Robust moment identification for nonlinear PDEs via a neural ODE approach.

Area of Science:

• Computational Physics
• Applied Mathematics
• Machine Learning

Background:

• Partial Differential Equations (PDEs) govern complex physical systems.
• Traditional methods for learning dynamics from data often require dense, clean observations.
• Reduced-order modeling is crucial for efficient simulation and analysis of high-dimensional systems.

Purpose of the Study:

• To develop a data-driven framework for learning reduced-order moment dynamics from PDE-governed systems.
• To enable robust dynamic modeling from sparse, irregular, and noisy time-series data.
• To discover interpretable low-dimensional representations for complex systems.

Main Methods:

• Utilized neural Ordinary Differential Equations (neural ODEs) to directly model moment trajectories.
• Employed Stiefel manifold optimization for data-driven coordinate transformations in systems lacking analytical closure.
• Applied the framework to nonlinear Schrödinger and Fisher-Kolmogorov-Petrovskii-Piskounov reaction-diffusion systems.

Main Results:

• The neural ODE framework accurately recovers moment dynamics from limited, irregular, and noisy data.
• Successfully discovered closed moment dynamics in low-dimensional representations for systems without analytical closure.
• Demonstrated superior extrapolation accuracy compared to physics-based models in data-limited scenarios.

Conclusions:

• The proposed neural ODE framework offers a powerful and flexible approach for learning interpretable, low-dimensional moment dynamics.
• The method shows significant robustness in data-limited settings, outperforming traditional techniques.
• Enables reliable modeling and analysis of complex PDE-governed systems even with incomplete observations.