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.
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
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
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.
Vikas Palakonda, Samira Ghorbanpour, Sangseok Yun et al.· Archives of Computational Me...· 0 citations
Symbolic Regression (SR) seeks to find succinct mathematical expressions that represent the fundamental relationships within data, providing interpretability and scientific understanding that exceeds that of black-box models. Nevertheless, traditional methods like Genetic Programming face challenges with scalability and are highly sensitive to noise, while sparse regression techniques such as SINDy rely significantly on predetermined feature libraries. In this work, we present a Neural Symbolic Regression (NSR) framework that treats neural networks as functional preconditioners for symbolic discovery. Our approach uses a decoupled pipeline: a neural network first learns a smooth, noise-robust approximation of the target function in an interaction- aware nonlinear feature space. LASSO is then applied to extract sparse, interpretable closed-form expressions. To improve predictive accuracy and symbolic fidelity by integrating distributed hyperparameter optimization with Ray Tune and ASHA scheduling. Experiments on the Nguyen benchmark suite show that our approach consistently outperforms SINDy and non-tuned neural baselines in RMSE, noise robustness, and out-of-distribution generalization. Ablation studies confirm the significance of feature interactions, neural depth, and tuning strategies. In general, this study presents a scalable and understandable neural-symbolic framework, creating a solid link between neural approximation and the discovery of sparse equations for scientific machine learning.
Symbolic regression (SR) seeks to discover parsimonious mathematical laws from observational data, yet conventional approaches often struggle with the vast combinatorial search space of physically meaningful expressions. We present InsightSR, a framework that embeds Large Language Models (LLMs) as a guiding layer around the PySR genetic programming engine. Rather than relying on LLMs to generate expressions directly, InsightSR uses LLMs to progressively transform the search space itself through two complementary pathways: a Semantic Seed Pathway that proposes dimensionally consistent functional skeletons, and a Structural Feature Pathway that recommends nonlinear feature transformations. These transformations accumulate over iterations, broadening the input space and shifting the symbolic search from constructing deep expression trees over raw variables to assembling shallow trees over a rich, semantically informed feature set. A post-generation feedback loop evaluates candidates, categorizes features by their empirical utility, and refines the guidance for the next iteration, transforming the discovery process from open-ended generation into iterative, self-correcting refinement. Across three benchmarks, InsightSR achieves a 95% exact recovery rate on the Feynman benchmark and 80.18% accuracy on the LLM-SRBench LSR-Transform task, substantially outperforming state-of-the-art genetic programming and neural-symbolic methods while maintaining strong out-of-distribution generalization on real-world datasets.
Artificial Neural networks (ANNs) are often treated as black-box models, making explainability a central challenge in deep learning. Many engineering methods have been proposed to approximately explain the ANN from various perspectives, such as feature attribution and visualization. However, it remains a long-standing open question whether the complex inference logic of an ANN can be explained exhaustively and concisely as sparse symbolic patterns. This raises a deeper inquiry: does the emergence of symbolic patterns reflect a natural law rather than chance? Here, we show that across a broad class of ANNs trained on diverse tasks, their inference logic can indeed be reformulated as sparse symbolic interactions. We further prove that two common mathematical criteria, which are implicitly required across tasks, lead to the emergence of such sparse symbolic interactions. Empirical evidence confirms that the two criteria hold for the majority of input samples in diverse models. Furthermore, the faithfulness of these interactions is also demonstrated by their strong sample-to-sample and model-to-model transferability, as well as their ability to explain the overall generalization power of ANNs. Our theoretical analysis and extensive experiments provide a solid foundation for symbolic explanations of ANNs, and offer novel insights into the ANN's generalization power. Our findings also highlight the potential of communicative learning, a paradigm in which the inference logic of an ANN can be directly inspected and tuned at the level of symbolic patterns, thus complementing traditional end-to-end learning paradigm. Finally, the observed emergence of symbolic patterns in ANNs suggests that similar symbolic representations may also emerge in other types of black-box systems under certain conditions, because our proof does not depend on any specific ANN architecture.
A weeklong summer workshop brought higher education faculty to campus to explore how AI and machine learning materials can be adapted for their classrooms.
