\)\)\)\)0\.5\(f1\(x5\)\)\*\(f2\(x1,x2,x3,x4,x6,x7,x8,x9\)\)Table 10:Detailed Performance \(R2R^\{2\}and Recovery Rate\) in comprehensive experiments\.DatasetGP\-GOMEAE2EPySRBase\+AIF\+DivBase\+AIF\+DivBase\+AIF\+DivR2R^\{2\}R2R^\{2\}RecR2R^\{2\}RecR2R^\{2\}R2R^\{2\}RecR2R^\{2\}RecR2R^\{2\}R2R^\{2\}RecR2R^\{2\}RecKeijzer\-101\.00001\.00001\.001\.00001\.000\.71490\.98000\.000\.99730\.000\.97840\.99250\.001\.00001\.00Keijzer\-110\.98480\.98180\.000\.98280\.000\.99410\.99230\.000\.99770\.000\.99480\.99560\.000\.99940\.00Keijzer\-120\.96111\.00001\.001\.00001\.000\.98160\.99390\.000\.99870\.000\.99600\.99730\.000\.99910\.00Keijzer\-130\.99921\.00001\.001\.00001\.000\.99930\.99810\.000\.99710\.000\.99990\.99950\.600\.99980\.40Keijzer\-140\.99300\.98230\.000\.98670\.000\.94580\.93400\.000\.93580\.000\.99970\.99830\.000\.99950\.00Keijzer\-150\.99991\.00001\.001\.00001\.000\.99380\.99970\.000\.99930\.000\.99880\.99950\.000\.99920\.00Nguyen\-90\.99681\.00001\.001\.00001\.000\.98910\.99840\.000\.99940\.401\.00001\.00001\.001\.00001\.00Nguyen\-100\.99880\.99990\.801\.00001\.000\.99920\.99880\.000\.99970\.001\.00001\.00001\.000\.99990\.60Nguyen\-111\.00001\.00001\.001\.00001\.000\.81170\.94140\.000\.99850\.000\.98930\.99660\.000\.99970\.20Nguyen\-120\.97431\.00001\.001\.00001\.000\.95620\.99400\.000\.99970\.200\.99190\.99410\.000\.99860\.00Nguyen\-10c0\.99841\.00001\.001\.00001\.000\.99740\.99960\.000\.99960\.000\.99910\.99980\.601\.00001\.00Constant\-30\.99561\.00001\.001\.00001\.000\.99660\.99960\.000\.99950\.000\.99840\.99980\.001\.00001\.00Constant\-41\.00001\.00001\.001\.00001\.000\.89520\.87510\.000\.99450\.000\.76530\.99940\.001\.00001\.00Constant\-70\.99801\.00001\.001\.00001\.000\.99820\.99880\.000\.99920\.000\.99990\.99900\.001\.00001\.00Livermore\-50\.90421\.00001\.001\.00001\.000\.89730\.99960\.000\.99910\.000\.98800\.99330\.000\.99350\.00Livermore\-100\.99941\.00001\.001\.00001\.000\.99870\.99970\.000\.99920\.001\.00000\.99970\.400\.99980\.00Livermore\-110\.99991\.00001\.001\.00001\.000\.99030\.99140\.000\.99200\.000\.99590\.99730\.000\.99780\.00Livermore\-120\.90131\.00001\.001\.00001\.000\.96420\.99150\.000\.99910\.001\.00000\.99990\.801\.00001\.00Livermore\-140\.99981\.00001\.001\.00001\.000\.99740\.99960\.000\.99960\.001\.00001\.00001\.001\.00001\.00Livermore\-170\.99931\.00001\.001\.00001\.000\.99830\.99890\.000\.99860\.001\.00000\.99960\.600\.99990\.80Jin\-10\.98751\.00001\.001\.00001\.000\.97961\.00001\.001\.00001\.000\.99990\.99970\.000\.99980\.00Jin\-20\.99951\.00001\.001\.00001\.000\.98520\.99620\.000\.99800\.001\.00001\.00001\.001\.00001\.00Jin\-30\.99601\.00001\.001\.00001\.000\.94320\.99820\.000\.99980\.000\.99910\.99970\.000\.99880\.00Jin\-40\.99780\.99990\.801\.00001\.000\.99840\.99970\.000\.99970\.001\.00000\.99990\.000\.99990\.00Jin\-50\.99941\.00001\.001\.00001\.000\.99520\.99950\.000\.99910\.001\.00001\.00001\.001\.00001\.00Jin\-60\.94660\.94060\.000\.94380\.970\.99400\.99670\.000\.99560\.000\.99980\.99950\.000\.99990\.00bacterial respiration 10\.99990\.99150\.000\.99750\.000\.91160\.99170\.000\.98660\.000\.99930\.99970\.200\.99950\.00bacterial respiration 30\.99090\.99670\.001\.00001\.000\.99820\.94090\.000\.99950\.000\.99900\.99800\.000\.99980\.00bar magnets 10\.99800\.99770\.000\.99770\.000\.99300\.99850\.000\.99750\.000\.99520\.99820\.000\.99790\.00bar magnets 20\.99680\.99850\.000\.99840\.000\.99470\.99410\.000\.99800\.000\.99770\.99540\.000\.99620\.00glider 10\.99991\.00001\.001\.00001\.000\.99940\.99960\.200\.99980\.201\.00000\.99870\.401\.00001\.00glider 20\.99940\.99910\.400\.99970\.600\.49770\.94100\.000\.96600\.001\.00001\.00001\.001\.00001\.00Lotka\-Volterra 10\.99850\.99840\.600\.99940\.000\.99940\.99930\.000\.99940\.001\.00000\.99980\.600\.99990\.60Lotka\-Volterra 20\.99991\.00000\.001\.00001\.000\.99920\.99950\.000\.99950\.000\.99880\.99960\.000\.99990\.00predator\-prey 10\.99990\.99980\.000\.99950\.400\.99930\.99940\.000\.99990\.001\.00000\.99990\.600\.99980\.00predator\-prey 20\.99720\.99860\.000\.99890\.000\.99250\.99100\.000\.99480\.000\.98630\.99820\.000\.99840\.00shear flow 10\.99861\.00001\.001\.00001\.000\.99110\.99740\.000\.99260\.001\.00000\.99990\.600\.99990\.80shear flow 20\.99991\.00001\.001\.00001\.000\.99080\.99770\.200\.99980\.400\.99970\.99930\.000\.99940\.00van der Pol 10\.99991\.00001\.001\.00001\.000\.99960\.99960\.000\.99980\.201\.00001\.00001\.001\.00001\.00van der Pol 21\.00001\.00001\.001\.00001\.000\.99991\.00000\.801\.00001\.001\.00000\.99980\.400\.99880\.40Korns\-120\.35391\.00001\.001\.00001\.000\.58150\.46080\.000\.92710\.000\.99620\.99990\.000\.99990\.60Neat\-50\.99810\.99990\.600\.99990\.600\.99910\.99960\.000\.99980\.000\.99950\.99980\.600\.99990\.20Neat\-70\.62120\.72320\.000\.91430\.00\-0\.00040\.88770\.000\.99910\.001\.00000\.99980\.201\.00001\.00Neat\-80\.99991\.00001\.001\.00001\.000\.97870\.99950\.000\.99950\.000\.98560\.99940\.000\.99970\.00Neat\-90\.99170\.99990\.401\.00001\.000\.98830\.99620\.000\.98600\.000\.99040\.99710\.000\.99540\.00Vladislavleva\-10\.98090\.99980\.201\.00001\.000\.98960\.99980\.000\.99900\.000\.86450\.99920\.000\.99320\.00Vladislavleva\-30\.99941\.00001\.001\.00001\.000\.99280\.99840\.000\.98840\.000\.98460\.99790\.000\.99940\.00Vladislavleva\-60\.99981\.00001\.001\.00001\.000\.99900\.99930\.000\.99940\.000\.94211\.00001\.000\.99990\.80Vladislavleva\-70\.25060\.86360\.000\.90380\.000\.71380\.97780\.000\.99070\.001\.00000\.99990\.801\.00001\.00Vladislavleva\-80\.98621\.00001\.001\.00001\.000\.99120\.98330\.000\.99740\.000\.94570\.92580\.000\.96820\.00Pagie\-10\.99850\.99780\.000\.99940\.000\.94780\.98880\.000\.99180\.000\.97660\.99500\.000\.99800\.00Meier\-30\.99971\.00001\.001\.00001\.000\.99460\.99530\.000\.99480\.000\.99020\.99570\.000\.99230\.00Meier\-40\.99991\.00001\.001\.00001\.000\.98040\.99500\.000\.99990\.001\.00001\.00001\.001\.00001\.00Feynman\-I\.6\.20\.99660\.99480\.000\.99680\.000\.42400\.60390\.000\.72650\.000\.99760\.99830\.000\.99650\.00Feynman\-I\.12\.11\.00001\.00001\.001\.00001\.000\.99991\.00001\.001\.00001\.000\.99741\.00001\.001\.00001\.00Feynman\-I\.12\.51\.00001\.00001\.001\.00001\.000\.99991\.00001\.001\.00001\.001\.00001\.00001\.001\.00001\.00Feynman\-I\.14\.41\.00001\.00001\.001\.00001\.000\.99981\.00001\.001\.00001\.000\.98720\.99860\.801\.00001\.00Feynman\-I\.25\.131\.00001\.00001\.001\.00001\.000\.99180\.99890\.600\.99970\.801\.00001\.00001\.001\.00001\.00Feynman\-I\.29\.41\.00001\.00001\.001\.00001\.000\.99890\.99990\.200\.99820\.001\.00001\.00001\.001\.00001\.00Feynman\-I\.34\.271\.00001\.00001\.001\.00001\.000\.99991\.00001\.001\.00001\.001\.00000\.99180\.200\.99400\.00Feynman\-I\.39\.11\.00001\.00001\.001\.00001\.000\.99971\.00001\.001\.00001\.001\.00001\.00001\.001\.00001\.00Feynman\-II\.8\.311\.00001\.00001\.001\.00001\.000\.99971\.00001\.001\.00001\.000\.99990\.99940\.001\.00001\.00Feynman\-II\.11\.280\.99961\.00001\.001\.00001\.000\.99670\.99700\.000\.99910\.000\.98841\.00001\.000\.99980\.00Feynman\-II\.27\.181\.00001\.00001\.001\.00001\.000\.99990\.99990\.000\.99940\.401\.00000\.99930\.401\.00001\.00Feynman\-II\.38\.141\.00001\.00001\.001\.00001\.000\.99991\.00001\.001\.00001\.000\.99740\.99920\.000\.99880\.00Feynman\-III\.12\.431\.00001\.00001\.001\.00001\.000\.99981\.00001\.001\.00001\.000\.95700\.99910\.600\.99950\.40Korns\-20\.99991\.00001\.001\.00001\.000\.96710\.99810\.001\.00001\.001\.00001\.00001\.001\.00001\.00Korns\-30\.99971\.00001\.001\.00001\.000\.97380\.74710\.000\.97700\.001\.00001\.00001\.001\.00001\.00Korns\-80\.97910\.99980\.801\.00001\.000\.99390\.99820\.000\.99970\.000\.98980\.99970\.000\.99990\.40Korns\-90\.99991\.00001\.001\.00001\.000\.99970\.99970\.000\.99960\.000\.99740\.99990\.800\.99980\.00Vladislavleva\-50\.98831\.00001\.001\.00001\.000\.99960\.99970\.001\.00001\.000\.99930\.99950\.000\.99980\.00Feynman\-I\.6\.20b0\.68670\.78310\.000\.88470\.000\.38470\.34870\.000\.71170\.000\.57910\.99460\.000\.99930\.00Feynman\-I\.12\.40\.99991\.00001\.001\.00001\.000\.84850\.52950\.001\.00001\.001\.00000\.99990\.801\.00001\.00Feynman\-I\.14\.31\.00001\.00001\.001\.00001\.000\.99971\.00001\.000\.99990\.801\.00001\.00001\.001\.00001\.00Feynman\-I\.16\.60\.97030\.99250\.000\.98760\.000\.54320\.90320\.000\.84890\.001\.00000\.95550\.001\.00001\.00Feynman\-I\.18\.120\.99991\.00001\.001\.00001\.000\.99830\.99970\.001\.00001\.000\.99691\.00001\.000\.99940\.20Feynman\-I\.27\.60\.99991\.00001\.001\.00001\.000\.83410\.92510\.000\.97110\.001\.00000\.99790\.800\.99930\.60Feynman\-I\.30\.30\.99981\.00001\.001\.00001\.000\.99930\.99840\.000\.99910\.000\.87060\.99760\.000\.99960\.40Feynman\-I\.34\.100\.99991\.00001\.001\.00001\.000\.22130\.99360\.000\.95720\.001\.00000\.99980\.801\.00001\.00Feynman\-I\.34\.140\.99991\.00001\.001\.00001\.000\.88920\.85280\.000\.99780\.000\.86980\.99930\.000\.99920\.00Feynman\-I\.37\.40\.98530\.99600\.000\.99780\.000\.97630\.99380\.000\.99490\.000\.99930\.99940\.000\.99560\.00Feynman\-I\.39\.110\.99991\.00001\.001\.00001\.000\.99070\.98780\.200\.98050\.000\.99990\.99980\.001\.00001\.00Feynman\-I\.43\.311\.00001\.00001\.001\.00001\.000\.99960\.99990\.601\.00000\.000\.99670\.99950\.801\.00001\.00Feynman\-I\.48\.200\.97840\.97730\.000\.99911\.000\.72860\.95680\.000\.82310\.000\.81830\.99710\.000\.99950\.00Feynman\-II\.4\.230\.99991\.00001\.001\.00001\.000\.16570\.97350\.200\.99990\.201\.00001\.00001\.001\.00001\.00Feynman\-II\.8\.70\.98961\.00001\.001\.00001\.000\.49250\.98950\.001\.00001\.001\.00001\.00001\.000\.99990\.80Feynman\-II\.10\.90\.99991\.00001\.001\.00001\.000\.99880\.99390\.400\.99990\.001\.00000\.99990\.401\.00001\.00Feynman\-II\.15\.40\.99991\.00001\.001\.00001\.000\.99900\.99990\.001\.00001\.001\.00001\.00001\.001\.00001\.00Feynman\-II\.15\.50\.99991\.00001\.001\.00001\.000\.99130\.99960\.000\.99990\.001\.00001\.00001\.000\.99990\.80Feynman\-II\.24\.170\.99991\.00001\.000\.99971\.000\.95700\.98310\.000\.99410\.001\.00001\.00001\.001\.00001\.00Feynman\-II\.27\.161\.00001\.00001\.001\.00001\.000\.99970\.99970\.001\.00001\.000\.99521\.00001\.000\.99990\.80Feynman\-II\.34\.2a0\.99991\.00001\.001\.00001\.000\.99660\.99990\.801\.00001\.001\.00001\.00001\.001\.00001\.00Feynman\-II\.34\.21\.00001\.00001\.001\.00001\.000\.99851\.00001\.001\.00001\.000\.99531\.00001\.001\.00001\.00Feynman\-II\.34\.110\.99991\.00001\.001\.00001\.000\.20711\.00001\.000\.99920\.001\.00001\.00001\.001\.00001\.00Feynman\-II\.34\.29a0\.99991\.00001\.001\.00001\.000\.96431\.00001\.001\.00001\.001\.00001\.00001\.001\.00001\.00Feynman\-II\.37\.11\.00001\.00001\.001\.00001\.000\.99990\.99990\.201\.00001\.000\.99991\.00001\.001\.00001\.00Feynman\-III\.8\.540\.0131\-1\.31710\.00\-0\.78710\.00\-1\.13240\.34540\.000\.23750\.001\.00001\.00001\.001\.00001\.00Feynman\-III\.15\.120\.99971\.00001\.001\.00001\.000\.99400\.99950\.001\.00001\.000\.99800\.99990\.601\.00001\.00Feynman\-III\.15\.14\-0\.01891\.00001\.001\.00001\.00\-0\.04190\.83730\.000\.99990\.201\.00001\.00001\.001\.00001\.00Feynman\-III\.15\.270\.99991\.00001\.001\.00001\.000\.93221\.00001\.000\.99980\.001\.00001\.00001\.001\.00001\.00Feynman\-III\.17\.370\.99991\.00001\.001\.00001\.000\.99940\.99940\.001\.00001\.000\.99530\.99990\.801\.00001\.00simple\_test\-11\.00001\.00001\.001\.00001\.000\.99981\.00001\.001\.00001\.001\.00001\.00001\.001\.00001\.00simple\_test\-20\.99801\.00001\.001\.00001\.000\.99690\.99580\.200\.99980\.000\.99931\.00001\.000\.99980\.60simple\_test\-30\.98540\.99190\.001\.00001\.000\.99440\.99340\.000\.99970\.000\.99950\.99980\.000\.99990\.80simple\_test\-41\.00001\.00001\.001\.00001\.000\.99900\.99810\.000\.99980\.000\.99991\.00001\.000\.99980\.40simple\_test\-50\.99530\.99730\.000\.99880\.000\.99470\.99880\.000\.99950\.001\.00000\.99990\.800\.99940\.00simple\_test\-60\.99660\.98680\.000\.99990\.000\.97500\.99330\.000\.99810\.000\.99970\.99910\.000\.99960\.40simple\_test\-71\.00001\.00001\.001\.00001\.000\.94250\.99840\.000\.99790\.200\.99910\.99930\.601\.00001\.00simple\_test\-81\.00001\.00001\.001\.00001\.000\.98870\.99360\.000\.99480\.001\.00001\.00001\.001\.00001\.00simple\_test\-90\.99991\.00001\.001\.00001\.000\.99870\.99980\.400\.99990\.400\.99791\.00001\.001\.00001\.00simple\_test\-100\.99481\.00001\.001\.00001\.000\.98370\.99940\.000\.99990\.801\.00000\.99980\.400\.99980\.60Korns\-10\-1\.97410\.14720\.000\.78290\.000\.65880\.56970\.000\.65230\.001\.00000\.99640\.000\.99860\.00Korns\-130\.89900\.92800\.000\.99920\.000\.93700\.75870\.000\.86460\.000\.90370\.99170\.000\.99710\.00Korns\-140\.69160\.97790\.001\.00001\.000\.59200\.45820\.000\.99420\.000\.99390\.99440\.000\.99990\.60Korns\-150\.80730\.97360\.000\.99990\.800\.90620\.92600\.000\.93580\.000\.99850\.99880\.000\.99980\.00Feynman\-I\.8\.140\.59031\.00001\.001\.00001\.000\.33090\.99540\.000\.99820\.000\.98160\.99560\.000\.98610\.00Feynman\-I\.12\.20\.99991\.00001\.001\.00001\.000\.40250\.98480\.000\.98170\.200\.34201\.00001\.001\.00001\.00Feynman\-I\.13\.40\.93001\.00001\.001\.00001\.000\.94360\.98750\.201\.00001\.000\.99850\.99960\.000\.99960\.40Feynman\-I\.15\.3x0\.50450\.91060\.000\.52670\.000\.16040\.62080\.000\.97310\.000\.98040\.81560\.000\.99730\.00Feynman\-I\.15\.3t0\.8662\-0\.53690\.000\.99340\.000\.68950\.61140\.000\.94430\.000\.99700\.98260\.000\.99560\.00Feynman\-I\.18\.40\.99991\.00001\.001\.00001\.000\.76960\.46130\.000\.98220\.000\.99880\.99790\.000\.99940\.00Feynman\-I\.18\.160\.99991\.00001\.001\.00001\.000\.99310\.99980\.001\.00001\.000\.96300\.99990\.800\.99990\.60Feynman\-I\.24\.60\.96371\.00001\.001\.00001\.000\.96540\.99060\.001\.00001\.000\.74980\.99850\.000\.99200\.00Feynman\-I\.29\.160\.97810\.99160\.000\.99380\.000\.94920\.65880\.000\.92820\.000\.99890\.93930\.000\.99710\.00Feynman\-I\.32\.50\.97351\.00001\.001\.00001\.00\-0\.0283\-0\.04800\.000\.99880\.000\.45761\.00001\.001\.00001\.00Feynman\-I\.34\.80\.99991\.00001\.001\.00001\.000\.95451\.00001\.001\.00001\.001\.00001\.00001\.001\.00001\.00Feynman\-I\.38\.120\.99991\.00001\.001\.00001\.00\-0\.64870\.99870\.000\.98790\.001\.00001\.00001\.001\.00001\.00Feynman\-I\.39\.220\.99991\.00001\.001\.00001\.000\.75781\.00001\.000\.99890\.001\.00001\.00001\.001\.00001\.00Feynman\-I\.43\.160\.99991\.00001\.001\.00001\.000\.63751\.00001\.001\.00001\.001\.00001\.00001\.001\.00001\.00Feynman\-I\.43\.430\.99991\.00001\.001\.00001\.000\.38140\.88850\.000\.79530\.001\.00001\.00001\.001\.00001\.00Feynman\-I\.47\.230\.99991\.00001\.001\.00001\.000\.88230\.98720\.000\.99930\.000\.91380\.99810\.000\.99720\.00Feynman\-I\.50\.260\.99741\.00001\.001\.00001\.000\.99390\.99970\.000\.99860\.000\.99960\.99990\.601\.00001\.00Feynman\-II\.6\.110\.99941\.00001\.001\.00001\.000\.26150\.00830\.000\.01550\.001\.00001\.00001\.001\.00001\.00Feynman\-II\.6\.15b0\.27111\.00001\.001\.00001\.000\.20840\.09440\.000\.85840\.000\.94361\.00001\.001\.00001\.00Feynman\-II\.11\.270\.99991\.00001\.001\.00001\.000\.94690\.99950\.000\.99990\.000\.99190\.99950\.400\.99940\.00Feynman\-II\.13\.170\.62041\.00001\.001\.00001\.000\.12990\.07340\.000\.99810\.001\.00001\.00001\.001\.00001\.00Feynman\-II\.13\.230\.98801\.00001\.001\.00001\.000\.94040\.72010\.000\.99060\.000\.99710\.97930\.000\.99640\.00Feynman\-II\.13\.340\.99961\.00001\.001\.00001\.000\.98260\.90460\.000\.78760\.000\.92440\.99390\.000\.99300\.00Feynman\-II\.34\.29b1\.00001\.00001\.001\.00001\.000\.94850\.99990\.601\.00001\.001\.00001\.00001\.001\.00001\.00Feynman\-II\.38\.31\.00001\.00001\.001\.00001\.000\.96750\.99970\.200\.99990\.401\.00001\.00001\.001\.00001\.00Feynman\-III\.4\.320\.99991\.00001\.001\.00001\.000\.13920\.80230\.000\.76190\.000\.99991\.00001\.001\.00001\.00Feynman\-III\.4\.330\.99951\.00001\.001\.00001\.000\.98380\.98930\.000\.95890\.000\.99580\.99500\.000\.99860\.00Feynman\-III\.7\.380\.99991\.00001\.001\.00001\.000\.38310\.99870\.401\.00001\.001\.00001\.00001\.001\.00001\.00Feynman\-III\.10\.190\.99990\.99160\.000\.99770\.200\.97800\.95650\.000\.96420\.000\.99350\.99780\.000\.99570\.00Feynman\-III\.13\.180\.99811\.00001\.001\.00001\.000\.39290\.99960\.001\.00001\.001\.00001\.00001\.001\.00001\.00Vladislavleva\-40\.92820\.94860\.000\.93970\.000\.98490\.98320\.000\.99020\.000\.88810\.90100\.000\.95420\.00Feynman\-I\.12\.110\.99921\.00001\.001\.00001\.000\.98170\.99440\.401\.00001\.000\.99830\.99970\.601\.00001\.00Feynman\-I\.13\.120\.99991\.00001\.001\.00001\.00\-0\.22750\.77910\.000\.99340\.001\.00001\.00001\.000\.99990\.20Feynman\-I\.41\.160\.22660\.99780\.000\.99800\.00\-2\.44000\.96280\.000\.95680\.00\-0\.05300\.98740\.000\.97050\.00Feynman\-I\.44\.40\.99991\.00001\.001\.00001\.000\.45340\.76020\.000\.93650\.000\.96581\.00001\.001\.00001\.00Feynman\-II\.2\.420\.99991\.00001\.001\.00001\.000\.74451\.00001\.001\.00001\.001\.00001\.00001\.001\.00001\.00Feynman\-II\.11\.30\.81421\.00001\.001\.00001\.00\-0\.0190\-0\.08500\.000\.95390\.000\.95191\.00001\.001\.00001\.00Feynman\-II\.11\.200\.99991\.00001\.001\.00001\.000\.92390\.99370\.000\.99920\.001\.00001\.00001\.001\.00001\.00Feynman\-II\.21\.32\-0\.02191\.00001\.001\.00001\.00\-0\.2049\-0\.35540\.000\.63720\.00\-0\.31600\.99950\.600\.99980\.40Feynman\-II\.35\.180\.68701\.00001\.001\.00001\.000\.65690\.99860\.000\.99680\.000\.99760\.99770\.000\.99970\.00Feynman\-II\.35\.210\.99140\.99280\.001\.00001\.000\.87380\.93040\.000\.89170\.000\.92420\.99890\.000\.99950\.00Feynman\-III\.14\.140\.99991\.00001\.001\.00001\.000\.76520\.99910\.000\.99420\.001\.00001\.00001\.001\.00001\.00Feynman\-III\.19\.51\-0\.05891\.00001\.001\.00001\.00\-1\.46320\.90150\.000\.70440\.000\.43810\.97790\.201\.00001\.00Feynman\-III\.21\.200\.99991\.00001\.001\.00001\.000\.73351\.00001\.000\.99970\.201\.00001\.00001\.001\.00001\.00simple\_test\-110\.98961\.00001\.001\.00001\.000\.97330\.99660\.001\.00001\.001\.00001\.00001\.001\.00001\.00complex\-test10\.99380\.99570\.001\.00001\.000\.77460\.85740\.000\.98510\.000\.99081\.00001\.000\.99890\.60complex\-test20\.97230\.99980\.601\.00001\.000\.73980\.96420\.000\.99990\.800\.89690\.96960\.000\.98740\.00complex\-test30\.94970\.89460\.001\.00001\.000\.26940\.73250\.000\.99960\.000\.99600\.99600\.001\.00001\.00complex\-test40\.95340\.98220\.000\.99890\.600\.96810\.98610\.000\.99970\.000\.99920\.99100\.001\.00001\.00complex\-test50\.95680\.89520\.000\.95560\.000\.86220\.97740\.000\.99960\.000\.99980\.99960\.401\.00001\.00Feynman\-I\.11\.190\.86481\.00001\.001\.00001\.000\.82790\.24420\.001\.00001\.001\.00000\.99940\.600\.99930\.20Feynman\-I\.32\.17\-0\.00860\.26210\.001\.00001\.00\-2\.09330\.96040\.000\.28870\.00\-0\.28471\.00001\.001\.00001\.00Feynman\-I\.40\.10\.17841\.00001\.001\.00001\.00\-0\.01780\.97900\.000\.99990\.001\.00001\.00001\.001\.00001\.00Feynman\-II\.6\.15a\-1\.86621\.00001\.000\.99970\.600\.00900\.57420\.000\.98910\.000\.92280\.99640\.000\.99280\.00Feynman\-II\.11\.70\.93261\.00001\.001\.00001\.000\.61700\.99640\.000\.99550\.000\.82411\.00001\.001\.00001\.00Feynman\-III\.9\.520\.99991\.00001\.001\.00001\.000\.77330\.90170\.200\.99990\.801\.00000\.99581\.001\.00001\.00complex\-test60\.70230\.82770\.001\.00001\.000\.86060\.75070\.000\.89430\.000\.99250\.96230\.001\.00001\.00Feynman\-II\.36\.28\-5\.67701\.00001\.001\.00001\.00\-0\.00530\.97590\.001\.00001\.000\.09021\.00001\.001\.00001\.00complex\-test70\.2409\-0\.19000\.000\.96710\.00\-0\.79620\.12860\.000\.85840\.000\.02310\.31100\.000\.89160\.00Feynman\-I\.9\.180\.02300\.75390\.000\.73610\.00\-0\.01210\.63510\.000\.69590\.00\-0\.06560\.89550\.000\.99350\.60Feynman\-I\.10\.70\.0155\-1\.51120\.00\-0\.54360\.00\-0\.03930\.16740\.000\.72940\.000\.11590\.41780\.000\.99170\.40Poly\-100\.40240\.67130\.001\.00001\.000\.78050\.75540\.001\.00001\.001\.00000\.91050\.000\.99970\.40complex\-test80\.1614\-0\.07350\.000\.77830\.000\.1992\-0\.03030\.000\.49050\.000\.13430\.66700\.000\.99340\.00complex\-test90\.43560\.69890\.001\.00001\.00\-0\.53590\.54900\.001\.00001\.000\.77070\.56220\.000\.99990\.20Table 11:Detailed Separation of DDRSR and AI Feynman in Ablation Experiments of Expression Decomposition\.DatasetDDRSR ScoreDDRSR FormAIF ScoreAIF FormKeijzer\-101\.0exp\(f1\(x1\)\*f2\(x2\)\)0\.0f1\(x1,x2\)Keijzer\-110\.0f1\(x1,x2\)0\.0f1\(x1,x2\)Keijzer\-121\.0f1\(x1\)\+f2\(x2\)1\.0f1\(x1\)\+f2\(x2\)Keijzer\-131\.0\(f1\(x1\)\)\*\(f2\(x2\)\)1\.0\(f1\(x1\)\)\*\(f2\(x2\)\)Keijzer\-140\.0f1\(x1,x2\)0\.0f1\(x1,x2\)Keijzer\-151\.0f1\(x1\)\+f2\(x2\)1\.0f1\(x1\)\+f2\(x2\)Nguyen\-91\.0f1\(x1\)\+f2\(x2\)1\.0f1\(x1\)\+f2\(x2\)Nguyen\-101\.0\(f1\(x1\)\)\*\(f2\(x2\)\)1\.0\(f1\(x1\)\)\*\(f2\(x2\)\)Nguyen\-111\.0exp\(f1\(x1\)\*f2\(x2\)\)0\.0f1\(x1,x2\)Nguyen\-121\.0f1\(x1\)\+f2\(x2\)1\.0f1\(x1\)\+f2\(x2\)Nguyen\-10c1\.0\(f1\(x1\)\)\*\(f2\(x2\)\)1\.0\(f1\(x1\)\)\*\(f2\(x2\)\)Constant\-31\.0\(f1\(x1\)\)\*\(f2\(x2\)\)1\.0\(f1\(x1\)\)\*\(f2\(x2\)\)Constant\-41\.0exp\(f1\(x1\)\*f2\(x2\)\+0\.9933\)0\.0f1\(x1,x2\)Constant\-71\.0\(f1\(x1\)\)\*\(f2\(x2\)\)1\.0\(f1\(x1\)\)\*\(f2\(x2\)\)Livermore\-51\.0f1\(x1\)\+f2\(x2\)1\.0f1\(x1\)\+f2\(x2\)Livermore\-101\.0\(f1\(x1\)\)\*\(f2\(x2\)\)1\.0\(f1\(x1\)\)\*\(f2\(x2\)\)Livermore\-110\.0f1\(x1,x2\)0\.0f1\(x1,x2\)Livermore\-121\.0\(f1\(x1\)\)\*\(f2\(x2\)\)0\.0f1\(x1,x2\)Livermore\-141\.0f1\(x1\)\+f2\(x2\)1\.0f1\(x1\)\+f2\(x2\)Livermore\-171\.0\(f1\(x1\)\)\*\(f2\(x2\)\)1\.0\(f1\(x1\)\)\*\(f2\(x2\)\)Jin\-11\.0f1\(x1\)\+f2\(x2\)1\.0f1\(x1\)\+f2\(x2\)Jin\-21\.0f1\(x1\)\+f2\(x2\)1\.0f1\(x1\)\+f2\(x2\)Jin\-31\.0f1\(x1\)\+f2\(x2\)1\.0f1\(x1\)\+f2\(x2\)Jin\-41\.0f1\(x1\)\+f2\(x2\)1\.0f1\(x1\)\+f2\(x2\)Jin\-51\.0\(f1\(x1\)\)\*\(f2\(x2\)\)1\.0\(f1\(x1\)\)\*\(f2\(x2\)\)Jin\-60\.0f1\(x1,x2\)0\.0f1\(x1,x2\)bacterial respiration 10\.0f1\(x1,x2\)0\.0f1\(x1,x2\)bacterial respiration 31\.0\(f1\(x1\)\)\*\(f2\(x2\)\)\+10\.000\.0f1\(x1,x2\)bar magnets 10\.0f1\(x1,x2\)0\.0f1\(x1,x2\)bar magnets 20\.0f1\(x1,x2\)0\.0f1\(x1,x2\)glider 11\.0f1\(x1\)\+f2\(x2\)1\.0f1\(x1\)\+f2\(x2\)glider 20\.0f1\(x1,x2\)0\.0f1\(x1,x2\)Lotka\-Volterra 10\.0f1\(x1,x2\)0\.0f1\(x1,x2\)Lotka\-Volterra 20\.0f1\(x1,x2\)0\.0f1\(x1,x2\)predator\-prey 10\.0f1\(x1,x2\)0\.0f1\(x1,x2\)predator\-prey 20\.0f1\(x1,x2\)0\.0f1\(x1,x2\)shear flow 11\.0\(f1\(x1\)\)\*\(f2\(x2\)\)1\.0\(f1\(x1\)\)\*\(f2\(x2\)\)shear flow 21\.0\(f1\(x1\)\)\*\(f2\(x2\)\)1\.0\(f1\(x1\)\)\*\(f2\(x2\)\)van der Pol 11\.0f1\(x1\)\+f2\(x2\)1\.0f1\(x1\)\+f2\(x2\)van der Pol 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I\.12\.111\.0\(f1\(x2\)\)\*\(f2\(x1\)\)\+\(f3\(x3\)\)\*\(f4\(x1\)\)\*\(f5\(x5\)\)\*\(f6\(x4\)\)1\.0\(f1\(x1\)\)\*\(f2\(x2\)\+\(f3\(x3\)\)\*\(\(f4\(x4\)\)\*\(f5\(x5\)\)\)\)Feynman I\.13\.121\.0\(f1\(x4\)\)\*\(f2\(x3\)\)\*\(f3\(x2\)\)\*\(f4\(x1\)\)\+\(f5\(x5\)\)\*\(f6\(x3\)\)\*\(f7\(x2\)\)\*\(f8\(x1\)\)1\.0\(f1\(x1\)\)\*\(\(f2\(x2\)\)\*\(\(f3\(x3\)\)\*\(f4\(x4\)\+f5\(x5\)\)\)\)Feynman I\.41\.161\.0\(f1\(x1,x2,x4,x5\)\)\*\(f2\(x3\)\)0\.0f1\(x1,x2,x3,x4,x5\)Feynman I\.44\.41\.0\(f1\(x4\)\)\*\(f2\(x3\)\)\*\(f3\(x2\)\)\*\(f4\(x1\)\)\+\(f5\(x5\)\)\*\(f6\(x3\)\)\*\(f7\(x2\)\)\*\(f8\(x1\)\)1\.0\(f1\(x1\)\)\*\(\(f2\(x2\)\)\*\(\(f3\(x3\)\)\*\(f4\(x4\)\+f5\(x5\)\)\)\)Feynman II\.2\.421\.0\(f1\(x2\)\)\*\(f2\(x5\)\)\*\(f3\(x4\)\)\*\(f4\(x1\)\)\+\(f5\(x3\)\)\*\(f6\(x5\)\)\*\(f7\(x4\)\)\*\(f8\(x1\)\)1\.0\(f1\(x1\)\)\*\(\(f2\(x4\)\)\*\(\(f3\(x5\)\)\*\(f4\(x2\)\+f5\(x3\)\)\)\)Feynman II\.11\.31\.0\(f1\(x1\)\)\*\(f2\(x4,x5\)\)\*\(f3\(x3\)\)\*\(f4\(x2\)\)1\.0\(f1\(x1\)\)\*\(f2\(x3\)\)\*\(f3\(x3\)\)\*\(f4\(x4,x5\)\)Feynman II\.11\.201\.0\(f1\(x1\)\)\*\(f2\(x5\)\)\*\(f3\(x4\)\)\*\(f4\(x3\)\)\*\(f5\(x2\)\)1\.0\(f1\(x1\)\)\*\(\(f2\(x2\)\)\*\(\(f3\(x3\)\*\(\(f4\(x4\)\)\*\(f5\(x5\)\)\)\)\)Feynman II\.21\.321\.0\(f1\(x1\)\)\*\(f2\(x4,x5\)\)\*\(f3\(x3\)\)\*\(f4\(x2\)\)1\.0\(f1\(x1\)\)\*\(f2\(x3\)\)\*\(f3\(x3\)\)\*\(f4\(x4,x5\)\)Feynman II\.35\.181\.0\(f1\(x1\)\)\*\(f2\(x2,x3,x4,x5\)\)0\.0f1\(x1,x2,x3,x4,x5\)Feynman II\.35\.211\.0\(f1\(x1\)\)\*\(f2\(x2,x3,x4,x5\)\)1\.0\(f1\(x1\)\)\*\(f2\(x2,x3,x4,x5\)\)Feynman III\.14\.141\.0\(f1\(x1\)\+f2\(x2\)\)\*\(f3\(x3,x4,x5,x2\)\)0\.5\(f1\(x1\)\)\*\(f2\(x2,x3,x4,x5\)\)Feynman III\.19\.511\.0\(f1\(x1\)\)\*\(f2\(x5\)\)\*\(f3\(x4\)\)\*\(f4\(x3\)\)\*\(f5\(x2\)\)1\.0\(f1\(x1\)\)\*\(\(f2\(x2\)\)\*\(\(f3\(x3\)\)\*\(\(f4\(x4\)\)\*\(f5\(x5\)\)\)\)\)Feynman III\.21\.201\.0\(f1\(x1\)\)\*\(f2\(x5\)\)\*\(f3\(x4\)\)\*\(f4\(x3\)\)\*\(f5\(x2\)\)1\.0\(f1\(x1\)\)\*\(\(f2\(x2\)\)\*\(\(f3\(x3\)\)\*\(\(f4\(x4\)\)\*\(f5\(x5\)\)\)\)\)Feynman\-I\.11\.191\.0\(f1\(x1\)\)\*\(f2\(x4\)\)\+\(f3\(x3\)\)\*\(f4\(x6\)\)\+\(f5\(x2\)\)\*\(f6\(x5\)\)1\.0\(f1\(x1\)\)\*\(f2\(x4\)\)\+\(f3\(x2\)\)\*\(f4\(x5\)\)\+\(f5\(x3\)\)\*\(f6\(x6\)\)Feynman\-I\.32\.171\.0\(f1\(x1\)\)\*\(f2\(x5,x6\)\)\*\(f3\(x4\)\)\*\(f4\(x3\)\)\*\(f5\(x2\)\)0\.5\(f1\(x3\)\)\*\(f2\(x1,x2,x4,x5,x6\)\)Feynman\-I\.40\.11\.0\(f1\(x1\)\+f2\(x6\)\+f3\(x3\)\+f4\(x2\)\)\*\(f5\(x4,x5,x2,x3,x6\)\)0\.5\(f1\(x1\)\)\*\(f2\(x2,x3,x4,x5,x6\)\)Feynman\-II\.6\.15a1\.0\(f1\(x1\)\)\*\(f2\(x6\)\)\*\(f3\(x5\)\)\*\(f4\(x3,x4\)\)\*\(f5\(x2\)\)1\.0\(f1\(x1\)\)\*\(\(f2\(x2\)\)\*\(\(f3\(x5\)\)\*\(\(f4\(x6\)\)\*\(f5\(x3,x4\)\)\)\)\)Feynman\-II\.11\.71\.0\(f1\(x1\)\)\*\(\(f2\(x2\)\)\*\(f3\(x6\)\)\*\(f4\(x5\)\)\*\(f5\(x4\)\)\*\(f6\(x3\)\)\+1\.\)0\.5\(f1\(x1\)\)\*\(f2\(x2,x3,x4,x5,x6\)\)Feynman\-III\.9\.521\.0\(f1\(x1\)\)\*\(f2\(x6\)\)\*\(f3\(x3,x5\)\)\*\(f4\(x2\)\)\+f5\(x4\)\+f6\(x6\)\+f7\(x5\)\+f8\(x3\)\+f9\(x2\)0\.5\(f1\(x1\)\)\*\(f2\(x2\)\)\*\(f3\(x3,x4,x5\)\)\*\(f4\(x6\)\)Feynman\-II\.36\.281\.0\(f1\(x2\)\)\*\(f2\(x4\)\)\*\(f3\(x3\)\)\*\(f4\(x1\)\)\+\(f5\(x5,x6,x7,x8\)\)\*\(f6\(x4\)\)\*\(f7\(x3\)\)\*\(f8\(x1\)\)0\.5\(f1\(x1\)\)\*\(\(f2\(x3\)\)\*\(\(f3\(x4\)\)\*\(f4\(x2\)\+f5\(x5,x6,x7,x8\)\)\)\)Feynman\-I\.9\.181\.0\(f1\(x1\)\)\*\(f2\(x9\)\)\*\(f3\(x3,x4,x5,x6,x7,x8\)\)\*\(f4\(x2\)\)1\.0\(f1\(x1\)\)\*\(f2\(x2\)\)\*\(f3\(x9\)\)\*\(f4\(x3,x4,x5,x6,x7,x8\)\)Feynman\-I\.10\.71\.0\(f1\(x1\)\)\*\(f2\(x9\)\)\*\(f3\(x3,x4,x5,x6,x7,x8\)\)\*\(f4\(x2\)\)1\.0\(f1\(x1\)\)\*\(f2\(x2\)\)\*\(f3\(x9\)\)\*\(f4\(x3,x4,x5,x6,x7,x8\)\)simple\_test\-11\.0\(f1\(x1\)\)\*\(f2\(x2\)\)\+\(f3\(x3\)\)\*\(f4\(x2\)\)1\.0\(f1\(x2\)\)\*\(f2\(x1\)\+f3\(x3\)\)simple\_test\-21\.0\(f1\(x1\)\)\*\(f2\(x2\)\)\+\(f3\(x3\)\)\*\(f4\(x2\)\)1\.0\(f1\(x2\)\)\*\(f2\(x1\)\+f3\(x3\)\)simple\_test\-31\.0\(f1\(x1\)\)\*\(f2\(x2\)\)\+\(f3\(x3\)\)\*\(f4\(x2\)\)0\.0f1\(x1,x2,x3\)simple\_test\-41\.0\(f1\(x1,x2\)\)\*\(f2\(x3,x2\)\)0\.0f1\(x1,x2,x3\)simple\_test\-51\.0\(f1\(x1,x2\)\)\*\(f2\(x3,x2\)\)0\.0f1\(x1,x2,x3\)simple\_test\-61\.0\(f1\(x1,x2\)\)\*\(f2\(x3,x2\)\)0\.0f1\(x1,x2,x3\)simple\_test\-71\.0exp\(f1\(x1\)\*f2\(x3\)\*f3\(x2\)\)0\.0f1\(x1,x2,x3\)simple\_test\-81\.0log\(f1\(x1\)\+f2\(x3\)\+f3\(x2\)\)0\.0f1\(x1,x2,x3\)simple\_test\-91\.0sin\(f1\(x1,x2,x3\)\)0\.0f1\(x1,x2,x3\)simple\_test\-101\.0\(f1\(x1\)\)\*\(f2\(x3\)\)\*\(f3\(x2\)\)\+1\.28460\.0f1\(x1,x2,x3\)simple\_test\-111\.0\(f1\(x2\)\)\*\(f2\(x3\)\)\*\(f3\(x1\)\)\+\(f4\(x5\)\)\*\(f5\(x3\)\)\*\(f6\(x1\)\)\+\(f7\(x4\)\)\*\(f8\(x5\)\)\*\(f9\(x3\)\)0\.5\(f1\(x1,x2,x4,x5\)\)\*\(f2\(x3\)\)Table 12:Detailed Performance \(R2R^\{2\}\) in Ablation Experiments of Expression Decomposition\.DatasetE2EGP\-GOMEADSOPySRMetaBase\+AIF\+DivBase\+AIF\+DivBase\+AIF\+DivBase\+AIF\+DivBase\+AIF\+DivKeijzer\-100\.71490\.93600\.99911\.00001\.00001\.00000\.99480\.99981\.00000\.97840\.99951\.00000\.95240\.95620\.9972Keijzer\-110\.99410\.99610\.99370\.98480\.96080\.94150\.99580\.97850\.99920\.99480\.99270\.99620\.25110\.66660\.6403Keijzer\-120\.98160\.97830\.99990\.96110\.99981\.00000\.98301\.00001\.00000\.99600\.99800\.99490\.97670\.99990\.9889Keijzer\-130\.99930\.99970\.99980\.99921\.00000\.99960\.99971\.00001\.00000\.99991\.00001\.00000\.99690\.99981\.0000Keijzer\-140\.94580\.93270\.94130\.99300\.98560\.99340\.99540\.99730\.99960\.99970\.91410\.99960\.97580\.93110\.9897Keijzer\-150\.99380\.99940\.99970\.99991\.00001\.00000\.99631\.00001\.00000\.99880\.99800\.99820\.96361\.00001\.0000Nguyen\-90\.98910\.99720\.99990\.99680\.99991\.00000\.99970\.99971\.00001\.00000\.99731\.00000\.98911\.00001\.0000Nguyen\-100\.99920\.99980\.99940\.99881\.00001\.00000\.99981\.00000\.98731\.00000\.99980\.99990\.98360\.97820\.9992Nguyen\-110\.81170\.79730\.99781\.00001\.00001\.00000\.99250\.99921\.00000\.98930\.98281\.00000\.95580\.95660\.9956Nguyen\-120\.95620\.99630\.99940\.97430\.99951\.00000\.97561\.00001\.00000\.99190\.99650\.99520\.98520\.99930\.9981Nguyen\-10c0\.99740\.99970\.99980\.99841\.00000\.99990\.99991\.00001\.00000\.99910\.99970\.99930\.99770\.99811\.0000Constant\-30\.99660\.99960\.99930\.99560\.99891\.00000\.99941\.00001\.00000\.99840\.99930\.99960\.99640\.99971\.0000Constant\-40\.89520\.92840\.99681\.00001\.00001\.00000\.99511\.00001\.00000\.76530\.99891\.00000\.93110\.98580\.9974Constant\-70\.99820\.99700\.99780\.99800\.99991\.00000\.99990\.99981\.00000\.99990\.99930\.99830\.97991\.00000\.9999Livermore\-50\.89730\.99180\.99860\.90420\.99931\.00000\.93970\.94821\.00000\.98800\.99560\.99630\.87190\.99850\.9917Livermore\-100\.99870\.99890\.99950\.99941\.00001\.00000\.99971\.00001\.00001\.00000\.99990\.99950\.96841\.00001\.0000Livermore\-110\.99030\.99530\.99610\.99991\.00001\.00000\.99990\.99940\.99920\.99590\.97560\.99270\.87260\.95550\.9750Livermore\-120\.96420\.98060\.97960\.90131\.00001\.00000\.98310\.99870\.98681\.00001\.00001\.0000\-0\.02560\.94810\.9534Livermore\-140\.99740\.99850\.99940\.99980\.99981\.00000\.99380\.95311\.00001\.00001\.00000\.99990\.99861\.00001\.0000Livermore\-170\.99830\.99970\.99790\.99931\.00001\.00000\.99991\.00001\.00001\.00000\.99950\.99980\.98021\.00000\.9992Jin\-10\.97960\.99751\.00000\.98750\.99991\.00000\.99330\.99421\.00000\.99990\.99980\.99910\.84740\.99981\.0000Jin\-20\.98520\.99930\.99910\.99951\.00001\.00000\.97550\.98161\.00001\.00001\.00001\.00000\.99990\.99961\.0000Jin\-30\.94320\.99840\.99960\.99600\.99951\.00000\.99521\.00000\.97190\.99910\.99650\.99980\.98290\.98710\.9999Jin\-40\.99840\.99910\.99970\.99781\.00001\.00000\.99891\.00001\.00001\.00000\.99990\.99990\.99660\.99991\.0000Jin\-50\.99520\.99990\.99960\.99941\.00001\.00000\.99961\.00001\.00001\.00001\.00000\.99880\.99991\.00000\.9998Jin\-60\.99400\.98870\.99660\.94660\.95130\.94380\.99990\.92790\.99730\.99980\.99990\.99990\.97390\.87460\.9721bacterial respiration 10\.91160\.98590\.99430\.99990\.99970\.99750\.99990\.99970\.99980\.99931\.00000\.99910\.99510\.99020\.9980bacterial respiration 30\.99820\.99720\.99980\.99090\.98861\.00000\.99991\.00001\.00000\.99900\.99870\.99960\.97020\.93241\.0000bar magnets 10\.99300\.99840\.99540\.99800\.98890\.99740\.99840\.99720\.99670\.99520\.99690\.99570\.96090\.99280\.9969bar magnets 20\.99470\.98950\.99320\.99680\.99410\.99600\.99500\.99720\.99820\.99770\.99690\.99770\.99100\.99410\.9872glider 10\.99940\.99970\.99960\.99991\.00001\.00000\.99990\.99911\.00001\.00000\.99671\.00000\.99981\.00001\.0000glider 20\.49770\.74190\.53910\.99940\.99611\.00000\.99831\.00001\.00001\.00001\.00001\.00000\.98940\.94340\.9944Lotka\-Volterra 10\.99940\.99950\.99980\.99850\.99730\.99990\.99990\.99971\.00001\.00000\.99990\.99950\.88880\.90390\.9694Lotka\-Volterra 20\.99920\.99850\.99900\.99991\.00001\.00000\.99991\.00001\.00000\.99880\.99970\.99980\.87990\.92730\.9309predator\-prey 10\.99930\.99970\.99970\.99990\.99980\.99980\.99981\.00001\.00001\.00000\.99980\.99980\.99160\.99910\.9953predator\-prey 20\.99250\.99210\.99360\.99720\.99930\.99730\.99950\.99841\.00000\.98630\.99750\.99700\.97310\.91130\.9938shear flow 10\.99110\.70920\.99570\.99861\.00000\.99890\.99981\.00001\.00001\.00001\.00000\.99990\.99780\.96410\.9994shear flow 20\.99080\.99910\.99750\.99991\.00000\.99990\.99831\.00001\.00000\.99970\.99870\.99970\.96500\.99870\.9998van der Pol 10\.99960\.99990\.99920\.99991\.00001\.00000\.99901\.00001\.00001\.00001\.00001\.00000\.99901\.00001\.0000van der Pol 20\.99991\.00000\.99991\.00001\.00001\.00000\.99991\.00001\.00001\.00000\.90611\.00000\.99920\.99981\.0000Korns\-120\.58150\.42760\.98280\.35390\.31730\.85750\.66060\.65831\.00000\.99620\.99991\.00000\.41580\.68090\.9838Neat\-50\.99910\.99820\.99980\.99810\.99990\.99920\.99980\.99961\.00000\.99950\.99990\.99870\.99761\.00000\.9662Neat\-7\-0\.00040\.96840\.99770\.62120\.52960\.62720\.57210\.85841\.00001\.00001\.00001\.00000\.82130\.74330\.8424Neat\-80\.97870\.99960\.99900\.99991\.00000\.99590\.99991\.00001\.00000\.98560\.99940\.99280\.98350\.99991\.0000Neat\-90\.98830\.97010\.98140\.99170\.99760\.98860\.97771\.00001\.00000\.99040\.99770\.99540\.95420\.99780\.9992Vladislavleva\-10\.98960\.99600\.99640\.98091\.00000\.99990\.99951\.00001\.00000\.86450\.94900\.99580\.95760\.99980\.9991Vladislavleva\-30\.99280\.99960\.99840\.99941\.00001\.00000\.99931\.00001\.00000\.98460\.99930\.99920\.99810\.99980\.9999Vladislavleva\-60\.99900\.99970\.99980\.99981\.00001\.00000\.99990\.99911\.00000\.94211\.00000\.99900\.96671\.00001\.0000Vladislavleva\-70\.71380\.66850\.99930\.25060\.61330\.93750\.68110\.99091\.00001\.00000\.99991\.00000\.18390\.95010\.9672Vladislavleva\-80\.99120\.99880\.99920\.98621\.00001\.00000\.99991\.00001\.00000\.94570\.99850\.99370\.95510\.99970\.9995Pagie\-10\.94780\.97360\.99680\.99850\.99880\.99990\.99710\.99950\.99780\.97660\.98990\.99540\.99390\.99110\.9949Meier\-30\.99460\.98590\.99490\.99971\.00001\.00000\.99990\.99820\.99840\.99020\.99610\.99040\.94600\.97410\.9590Meier\-40\.98040\.95560\.98700\.99990\.99981\.00000\.72530\.99821\.00001\.00001\.00001\.00000\.03380\.97520\.9816Feynman\-I\.6\.20\.42400\.63410\.27980\.99660\.89930\.93900\.97440\.96280\.99060\.99760\.94090\.99480\.78620\.81370\.7709Feynman\-I\.12\.10\.99991\.00001\.00001\.00001\.00001\.00000\.99990\.99991\.00000\.99741\.00001\.00000\.97600\.97241\.0000Feynman\-I\.12\.50\.99991\.00001\.00001\.00001\.00001\.00000\.99991\.00001\.00001\.00000\.99680\.99990\.96580\.97171\.0000Feynman\-I\.14\.40\.99981\.00001\.00001\.00001\.00001\.00000\.99991\.00001\.00000\.98721\.00000\.99830\.97920\.97870\.9956Feynman\-I\.25\.130\.99180\.99500\.99961\.00001\.00001\.00001\.00000\.99991\.00001\.00001\.00001\.00000\.80040\.99851\.0000Feynman\-I\.29\.40\.99890\.99970\.98961\.00001\.00001\.00000\.99991\.00001\.00001\.00000\.99921\.00000\.94610\.93301\.0000Feynman\-I\.34\.270\.99991\.00001\.00001\.00001\.00001\.00000\.99991\.00001\.00001\.00000\.99501\.00000\.94660\.96871\.0000Feynman\-I\.39\.10\.99971\.00001\.00001\.00001\.00001\.00000\.99991\.00001\.00001\.00001\.00001\.00000\.96350\.98411\.0000Feynman\-II\.8\.310\.99970\.99961\.00001\.00001\.00001\.00000\.99991\.00001\.00000\.99990\.99560\.99990\.94730\.97900\.9995Feynman\-II\.11\.280\.99670\.99630\.98600\.99960\.99970\.99980\.99991\.00001\.00000\.98840\.99940\.99940\.99430\.98690\.9841Feynman\-II\.27\.180\.99990\.99951\.00001\.00001\.00001\.00000\.99991\.00001\.00001\.00000\.99901\.00000\.89010\.99431\.0000Feynman\-II\.38\.140\.99991\.00000\.99991\.00001\.00001\.00000\.99991\.00001\.00000\.99740\.99330\.99680\.99521\.00001\.0000Feynman\-III\.12\.430\.99980\.99981\.00001\.00001\.00001\.00000\.99991\.00001\.00000\.95701\.00001\.00000\.95290\.93211\.0000Korns\-20\.96710\.98120\.99410\.99991\.00001\.00000\.99840\.99271\.00001\.00001\.00001\.00000\.83880\.79430\.8515Korns\-30\.97380\.90270\.99440\.99971\.00001\.00000\.96350\.97461\.00001\.00001\.00001\.00000\.54220\.52230\.9829Korns\-80\.99390\.95770\.99910\.97910\.99821\.00000\.94550\.93751\.00000\.98980\.98920\.99700\.59410\.87760\.9986Korns\-90\.99970\.99870\.99960\.99991\.00001\.00000\.95001\.00001\.00000\.99740\.98280\.99960\.89080\.99260\.9995Vladislavleva\-50\.99960\.99941\.00000\.98830\.99681\.00000\.68251\.00001\.00000\.99930\.99790\.99990\.74210\.99790\.9999Feynman\-I\.6\.20b0\.38470\.49300\.40800\.68670\.80170\.7412\-0\.22310\.27670\.01700\.57910\.83820\.98190\.20250\.32760\.1309Feynman\-I\.12\.40\.84850\.98500\.94380\.99991\.00001\.00000\.99991\.00001\.00001\.00000\.99991\.0000\-0\.02540\.91330\.9100Feynman\-I\.14\.30\.99970\.99891\.00001\.00001\.00001\.00000\.99991\.00001\.00001\.00000\.99661\.00000\.25080\.97051\.0000Feynman\-I\.16\.60\.54320\.86140\.70610\.97030\.96650\.98840\.64550\.65940\.68391\.00000\.90221\.00000\.75070\.79440\.8069Feynman\-I\.18\.120\.99830\.99930\.99970\.99991\.00001\.00001\.00000\.99690\.99320\.99691\.00000\.99950\.98581\.00000\.9982Feynman\-I\.27\.60\.83410\.64950\.95410\.99991\.00001\.00000\.99510\.99170\.99321\.00001\.00001\.00000\.72970\.68240\.8315Feynman\-I\.30\.30\.99930\.99910\.99910\.99980\.99941\.00000\.99971\.00001\.00000\.87060\.98960\.99980\.23280\.93530\.9576Feynman\-I\.34\.100\.2213\-0\.34130\.76310\.99991\.00001\.00000\.94440\.96590\.99381\.00000\.99981\.0000\-0\.2419\-0\.15030\.0082Feynman\-I\.34\.140\.88920\.98240\.89420\.99991\.00001\.00000\.51830\.99700\.97660\.86980\.83750\.99900\.52760\.53220\.6586Feynman\-I\.37\.40\.97630\.99470\.98280\.98530\.99050\.98710\.96840\.96650\.97480\.99930\.99900\.99970\.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1\.00000\.96370\.98701\.00000\.80620\.97481\.00000\.74980\.98271\.00000\.44980\.94680\.9952Feynman\-I\.29\.160\.94920\.88660\.92010\.97810\.99160\.72990\.85500\.94620\.91160\.99890\.99500\.97990\.54910\.90700\.8591Feynman\-I\.32\.5\-0\.02830\.91110\.99860\.97350\.96001\.00000\.07960\.90030\.99980\.45761\.00001\.00000\.66210\.45040\.4036Feynman\-I\.34\.80\.95450\.99970\.99970\.99991\.00001\.00000\.99990\.99611\.00001\.00001\.00001\.00000\.53320\.93680\.9791Feynman\-I\.38\.12\-0\.64870\.98550\.99570\.99991\.00001\.0000\-0\.01541\.00001\.00001\.00001\.00001\.00000\.09840\.73480\.5422Feynman\-I\.39\.220\.75780\.99700\.99950\.99991\.00001\.00000\.98991\.00001\.00001\.00001\.00001\.00000\.34220\.95971\.0000Feynman\-I\.43\.160\.63750\.99040\.99990\.99991\.00001\.00000\.84330\.98621\.00001\.00000\.99991\.0000\-0\.16930\.97690\.9990Feynman\-I\.43\.430\.38140\.90490\.98620\.99991\.00001\.00000\.55891\.00001\.00001\.00001\.00001\.00000\.73700\.99260\.9997Feynman\-I\.47\.230\.88230\.99980\.99620\.99991\.00001\.00000\.83221\.00000\.98950\.91380\.99990\.99980\.29540\.97950\.7977Feynman\-I\.50\.260\.99390\.99920\.99830\.99740\.98840\.99820\.97380\.99200\.99430\.99960\.99740\.99990\.72630\.97360\.9233Feynman\-II\.6\.110\.26150\.26290\.96170\.99941\.00001\.00000\.13711\.00001\.00001\.00001\.00000\.9980\-0\.47420\.26010\.9535Feynman\-II\.6\.15b0\.20840\.99920\.61530\.27111\.00000\.99990\.11781\.00001\.00000\.94361\.00000\.99980\.99550\.83530\.8027Feynman\-II\.11\.270\.94690\.97840\.99210\.99991\.00000\.99990\.94300\.94131\.00000\.99190\.99870\.99720\.56150\.97720\.9743Feynman\-II\.13\.170\.12990\.52530\.99630\.62041\.00001\.00000\.23651\.00001\.00001\.00001\.00001\.00000\.55540\.76320\.9202Feynman\-II\.13\.230\.94040\.79580\.50180\.98801\.00001\.00000\.58170\.87760\.99980\.99710\.99010\.99890\.70860\.68590\.4412Feynman\-II\.13\.340\.98260\.92010\.85620\.99960\.82671\.0000\-0\.39990\.74420\.83240\.92440\.92820\.99820\.12020\.87190\.9494Feynman\-II\.34\.29b0\.94850\.90380\.91521\.00001\.00001\.00000\.97981\.00001\.00001\.00001\.00001\.00000\.25660\.91671\.0000Feynman\-II\.38\.30\.96750\.95670\.93441\.00001\.00001\.00000\.99301\.00001\.00001\.00001\.00001\.0000\-0\.07900\.92760\.9212Feynman\-III\.4\.320\.1392\-0\.14020\.29980\.99991\.00001\.00000\.84370\.91060\.80920\.99991\.00001\.00000\.37990\.15450\.2231Feynman\-III\.4\.330\.98380\.98590\.94380\.99950\.99900\.99920\.90580\.84480\.91430\.99580\.98670\.99510\.81170\.79110\.8444Feynman\-III\.7\.380\.38310\.89780\.95080\.99991\.00001\.00000\.99040\.93461\.00001\.00001\.00001\.00000\.01010\.93340\.9994Feynman\-III\.10\.190\.97800\.97120\.98000\.99990\.99460\.99990\.98020\.98710\.99000\.99350\.99910\.99190\.82440\.94730\.9568Feynman\-III\.13\.180\.39290\.99740\.99440\.99810\.98641\.00000\.27220\.97141\.00001\.00001\.00001\.00000\.00540\.93161\.0000Vladislavleva\-40\.98490\.97540\.96600\.92820\.86010\.88200\.68510\.71590\.72070\.88810\.90810\.94010\.66900\.40380\.6548Feynman\-I\.12\.110\.98170\.94771\.00000\.99920\.99031\.00000\.91720\.97871\.00000\.99830\.98691\.00000\.78780\.97710\.9998Feynman\-I\.13\.12\-0\.22750\.90550\.99640\.99990\.92841\.0000\-0\.11210\.90221\.00001\.00001\.00001\.00000\.40310\.51060\.5677Feynman\-I\.41\.16\-2\.44000\.69090\.91790\.22660\.24670\.9931\-0\.02160\.1233\-0\.0246\-0\.05300\.46090\.7469\-0\.78930\.38990\.8288Feynman\-I\.44\.40\.45340\.97380\.98670\.99991\.00001\.00000\.51670\.91141\.00000\.96580\.97380\.9999\-0\.03640\.98150\.9216Feynman\-II\.2\.420\.74450\.98970\.99470\.99991\.00001\.0000\-0\.00880\.91631\.00001\.00001\.00000\.9998\-1\.13010\.95960\.9347Feynman\-II\.11\.3\-0\.01900\.02990\.25490\.81420\.42521\.0000\-0\.0141\-0\.02480\.86850\.95190\.62520\.9951\-0\.0616\-0\.93730\.3752Feynman\-II\.11\.200\.92390\.98980\.99760\.99991\.00001\.00000\.87400\.93531\.00001\.00001\.00000\.9956\-0\.04820\.95370\.9602Feynman\-II\.21\.32\-0\.2049\-0\.18980\.9853\-0\.02190\.45901\.0000\-0\.0267\-0\.05380\.8197\-0\.31600\.23381\.0000\-0\.02280\.01070\.4059Feynman\-II\.35\.180\.65690\.80660\.94040\.68700\.99160\.89310\.65760\.85640\.84500\.99760\.86350\.99860\.48020\.29760\.6934Feynman\-II\.35\.210\.87380\.71250\.52720\.99140\.98360\.65480\.68740\.66060\.83500\.92420\.92110\.96770\.43960\.64700\.8271Feynman\-III\.14\.140\.7652\-1\.27440\.36390\.99990\.95181\.0000\-1\.0318\-0\.1305\-0\.36421\.00001\.00001\.0000\-0\.05260\.02010\.2999Feynman\-III\.19\.51\-1\.46320\.73270\.4219\-0\.05890\.80921\.0000\-0\.01690\.94130\.96150\.43810\.93481\.0000\-0\.0179\-0\.0048\-0\.0266Feynman\-III\.21\.200\.73350\.96610\.96460\.99991\.00001\.00000\.69100\.92151\.00001\.00001\.00001\.00000\.73510\.99380\.9570test\-110\.97330\.99301\.00000\.98960\.99071\.00000\.64890\.93571\.00001\.00001\.00001\.00000\.52700\.87050\.9987Feynman\-I\.11\.190\.82790\.99871\.00000\.86481\.00001\.00000\.76261\.00001\.00001\.00000\.99851\.00000\.42730\.97940\.9890Feynman\-I\.32\.17\-2\.0933\-0\.00100\.3331\-0\.0086\-0\.34680\.9996\-0\.3821\-0\.10810\.0722\-0\.2847\-0\.21640\.9964\-0\.0461\-0\.1061\-0\.0373Feynman\-I\.40\.1\-0\.01780\.52280\.95190\.1784\-1\.72871\.00000\.08020\.03071\.00001\.0000\-0\.08871\.0000\-0\.02560\.48240\.3427Feynman\-II\.6\.15a0\.00900\.91680\.9620\-1\.86620\.92870\.9994\-1\.56370\.99740\.99530\.92281\.00000\.9991\-0\.48040\.86470\.9090Feynman\-II\.11\.70\.61700\.66570\.99980\.93260\.01511\.0000\-0\.02680\.81261\.00000\.82411\.00000\.9989\-0\.14730\.43500\.9462Feynman\-III\.9\.520\.77330\.29710\.98230\.99991\.00000\.24950\.76340\.99970\.97501\.00000\.99581\.00000\.09150\.34090\.8354Feynman\-II\.36\.28\-0\.00530\.25290\.9742\-5\.67700\.50741\.0000\-0\.1595\-0\.27030\.99960\.09021\.00000\.99990\.2811\-0\.95240\.5387Feynman\-I\.9\.18\-0\.0121\-0\.0376\-0\.94260\.0230\-0\.80970\.14280\.02860\.66210\.5038\-0\.0656\-0\.0344\-0\.0104\-0\.0322\-0\.1194\-0\.1540Feynman\-I\.10\.7\-0\.03930\.2757\-0\.45930\.0155\-1\.66010\.1826\-0\.0487\-0\.0901\-0\.04630\.1159\-1\.46810\.3807\-0\.05250\.0104\-0\.0843Poly\-100\.78050\.92981\.00000\.40240\.94161\.00000\.36650\.83221\.00001\.00001\.00001\.00000\.01080\.72630\.9999Table 13:Detailed Separation of DDRSR and AI Feynman in Ablation Experiments of Variable CompositionDatasetDDRSR ScoreDDRSR FormAIF ScoreAIF Formcomplex\-test10\.0f1\(x1,x2,x3,x4,x5\)0\.0f1\(x1,x2,x3,x4,x5\)complex\-test20\.0f1\(x1,x2,x3,x4,x5\)0\.0f1\(x1,x2,x3,x4,x5\)complex\-test31\.0f1\(\(x1^\(1\)\*x2^\(2\)\*x5^\(\-1\)\),x3,x4\)0\.5f1\(x1/x5,x2,x3,x4\)complex\-test41\.0f1\(\(x1\+\(1\.2000\)\*x2\),x3,\(x4\+\(1\.1667\)\*x5\)\)0\.0f1\(x1,x2,x3,x4,x5\)complex\-test51\.0f1\(\(x1\+\(2\.0000\)\*x2\+\(3\.0000\)\*x3\),\(x4^\(1\)\*x5^\(2\)\)\)0\.0f1\(x1,x2,x3,x4,x5\)complex\-test61\.0f1\(\(x1^\(1\)\*x2^\(2\)\),\(x3\+\(1\.5000\)\*x4\),\(x5^\(2\)\*x6^\(3\)\)\)0\.0f1\(x1,x2,x3,x4,x5,x6\)complex\-test71\.0f1\(x1,x2,\(x3\+\(2\.0000\)\*x4\),\(\(x5\+\(\-0\.5000\)\*x6\)^\(1\)\*x7^\(\-1\)\*x8^\(\-2\)\)\)0\.0f1\(x1,x2,x3,x4,x5,x6,x7,x8\)complex\-test81\.0f1\(x1,x2,\(x3\+\(\-2\.0000\)\*x4\),\(x5\+\(\-3\.0000\)\*x6\),\(x7\+\(\-4\.0000\)\*x8\),x9\)0\.0f1\(x1,x2,x3,x4,x5,x6,x7,x8,x9\)complex\-test91\.0f1\(x1,\(x2^\(1\)\*x4^\(2\)\),x3,x5,\(x6^\(2\)\*x8^\(1\)\),x7,x9\)0\.0f1\(x1,x2,x3,x4,x5,x6,x7,x8,x9\)Table 14:Detailed Performance \(R2R^\{2\}\) in Ablation Experiments of Variable Composition\.DatasetE2EGP\-GOMEADSOPySRMetaBase\+AIF\+DivBase\+AIF\+DivBase\+AIF\+DivBase\+AIF\+DivBase\+AIF\+Divcomplex\-test10\.77460\.74870\.71240\.99380\.99990\.99990\.65330\.71810\.67230\.99080\.99270\.98210\.62640\.67070\.5972complex\-test20\.73980\.80610\.88670\.97230\.97690\.97470\.15010\.16220\.22320\.89690\.93450\.92910\.65140\.66710\.6844complex\-test30\.26940\.93370\.99970\.94970\.88240\.9946\-0\.92340\.94450\.99500\.99600\.99301\.0000\-0\.09920\.06420\.1164complex\-test40\.96810\.97110\.99850\.95340\.98770\.98710\.85410\.78230\.95320\.99920\.99410\.99990\.65220\.80870\.9726complex\-test50\.86220\.76160\.99970\.95680\.91830\.98320\.70940\.78750\.21090\.99980\.99971\.00000\.77430\.84330\.9844complex\-test60\.86060\.62530\.95580\.70230\.75611\.0000\-0\.5006\-0\.80650\.97820\.99250\.97900\.9984\-0\.03560\.37490\.8935complex\-test7\-0\.7962\-0\.34750\.93640\.2409\-0\.54620\.8854\-0\.4746\-0\.00070\.91660\.0231\-0\.02110\.9167\-0\.3776\-0\.52730\.4535complex\-test80\.1992\-0\.04380\.23810\.16140\.26920\.3756\-0\.0297\-0\.48770\.11470\.13430\.52400\.69640\.0229\-0\.1689\-0\.4380complex\-test9\-0\.53590\.37030\.74300\.43560\.59840\.8925\-0\.1396\-0\.28760\.10610\.77070\.63310\.5517\-0\.5859\-0\.3428\-0\.0370
## NeurIPS Paper Checklist

1. 1\.Claims
2. Question: Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope?
3. Answer:\[Yes\]
4. Justification: The main claims made in the abstract and introduction do accurately reflect the paper’s contributions and scope\.
5. Guidelines: - •The answer\[N/A\]means that the abstract and introduction do not include the claims made in the paper\. - •The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations\. A\[No\]or\[N/A\]answer to this question will not be perceived well by the reviewers\. - •The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings\. - •It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper\.
6. 2\.Limitations
7. Question: Does the paper discuss the limitations of the work performed by the authors?
8. Answer:\[Yes\]
9. Justification: The paper discuss the limitations of the work in Appendix[A](https://arxiv.org/html/2608.02628#A1)\.
10. Guidelines: - •The answer\[N/A\]means that the paper has no limitation while the answer\[No\]means that the paper has limitations, but those are not discussed in the paper\. - •The authors are encouraged to create a separate “Limitations” section in their paper\. - •The paper should point out any strong assumptions and how robust the results are to violations of these assumptions \(e\.g\., independence assumptions, noiseless settings, model well\-specification, asymptotic approximations only holding locally\)\. The authors should reflect on how these assumptions might be violated in practice and what the implications would be\. - •The authors should reflect on the scope of the claims made, e\.g\., if the approach was only tested on a few datasets or with a few runs\. In general, empirical results often depend on implicit assumptions, which should be articulated\. - •The authors should reflect on the factors that influence the performance of the approach\. For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting\. Or a speech\-to\-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon\. - •The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size\. - •If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness\. - •While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren’t acknowledged in the paper\. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community\. Reviewers will be specifically instructed to not penalize honesty concerning limitations\.
11. 3\.Theory assumptions and proofs
12. Question: For each theoretical result, does the paper provide the full set of assumptions and a complete \(and correct\) proof?
13. Answer:\[Yes\]
14. Justification: Detailed mathematical proofs and derivations are provided in the appendix\.
15. Guidelines: - •The answer\[N/A\]means that the paper does not include theoretical results\. - •All the theorems, formulas, and proofs in the paper should be numbered and cross\-referenced\. - •All assumptions should be clearly stated or referenced in the statement of any theorems\. - •The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch to provide intuition\. - •Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material\. - •Theorems and Lemmas that the proof relies upon should be properly referenced\.
16. 4\.Experimental result reproducibility
17. Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper \(regardless of whether the code and data are provided or not\)?
18. Answer:\[Yes\]
19. Justification: The information provided in the main paper and the appendix ensures the reproducibility of the experimental results\.
20. Guidelines: - •The answer\[N/A\]means that the paper does not include experiments\. - •If the paper includes experiments, a\[No\]answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not\. - •If the contribution is a dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable\. - •Depending on the contribution, reproducibility can be accomplished in various ways\. For example, if the contribution is a novel architecture, describing the architecture fully might suffice, or if the contribution is a specific model and empirical evaluation, it may be necessary to either make it possible for others to replicate the model with the same dataset, or provide access to the model\. In general\. releasing code and data is often one good way to accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access to a hosted model \(e\.g\., in the case of a large language model\), releasing of a model checkpoint, or other means that are appropriate to the research performed\. - •While NeurIPS does not require releasing code, the conference does require all submissions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution\. For example 1. \(a\)If the contribution is primarily a new algorithm, the paper should make it clear how to reproduce that algorithm\. 2. \(b\)If the contribution is primarily a new model architecture, the paper should describe the architecture clearly and fully\. 3. \(c\)If the contribution is a new model \(e\.g\., a large language model\), then there should either be a way to access this model for reproducing the results or a way to reproduce the model \(e\.g\., with an open\-source dataset or instructions for how to construct the dataset\)\. 4. \(d\)We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reproducibility\. In the case of closed\-source models, it may be that access to the model is limited in some way \(e\.g\., to registered users\), but it should be possible for other researchers to have some path to reproducing or verifying the results\.
21. 5\.Open access to data and code
22. Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?
23. Answer:\[No\]
24. Justification: The complete codebase and associated resources will be open\-sourced following the acceptance of this manuscript\.
25. Guidelines: - •The answer\[N/A\]means that paper does not include experiments requiring code\. - • - •While we encourage the release of code and data, we understand that this might not be possible, so\[No\]is an acceptable answer\. Papers cannot be rejected simply for not including code, unless this is central to the contribution \(e\.g\., for a new open\-source benchmark\)\. - •The instructions should contain the exact command and environment needed to run to reproduce the results\. See the NeurIPS code and data submission guidelines \([https://neurips\.cc/public/guides/CodeSubmissionPolicy](https://neurips.cc/public/guides/CodeSubmissionPolicy)\) for more details\. - •The authors should provide instructions on data access and preparation, including how to access the raw data, preprocessed data, intermediate data, and generated data, etc\. - •The authors should provide scripts to reproduce all experimental results for the new proposed method and baselines\. If only a subset of experiments are reproducible, they should state which ones are omitted from the script and why\. - •At submission time, to preserve anonymity, the authors should release anonymized versions \(if applicable\)\. - •Providing as much information as possible in supplemental material \(appended to the paper\) is recommended, but including URLs to data and code is permitted\.
26. 6\.Experimental setting/details
27. Question: Does the paper specify all the training and test details \(e\.g\., data splits, hyperparameters, how they were chosen, type of optimizer\) necessary to understand the results?
28. Answer:\[Yes\]
29. Justification: The paper specify all the training and test details\.
30. Guidelines: - •The answer\[N/A\]means that the paper does not include experiments\. - •The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them\. - •The full details can be provided either with the code, in appendix, or as supplemental material\.
31. 7\.Experiment statistical significance
32. Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
33. Answer:\[Yes\]
34. Justification: The paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments
35. Guidelines: - •The answer\[N/A\]means that the paper does not include experiments\. - •The authors should answer\[Yes\]if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper\. - •The factors of variability that the error bars are capturing should be clearly stated \(for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions\)\. - •The method for calculating the error bars should be explained \(closed form formula, call to a library function, bootstrap, etc\.\) - •The assumptions made should be given \(e\.g\., Normally distributed errors\)\. - •It should be clear whether the error bar is the standard deviation or the standard error of the mean\. - •It is OK to report 1\-sigma error bars, but one should state it\. The authors should preferably report a 2\-sigma error bar than state that they have a 96% CI, if the hypothesis of Normality of errors is not verified\. - •For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range \(e\.g\., negative error rates\)\. - •If error bars are reported in tables or plots, the authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text\.
36. 8\.Experiments compute resources
37. Question: For each experiment, does the paper provide sufficient information on the computer resources \(type of compute workers, memory, time of execution\) needed to reproduce the experiments?
38. Answer:\[Yes\]
39. Justification: The paper provide sufficient information on the computer resources\.
40. Guidelines: - •The answer\[N/A\]means that the paper does not include experiments\. - •The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage\. - •The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute\. - •The paper should disclose whether the full research project required more compute than the experiments reported in the paper \(e\.g\., preliminary or failed experiments that didn’t make it into the paper\)\.
41. 9\.Code of ethics
43. Answer:\[Yes\]
44. Justification: The research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics\.
45. Guidelines: - •The answer\[N/A\]means that the authors have not reviewed the NeurIPS Code of Ethics\. - •If the authors answer\[No\], they should explain the special circumstances that require a deviation from the Code of Ethics\. - •The authors should make sure to preserve anonymity \(e\.g\., if there is a special consideration due to laws or regulations in their jurisdiction\)\.
46. 10\.Broader impacts
47. Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
48. Answer:\[N/A\]
49. Justification: The paper has neither potential positive societal impacts or negative societal impacts of the work performed\.
50. Guidelines: - •The answer\[N/A\]means that there is no societal impact of the work performed\. - •If the authors answer\[N/A\]or\[No\], they should explain why their work has no societal impact or why the paper does not address societal impact\. - •Examples of negative societal impacts include potential malicious or unintended uses \(e\.g\., disinformation, generating fake profiles, surveillance\), fairness considerations \(e\.g\., deployment of technologies that could make decisions that unfairly impact specific groups\), privacy considerations, and security considerations\. - •The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments\. However, if there is a direct path to any negative applications, the authors should point it out\. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate Deepfakes for disinformation\. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster\. - •The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from \(intentional or unintentional\) misuse of the technology\. - •If there are negative societal impacts, the authors could also discuss possible mitigation strategies \(e\.g\., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML\)\.
51. 11\.Safeguards
52. Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse \(e\.g\., pre\-trained language models, image generators, or scraped datasets\)?
53. Answer:\[N/A\]
54. Justification: The paper poses no such risks\.
55. Guidelines: - •The answer\[N/A\]means that the paper poses no such risks\. - •Released models that have a high risk for misuse or dual\-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters\. - •Datasets that have been scraped from the Internet could pose safety risks\. The authors should describe how they avoided releasing unsafe images\. - •We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort\.
56. 12\.Licenses for existing assets
57. Question: Are the creators or original owners of assets \(e\.g\., code, data, models\), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?
58. Answer:\[Yes\]
59. Justification: All code and data used in the paper are properly cited\.
60. Guidelines: - •The answer\[N/A\]means that the paper does not use existing assets\. - •The authors should cite the original paper that produced the code package or dataset\. - •The authors should state which version of the asset is used and, if possible, include a URL\. - •The name of the license \(e\.g\., CC\-BY 4\.0\) should be included for each asset\. - •For scraped data from a particular source \(e\.g\., website\), the copyright and terms of service of that source should be provided\. - •If assets are released, the license, copyright information, and terms of use in the package should be provided\. For popular datasets,[paperswithcode\.com/datasets](https://arxiv.org/html/2608.02628v1/paperswithcode.com/datasets)has curated licenses for some datasets\. Their licensing guide can help determine the license of a dataset\. - •For existing datasets that are re\-packaged, both the original license and the license of the derived asset \(if it has changed\) should be provided\. - •If this information is not available online, the authors are encouraged to reach out to the asset’s creators\.
61. 13\.New assets
62. Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
63. Answer:\[Yes\]
64. Justification: We will provide the full details upon acceptance of the paper\.
65. Guidelines: - •The answer\[N/A\]means that the paper does not release new assets\. - •Researchers should communicate the details of the dataset/code/model as part of their submissions via structured templates\. This includes details about training, license, limitations, etc\. - •The paper should discuss whether and how consent was obtained from people whose asset is used\. - •At submission time, remember to anonymize your assets \(if applicable\)\. You can either create an anonymized URL or include an anonymized zip file\.
66. 14\.Crowdsourcing and research with human subjects
67. Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation \(if any\)?
68. Answer:\[N/A\]
69. Justification: The paper does not involve crowdsourcing nor research with human subjects\.
70. Guidelines: - •The answer\[N/A\]means that the paper does not involve crowdsourcing nor research with human subjects\. - •Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper\. - •According to the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector\.
71. 15\.Institutional review board \(IRB\) approvals or equivalent for research with human subjects
72. Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board \(IRB\) approvals \(or an equivalent approval/review based on the requirements of your country or institution\) were obtained?
73. Answer:\[N/A\]
74. Justification: The paper does not involve crowdsourcing nor research with human subjects\.
75. Guidelines: - •The answer\[N/A\]means that the paper does not involve crowdsourcing nor research with human subjects\. - •Depending on the country in which research is conducted, IRB approval \(or equivalent\) may be required for any human subjects research\. If you obtained IRB approval, you should clearly state this in the paper\. - •We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution\. - •For initial submissions, do not include any information that would break anonymity \(if applicable\), such as the institution conducting the review\.
76. 16\.Declaration of LLM usage
77. Question: Does the paper describe the usage of LLMs if it is an important, original, or non\-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does*not*impact the core methodology, scientific rigor, or originality of the research, declaration is not required\.
78. Answer:\[N/A\]
79. Justification: The core method development in this research does not involve LLMs as any important, original, or non\-standard components\.
80. Guidelines: - •The answer\[N/A\]means that the core method development in this research does not involve LLMs as any important, original, or non\-standard components\. - •Please refer to our LLM policy in the NeurIPS handbook for what should or should not be described\.