Learned Lagrangian Models of PDEs via Euler-Lagrange Residual Minimization

TL;DR

This paper introduces a novel method for forecasting PDE system dynamics using learned Lagrangian models, leveraging an optimization-based integrator to ensure stability and conservation.

Contribution

It develops a mesh-free, optimization-based integrator that minimizes Euler-Lagrange residuals, enabling stable long-range predictions for learned PDE models without structural constraints.

Findings

Achieves error comparable to classical symplectic methods.

Generalizes to spatially varying dynamics and boundary conditions.

Scales linearly with domain size via Jacobi iteration.

Abstract

We present the first method to directly use a learned continuous Lagrangian to forecast the dynamics of systems governed by partial differential equations, exploiting the inherent conservative structure to achieve stable long-range predictions. We develop an optimization-based integrator that minimizes the squared Euler--Lagrange residual via a mesh-free near-symplectic construction on local space-time patches. Different from integrators for analytical models, integrators for learned models should decouple model error (phase error) from integration error (conservation error). By relying on optimization rather than time-stepping, we bypass the global coupling inherent to fixed discretizations, which slows time- and space-stepping and complicates learning. Our method scales linearly with domain size via Jacobi iteration, and places no structural requirements on the learned network, allowing it to be coupled with existing physics-guided machine learning (ML) methods. We validate our approach on a learned representation of a double pendulum, a one-dimensional wave equation, and a two-dimensional wave equation. Our method achieves error comparable to classical symplectic methods while generalizing to spatially varying dynamics and arbitrary boundary conditions without retraining.