Physics-informed learning of governing equations from scarce data.

Area of Science:

• Physics
• Applied Mathematics
• Data Science

Background:

• Discovering governing equations is crucial for modeling complex physical systems.
• Existing methods often require large, accurate datasets, which are not always available.
• Nonlinear spatiotemporal systems present significant challenges for equation discovery.

Purpose of the Study:

• To introduce a novel approach for discovering governing partial differential equations (PDEs) from scarce and noisy data.
• To develop a computational framework that integrates deep neural networks and sparse regression for equation discovery.
• To demonstrate the method's efficacy and robustness in identifying PDEs for nonlinear spatiotemporal systems.

Main Methods:

• Utilizing physics-informed neural networks (PINNs) for representation learning and physics embedding.
• Employing automatic differentiation to compute system variable derivatives.
• Integrating sparse regression to identify key terms and parameters for PDE structure.
• Validating the approach numerically and experimentally on diverse PDE systems.

Main Results:

• Successfully discovered governing PDEs from scarce and noisy datasets.
• Demonstrated robustness across various initial and boundary conditions.
• Showcased the ability to handle nonlinear spatiotemporal dynamics.
• Validated the method's effectiveness in data-limited scenarios.

Conclusions:

• The proposed physics-informed neural network with sparse regression offers a powerful tool for closed-form model discovery.
• This approach is particularly valuable for applications where large, precise datasets are difficult to obtain.
• The framework advances the ability to uncover underlying physical laws from experimental or simulation data.