Nonlinear dynamics is ubiquitous in nature and commonly seen in various science and engineering disciplines. Distilling analytical expressions that govern nonlinear dynamics from limited data remains vital but challenging. To tackle this fundamental issue, we propose a novel Symbolic Physics Learner (SPL) machine to discover the mathematical structure of nonlinear dynamics. The key concept is to interpret mathematical operations and system state variables by computational rules and symbols, establish symbolic reasoning of mathematical formulas via expression trees, and employ a Monte Carlo tree search (MCTS) agent to explore optimal expression trees based on measurement data. The MCTS agent obtains an optimistic selection policy through the traversal of expression trees, featuring the one that maps to the arithmetic expression of underlying physics. Salient features of the proposed framework include search flexibility and enforcement of parsimony for discovered equations. The efficacy and superiority of the SPL machine are demonstrated by numerical examples, compared with state-of-the-art baselines.

翻译：非线性动力学在自然界中普遍存在，常见于各类科学与工程学科。从有限数据中提炼出控制非线性动力学的解析表达式至关重要但仍具挑战性。为解决这一根本性问题，我们提出了一种新颖的符号物理学习器（SPL）机制，用于发现非线性动力学的数学结构。其核心理念是将数学运算和系统状态变量解读为计算规则与符号，通过表达式树建立数学公式的符号推理，并采用蒙特卡洛树搜索（MCTS）智能体基于测量数据探索最优表达式树。MCTS智能体通过遍历表达式树获得乐观选择策略，最终选出映射至底层物理算术表达式的表达式树。该框架的显著特征包括搜索灵活性与强制方程简洁性。通过数值算例对比最新基线方法，验证了SPL机制的有效性与优越性。