Machine learning of discrete field theories with guaranteed convergence and uncertainty quantification
Christian Offen
TL;DR
We address learning discrete variational structures for field theories from data by embedding them in a Gaussian process regression framework that learns a discrete Lagrangian density $L_d$ from stencil observations. The method enforces the discrete Euler–Lagrange equations $DEL(L_d)(\mathfrak{u})=0$ on training data and uses normalization to resolve gauge ambiguity, yielding a posterior mean $L_d^M$ with associated uncertainty $\mathcal{K}_{\Phi_b^M}$. A convergence theorem shows that as data become dense, the posterior mean converges to a true Lagrangian density $L_{d,(\infty)}$ consistent with the dynamics, and uncertainty quantification extends to any linear observable. Numerical experiments on the discrete wave equation and discrete Schrödinger equation demonstrate accurate extrapolation to unseen data and transparent model uncertainty, highlighting the approach's structure-preserving, meshless character and potential for discovering variational laws in PDEs.
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 Schrödinger 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.