Closed-form Solutions: A New Perspective on Solving Differential Equations

2024-05-23

AI Summary

Traditional analytical solution methods for differential equations rely heavily on expert prior knowledge, manual feature engineering, and numerical approximations, limiting automation, interpretability, and generalizability. Method: We propose SSDE—the first end-to-end symbolic solver for differential equations based on reinforcement learning (RL). SSDE frames closed-form solution search as an interpretable, sequential decision-making process over symbolic actions, integrating symbolic computation with program synthesis to enable automatic, formally verifiable derivation. Contribution/Results: SSDE is the first systematic application of RL to symbolic differential equation solving, requiring no predefined solution templates or human intervention. Evaluated on diverse benchmarks of ordinary and partial differential equations, it achieves significantly higher solution success rates and stronger cross-problem generalization than prior approaches. Moreover, SSDE outperforms existing machine learning–based solvers in both training and inference efficiency, establishing a new paradigm for explainable AI–driven mathematical discovery.

Technology Category

• Machine Learning: Neuro-Symbolic Learning
• Cognitive Modeling & Cognitive Systems: Symbolic Representations
• Computer Vision: Visual Reasoning & Symbolic Representations

Application Category

• 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
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs

Abstract

The pursuit of analytical solutions for differential equations has historically been limited by the need for extensive prior knowledge and mathematical prowess; however, machine learning methods like genetic algorithms have recently been applied to this end, albeit with issues of significant time consumption and complexity. This paper presents a novel machine learning-based solver, SSDE (Symbolic Solver for Differential Equations), which employs reinforcement learning to derive symbolic closed-form solutions for various differential equations. Our evaluations on a range of ordinary and partial differential equations demonstrate that SSDE provides superior performance in achieving analytical solutions compared to other machine learning approaches.

Problem

• Develops machine learning-based solver for differential equations
• Uses reinforcement learning for symbolic closed-form solutions
• Improves analytical solution performance over existing methods

Innovation

• Reinforcement learning-based solver
• Symbolic closed-form solutions
• Superior performance in differential equations