LieDiscover finds complex mathematical symmetries in data automatically
LieDiscover: Adaptive Symbolic Library Construction for Explicit Open-form Symmetry Discovery
Symbolic ComputationMachine Learning
Summary
Finding hidden patterns called symmetries in data helps scientists understand natural laws. Existing methods struggle to identify complicated symmetries that involve advanced math formulas. The authors propose LieDiscover, a new approach that builds and improves a library of mathematical expressions step-by-step using a learning method. This lets LieDiscover find very complex hidden symmetries that other methods miss and helps solve equations better.
What this means in practice
- •For applied mathematicians: Improve modeling of physical systems by uncovering hidden symmetries that simplify solving complex differential equations.
- •For software engineers in scientific computing: Enhance automated equation discovery tools with advanced symbolic search to identify complex mathematical structures from data.
Authors
Xinxin Li, Jianming Ma, Xingyu Cui, Da Li, Juan Zhang, Junping Yin
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.