Aug 2026· Archives of Computational Methods in Engineering· 0 citations· 231 references
TL;DR
This survey comprehensively analyzes contemporary symbolic regression methodologies by systematically integrating four paradigmatic approaches: deterministic methods, metaheuristic algorithms, neural-symbolic frameworks, and emerging hybrid strategies, establishing a unified taxonomic framework that bridges evolutionary computation, mathematical programming, and deep learning paradigms.
Abstract
Symbolic regression autonomously discovers interpretable mathematical expressions from data by optimizing functional structure and associated parameters. Unlike traditional regression approaches that rely on predefined model forms, symbolic regression generates explicit models that maintain high predictive accuracy while providing valuable scientific insights. This survey comprehensively analyzes contemporary symbolic regression methodologies by systematically integrating four paradigmatic approaches: deterministic methods, metaheuristic algorithms, neural-symbolic frameworks, and emerging hybrid strategies.
We establish a unified taxonomic framework that bridges evolutionary computation, mathematical programming, and deep learning paradigms. Our analysis reveals convergence patterns toward physics-informed discovery, multi-objective optimization, and human-collaborative frameworks. WWe examine hybrid integration strategies, semantic-aware operators, and constraint-handling mechanisms, and we critically assess evaluation methodologies, benchmarking practices, applications across scientific and engineering domains, and fundamental limitations, including search complexity, overfitting, and high-dimensional scaling challenges.
This work makes four key contributions. First, we provide a unified framework analyzing over 300 methods across four paradigms. Second, we offer comprehensive coverage of developments from 2020 to 2025, including transformer-based and large language model approaches. Third, we present a systematic analysis of hybrid strategies and convergence patterns. Fourth, we provide actionable guidance for method selection. This survey establishes a comprehensive contemporary reference for symbolic regression research, highlighting pathways toward robust, interpretable, and scalable automated scientific discovery systems.
Multi-Objective Tool-augmented Symbolic Regression (MOT-SR), a unified framework that integrates external analytical tools to extract structural priors and guide equation generation, while jointly optimizing for accuracy, complexity, and generalization via a multi-objective evaluation module that maintains a dynamic Pareto front is proposed.
Boxiao Wang, Runxian Wang, Kai Li et al.· arXiv.org· 0 citations
Symbolic regression (SR) discovers closed-form mathematical expressions from data, offering interpretability beyond black-box models. Existing methods suffer from slow convergence in combinatorial search spaces and lack mechanisms to exploit compositional structure in the data. We introduce SMILE (Sine, Multiplication, Identity, Logarithm, Exponential), a hybrid framework that unifies continuous gradient-based optimization with discrete symbolic recovery through three stages: structural analysis of the data to identify the compositional hierarchy of the target expression, continuous optimization to learn parameters of a network that encodes the target expression using interpretable activations, and symbolic recovery through structured pruning, coefficient optimization, and rounding. This final stage distills the learned network into a compact expression with exact symbolic constants. We evaluate SMILE on SRBench across ground-truth and black-box datasets, with ablation studies validating each component. SMILE achieves the highest symbolic solution rate at the largest noise levels, demonstrating strong robustness where competing methods degrade substantially. It consistently lies on the Pareto front of accuracy versus complexity, recovering significantly simpler expressions in a fraction of the time required by the competing methods.
Mansooreh Montazerin, Antonio Ortega, Ajitesh Srivastava· 0 citations
Deep Divide-and-Reduce in Symbolic Regression (DDRSR) is proposed, a principled extension derived from a formal analysis of a broader class of decomposition structures that fundamentally broadens the applicability of expression decomposition and reduction and ensures both wider versatility and sound analytical grounding.
Symbolic regression (SR), the automated discovery of mathematical expressions from data, is a cornerstone of scientific inquiry. However, it is often hindered by the combinatorial explosion of the search space and a tendency to overfit. Popular methods, rooted in genetic programming, explore this space syntactically, often yielding overly complex, uninterpretable models. This paper introduces IdeaSearchFitter, a framework that employs Large Language Models (LLMs) as semantic operators within an evolutionary search. By generating candidate expressions guided by natural-language rationales, our method biases discovery towards models that are not only accurate but also conceptually coherent and interpretable. We demonstrate IdeaSearchFitter's efficacy across diverse challenges: it achieves competitive, noise-robust performance on the Feynman Symbolic Regression Database (FSReD), outperforming several strong baselines; discovers mechanistically aligned models with good accuracy-complexity trade-offs on real-world data; and derives compact, physically-motivated parametrizations for Parton Distribution Functions in a frontier high-energy physics application. IdeaSearchFitter is a specialized module within our broader iterated agent framework, IdeaSearch, which is publicly available at \href{https://www.ideasearch.cn/}{https://www.ideasearch.cn/}.
Zhuo-Yang Song, Ze-Yu Cai, Shu-Tao Zhang et al.· Communications in Theoretica...· 0 citations
We investigate whether symbolic regression can discover explicit neural network weight-update rules that outperform standard hand-designed optimizers on small symbolic regression benchmarks. Candidate update rules are represented as fixed-depth symbolic expressions over operands derived from common optimizers, including gradient, momentum, adaptive-gradient, and moment-estimate quantities. Across 30 benchmark/neural network combinations, the symbolic regression procedure found an update rule outperforming the best hyperparameter-tuned established optimizer in 25 cases, with an aggregate MSE reduction of 44.47\% over the improved cases. The discovered rules do not all share a single common symbolic form, but many combine adaptive normalization, momentum-like quantities, nonlinear transformations, and rational expressions. These results suggest that symbolic regression can serve as a lightweight mechanism for discovering compact optimizer variants, while also highlighting the need for larger-scale validation.
Charles D. Brum, Edward Finkelstein· arXiv.org· 0 citations
We introduce Automatic Symbolic Regression (AutoSR), a fully automated system that instantiates Research-Space Symbolic Regression by searching persistent scientific investigations rather than isolated equations. Finite, noisy data often yield numerically competitive expressions that imply very different behavior outside the observed regime, making numerical fit and syntactic complexity insufficient measures of scientific credibility. Existing approaches largely focus on improving expressions, yet the search typically retains little beyond the resulting formula and score, losing the scientific record, such as motivations and probes, that inform what to try next. AutoSR preserves this record in a \textbf{Research State}, coupling each candidate equation with the reasoning, computational evidence, and independent review developed along its branch. Proposer--reviewer agents develop these states under progressive-widening Monte Carlo tree search (PW-MCTS), which allocates computation across competing investigations, while the accumulated research record is ultimately synthesized into a final report that explains the leading relation and the basis for its selection. Across nine selected challenges from two benchmark suites, AutoSR recovers algebraically equivalent relations in every case, including three cp3-bench problems that no published system recovers and six structurally diverse LSR-Transform problems. Overall, AutoSR extends symbolic regression from equation-level search toward automated scientific investigation, allowing scientific knowledge and accumulated evidence to shape both what is explored and how the resulting equation is justified.
Kejia Zhang, Youran Sun, Xinyu Ren et al.· 0 citations
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