Numerical Spectrum Linking: Identification of Governing PDE via Koopman-Chebyshev Approximation

TL;DR

This work addresses PDE identification from data by connecting Chebyshev spectral representations with Koopman operator theory, forming a numerical spectrum linking framework that yields finite-dimensional Koopman matrices from a Chebyshev basis. It derives a physics-informed Koopman matrix $\\mathbf{K}^\\star$ from the governing PDE via Chebyshev spectral discretization and obtains a data-driven matrix $\\hat{\\mathbf{K}}$ from observations; spectral comparison of eigenvalues and eigenvectors then validates the identified PDE and its spatial structure. The approach demonstrates reliable PDE identification on benchmark advection–diffusion-type equations at modest resolution, and it provides a pathway toward interpretable operator learning and data-driven symbolic regression to recover explicit PDEs. The framework promises practical impact for discovering governing laws from limited data and for extending PDE discovery to higher dimensions and more complex dynamics.

Abstract

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.

Figures (3)

• Figure 1: Establishing a foundation for data-driven symbolic regression of PDEs governing dynamics.
• Figure 2: Outline of Chebyshev polynomials and the Koopman operator
• Figure 3: Outline of the proposed numerical spectrum linking method