Learning of discrete models of variational PDEs from data.

The article focuses on a machine learning framework for discovering discrete field theories from observational data on space–time lattices by training neural network models of discrete Lagrangian densities. These models enforce consistency with discrete Euler–Lagrange equations and incorporate numerically informed regularization strategies to ensure well-conditioned and computationally efficient solutions. The approach preserves locality and variational structure, enabling the identification and preservation of highly symmetric solutions such as traveling waves—even when such solutions are absent from the training data—by leveraging Palais' principle of symmetric criticality. Numerical experiments on the discrete wave and Schrödinger equations demonstrate that this stencil-based learning method outperforms data-driven reduced order models (ROMs) in generalization and in capturing traveling waves. The framework highlights advantages in preserving geometric and physical structures directly on the lattice, with future work aimed at extending to high-dimensional problems and more complex dynamical features.