Machine learning of continuous and discrete variational ODEs with convergence guarantee and uncertainty quantification

TL;DR

This paper presents a Gaussian process-based method for learning dynamical systems governed by Euler--Lagrange equations, offering convergence guarantees, structure preservation, and uncertainty quantification for system identification.

Contribution

It introduces a novel structure-preserving learning approach for continuous and discrete Lagrangians with proven convergence and uncertainty quantification capabilities.

Findings

Convergence of the method is rigorously proven as data points become dense.

The approach provides efficient uncertainty quantification for energy and symplectic structures.

The method overcomes ill-posedness through geometric regularisation and convex minimisation techniques.

Abstract

The article introduces a method to learn dynamical systems that are governed by Euler--Lagrange equations from data. The method is based on Gaussian process regression and identifies continuous or discrete Lagrangians and is, therefore, structure preserving by design. A rigorous proof of convergence as the distance between observation data points converges to zero and lower bounds for convergence rates are provided. Next to convergence guarantees, the method allows for quantification of model uncertainty, which can provide a basis of adaptive sampling techniques. We provide efficient uncertainty quantification of any observable that is linear in the Lagrangian, including of Hamiltonian functions (energy) and symplectic structures, which is of interest in the context of system identification. The article overcomes major practical and theoretical difficulties related to the ill-posedness of the identification task of (discrete) Lagrangians through a careful design of geometric regularisation strategies and through an exploit of a relation to convex minimisation problems in reproducing kernel Hilbert spaces.