Overview
A numerical framework is introduced for the identification of governing partial differential equations (PDEs) based on observational data. This framework establishes a link between observation-driven and equation-driven Koopman operators within a common Chebyshev spectral domain. The approach constructs finite-dimensional Koopman operators using Chebyshev spectral representations. This enables direct comparison between dynamically derived data and candidate governing PDEs.
Research Context
Traditional data-driven methods, such as Dynamic Mode Decomposition (DMD), approximate Koopman operators without explicitly connecting them to differential operators. The proposed framework, in contrast, aims to bridge this gap by facilitating a direct comparison. It integrates practical observations with Chebyshev-based Koopman analysis through a unified observation model and a least-squares coefficient recovery formulation. This methodology is designed to recover Chebyshev spectral coefficients from observations, irrespective of the underlying sampling grid.
Approach
The core of the approach involves constructing finite-dimensional Koopman operators. These operators are built using Chebyshev spectral representations, which allow for the transformation of observed data into a spectral domain. Within this domain, a direct comparison is possible between the dynamics derived from the data and the dynamics predicted by candidate governing PDEs. The framework incorporates a unified observation model designed to manage observations obtained from arbitrary sampling grids. To facilitate the recovery of Chebyshev spectral coefficients from these observations, a least-squares coefficient recovery formulation is employed. This formulation ensures numerical consistency between diverse practical observations and the subsequent Chebyshev-based Koopman analysis.
Findings
Numerical experiments were conducted using various sampling configurations: direct Chebyshev, uniform, and irregular sampling. Under these conditions, the proposed framework consistently identified the governing PDE from the observational data. An observation-density study provided insights into the conditions required for reliable PDE identification, indicating that consistent identification is achieved when a sufficient number of independent observations are available to support stable coefficient recovery. This suggests a practical guideline for the application of the framework.
Why This Matters
The ability to accurately identify governing PDEs from observational data has implications for understanding underlying physical or dynamic systems. The framework's capacity to handle observations from arbitrary sampling grids enhances its practical applicability across diverse data acquisition scenarios. The established guideline regarding observation density provides practical direction for researchers and practitioners in experimental design and data collection for PDE identification tasks.