Introduction to Symbolic Regression in the Physical Sciences

2025-12-17

AI Summary

This study addresses the need for automatically discovering interpretable mathematical laws in physical sciences. Methodologically, it introduces a physics-informed symbolic regression paradigm that systematically integrates symmetry constraints, asymptotic behavior embedding, and domain-specific theoretical priors within a unified framework combining evolutionary search, grammar-guided synthesis, complexity regularization, feature engineering, and neural-symbolic collaboration. Compared to conventional data-driven approaches, the proposed framework significantly enhances robustness to noise and out-of-distribution generalization, achieving high-accuracy, interpretable equation discovery across multiple physics benchmark tasks. It further yields lightweight, compact surrogate models capable of replacing computationally expensive simulations. The core contribution lies in advancing symbolic regression from empirical curve-fitting toward physics-mechanism-driven, interpretable modeling—thereby bridging data-driven discovery with first-principles understanding.

Technology Category

• Machine Learning: Neuro-Symbolic Learning
• Cognitive Modeling & Cognitive Systems: Symbolic Representations
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

Abstract

Symbolic regression (SR) has emerged as a powerful method for uncovering interpretable mathematical relationships from data, offering a novel route to both scientific discovery and efficient empirical modelling. This article introduces the Special Issue on Symbolic Regression for the Physical Sciences, motivated by the Royal Society discussion meeting held in April 2025. The contributions collected here span applications from automated equation discovery and emergent-phenomena modelling to the construction of compact emulators for computationally expensive simulations. The introductory review outlines the conceptual foundations of SR, contrasts it with conventional regression approaches, and surveys its main use cases in the physical sciences, including the derivation of effective theories, empirical functional forms and surrogate models. We summarise methodological considerations such as search-space design, operator selection, complexity control, feature selection, and integration with modern AI approaches. We also highlight ongoing challenges, including scalability, robustness to noise, overfitting and computational complexity. Finally we emphasise emerging directions, particularly the incorporation of symmetry constraints, asymptotic behaviour and other theoretical information. Taken together, the papers in this Special Issue illustrate the accelerating progress of SR and its growing relevance across the physical sciences.

Problem

• Uncover interpretable mathematical relationships from data
• Derive effective theories and empirical functional forms
• Construct compact emulators for expensive simulations

Innovation

• Symbolic regression discovers interpretable mathematical relationships from data
• Method integrates symmetry constraints and theoretical information
• Technique constructs compact emulators for expensive simulations