Discovering symmetries from observational data provides interpretable insights into dynamical systems and priors for scientific machine learning. However, existing symmetry discovery methods face two limitations. Many methods represent symmetries implicitly, making it difficult to obtain explicit infinitesimal generators that can serve as physical priors for learning dynamical systems. Meanwhile, methods that explicitly identify generators often rely on predefined function libraries, limiting their ability to capture complex symmetries involving high-order polynomials or transcendental functions. To address these limitations, we propose LieDiscover, a framework that explicitly identifies infinitesimal generators through an iterative symbolic search process that dynamically expands the symbolic function library to capture complex functional forms. Specifically, at each search step, we design an encoder-decoder architecture to generate a candidate symbolic expression. An adaptive singular value decomposition (SVD)-based evaluation is introduced to assess how each candidate contributes to the recoverable symmetry subspace. The resulting evaluations are incorporated into step-wise rewards to guide reinforcement learning toward discovering richer symmetry structures with concise symbolic representations. Experiments on eight partial differential equation (PDE) benchmarks demonstrate that LieDiscover recovers explicit generators involving higher-order polynomial, exponential, and trigonometric terms, while achieving more accurate symmetry subspace recovery and improved robustness to noise. Furthermore, the recovered generators enable effective symmetry-based data augmentation, improving the accuracy of downstream neural PDE solvers.

翻译：暂无翻译