Learning of discrete models of variational PDEs from data

Area of Science:

• Computational Physics
• Machine Learning
• Theoretical Physics

Background:

• Learning discrete field theories from observational data is crucial for understanding complex systems.
• Traditional methods may struggle to identify underlying structures or simple solutions within data.

Purpose of the Study:

• To develop a novel machine learning framework for learning discrete field theories from spatiotemporal data.
• To ensure numerical regularity and robustness of learned theories for efficient simulations.
• To enable the identification of structurally simple solutions, like traveling waves, even if absent in training data.

Main Methods:

• Training a neural network to model a discrete Lagrangian density.
• Ensuring consistency with discrete Euler-Lagrange equations using observational data.
• Introducing regularizers to optimize numerical regularity and ensure robust simulations.
• Demonstrating the approach on wave and Schrödinger equations.

Main Results:

• A structure-preserving machine learning architecture for discrete field theories.
• Techniques for deriving regularizers that enhance numerical stability and efficiency.
• Successful identification of simple solutions (e.g., traveling waves) not present in training data.
• Comparison with data-driven model order reduction, highlighting advantages in solution identification.

Conclusions:

• The proposed method effectively learns discrete field theories from data, yielding robust and efficient models.
• The technique facilitates the discovery of fundamental solutions, offering insights beyond the training dataset.
• This approach advances data-driven scientific discovery in physics and related fields.