A numerical framework is proposed for identifying partial differential equations (PDEs) governing dynamical systems directly from their observation data using Chebyshev polynomial approximation. In contrast to data-driven approaches such as dynamic mode decomposition (DMD), which approximate the Koopman operator without a clear connection to differential operators, the proposed method constructs finite-dimensional Koopman matrices by projecting the dynamics onto a Chebyshev basis, thereby capturing both differential and nonlinear terms. This establishes a numerical link between the Koopman and differential operators. Numerical experiments on benchmark dynamical systems confirm the accuracy and efficiency of the approach, underscoring its potential for interpretable operator learning. The framework also lays a foundation for future integration with symbolic regression, enabling the construction of explicit mathematical models directly from data.

翻译：本文提出了一种数值框架，利用切比雪夫多项式近似直接从观测数据中识别控制动力系统的偏微分方程（PDEs）。与动态模态分解（DMD）等数据驱动方法不同（这些方法近似Koopman算子但未明确关联微分算子），本方法通过将动力学投影到切比雪夫基上构建有限维Koopman矩阵，从而同时捕捉微分项和非线性项。这建立了Koopman算子与微分算子之间的数值关联。在基准动力系统上的数值实验验证了该方法的准确性与效率，凸显了其在可解释算子学习方面的潜力。该框架还为未来与符号回归的集成奠定了基础，使得能够直接从数据构建显式数学模型。