Machine learning of discrete field theories with guaranteed convergence and uncertainty quantification

TL;DR

This paper presents a Gaussian process regression method to identify discrete variational principles from observed field solutions, ensuring convergence and enabling uncertainty quantification, extending previous work from ODEs to PDEs.

Contribution

It introduces a geometric machine learning approach for data-driven discovery of discrete Lagrangians with proven convergence and uncertainty quantification, applicable to PDEs.

Findings

Successfully applied to discrete wave and Schrödinger equations.

Provides rigorous convergence guarantees for the method.

Enables quantification of model uncertainty in discrete field theories.

Abstract

We introduce a method based on Gaussian process regression to identify discrete variational principles from observed solutions of a field theory. The method is based on the data-based identification of a discrete Lagrangian density. It is a geometric machine learning technique in the sense that the variational structure of the true field theory is reflected in the data-driven model by design. We provide a rigorous convergence statement of the method. The proof circumvents challenges posed by the ambiguity of discrete Lagrangian densities in the inverse problem of variational calculus. Moreover, our method can be used to quantify model uncertainty in the equations of motions and any linear observable of the discrete field theory. This is illustrated on the example of the discrete wave equation and Schr\"odinger equation. The article constitutes an extension of our previous article arXiv:2404.19626 for the data-driven identification of (discrete) Lagrangians for variational dynamics from an ode setting to the setting of discrete pdes.