LieDiscover: Adaptive Symbolic Library Construction for Explicit Open-form Symmetry Discovery

2026-09-27

AI Summary

This study addresses the limitations of existing methods in determining the number of unknown generators and capturing complex symmetries involving high-order polynomials or transcendental functions. To this end, symmetry discovery is formulated as a joint optimization problem over function libraries and coefficients. Methodologically, an encoder-decoder architecture is employed to dynamically generate symbolic expressions for library expansion, while reinforcement learning is innovatively introduced to accelerate exploration within the symbolic search space. A stepwise reward mechanism enables adaptive construction of the symbolic library, thereby overcoming the constraints of traditional predefined bases. Experimental results demonstrate that the proposed approach successfully uncovers open-form infinitesimal generators containing high-order polynomials and transcendental functions, significantly enhancing performance in downstream partial differential equation solving and discovery tasks.

Abstract

Discovering underlying symmetries from data has emerged as a crucial challenge in scientific discovery. Existing data-driven methods for symmetry discovery fail to determine the exact number and mathematical form of unknown infinitesimal generators. Recent explicit methods represent generators using a predefined function library and identify them through algebraic optimization, but they often struggle to capture complex symmetries involving high-order polynomials or transcendental functions. To address this limitation, we formulate symmetry discovery as a joint optimization problem over the function library and coefficients. We propose a novel framework that leverages an encoder-decoder architecture to dynamically generate symbolic expressions and expand the library. This generation process is optimized via reinforcement learning, which accelerates the exploration of the symbolic search space through step-wise rewards. Experiments demonstrate that LieDiscover can successfully uncover open-form infinitesimal generators involving high-order polynomials or transcendental functions, which remain intractable for existing methods. The discovered symmetries also improve performance in downstream PDE solving and discovery tasks.

Problem

• symmetry discovery
• infinitesimal generators
• symbolic library
• high-order polynomials
• transcendental functions

Innovation

• Symmetry Discovery
• Symbolic Library Construction
• Reinforcement Learning
• Encoder-Decoder Architecture
• Infinitesimal Generators