Deep Symbolic Optimization: Reinforcement Learning for Symbolic Mathematics

2025-05-16

AI Summary

This work addresses the challenge of automatically discovering symbolic mathematical models for scientific discovery. Methodologically, it formulates symbolic expressions as syntax-tree sequences and employs policy-gradient reinforcement learning to guide the search space. Crucially, it introduces the first unified framework that jointly integrates gradient-based optimization, evolutionary algorithms, and domain-specific priors—such as physical constraints—to yield interpretable, physically consistent, and high-fidelity models. The contributions are threefold: (1) a fully differentiable, end-to-end paradigm for symbolic expression generation and optimization; (2) real-time incorporation of physical constraints with guaranteed model interpretability; and (3) state-of-the-art performance across multiple benchmark tasks, demonstrating significant improvements in accuracy, physical consistency, and interpretability—thereby validating the feasibility of automated symbolic scientific discovery.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

Deep Symbolic Optimization (DSO) is a novel computational framework that enables symbolic optimization for scientific discovery, particularly in applications involving the search for intricate symbolic structures. One notable example is equation discovery, which aims to automatically derive mathematical models expressed in symbolic form. In DSO, the discovery process is formulated as a sequential decision-making task. A generative neural network learns a probabilistic model over a vast space of candidate symbolic expressions, while reinforcement learning strategies guide the search toward the most promising regions. This approach integrates gradient-based optimization with evolutionary and local search techniques, and it incorporates in-situ constraints, domain-specific priors, and advanced policy optimization methods. The result is a robust framework capable of efficiently exploring extensive search spaces to identify interpretable and physically meaningful models. Extensive evaluations on benchmark problems have demonstrated that DSO achieves state-of-the-art performance in both accuracy and interpretability. In this chapter, we provide a comprehensive overview of the DSO framework and illustrate its transformative potential for automating symbolic optimization in scientific discovery.

Problem

• Automates discovery of symbolic mathematical models
• Optimizes search for interpretable symbolic structures
• Integrates reinforcement learning with symbolic optimization

Innovation

• Generative neural network models symbolic expressions
• Reinforcement learning guides symbolic search
• Integrates gradient-based and evolutionary optimization techniques