Conservation laws are key theoretical and practical tools for understanding, characterizing, and modeling nonlinear dynamical systems. However, for many complex systems, the corresponding conserved quantities are difficult to identify, making it hard to analyze their dynamics and build stable predictive models. Current approaches for discovering conservation laws often depend on detailed dynamical information or rely on black box parametric deep learning methods. We instead reformulate this task as a manifold learning problem and propose a non-parametric approach for discovering conserved quantities. We test this new approach on a variety of physical systems and demonstrate that our method is able to both identify the number of conserved quantities and extract their values. Using tools from optimal transport theory and manifold learning, our proposed method provides a direct geometric approach to identifying conservation laws that is both robust and interpretable without requiring an explicit model of the system nor accurate time information.

翻译：守恒律是理解、表征与建模非线性动力系统的关键理论与实用工具。然而，对于许多复杂系统而言，相应的守恒量难以辨识，这导致其动力学分析困难且难以构建稳定的预测模型。现有守恒律发现方法通常依赖详细的动力学信息或采用黑箱式参数化深度学习方法。我们转而将该任务重构为流形学习问题，并提出一种非参数化的守恒量发现方法。我们在多种物理系统上测试了这一新方法，证明其既能识别守恒量的数量，也能提取其数值。借助最优传输理论与流形学习工具，所提方法提供了一种直接且可解释的几何化途径用于辨识守恒律，既无需显式系统模型，也无需精确时间信息，同时兼具鲁棒性。