Learned Lagrangian Models of PDEs via Euler-Lagrange Residual Minimization

2026-05-07

AI Summary

This work proposes a mesh-free, near-symplectic integrator based on a learned continuous Lagrangian for long-term stable prediction of dynamical systems governed by partial differential equations (PDEs). By minimizing the Euler–Lagrange residual over local space-time blocks, the method constructs an optimization-based integrator that decouples model error from integration error, circumventing the global coupling inherent in conventional time-stepping schemes. It naturally accommodates arbitrary boundary conditions without retraining. To the best of our knowledge, this is the first approach to directly employ a learned Lagrangian for PDE evolution. The method achieves accuracy comparable to classical symplectic integrators in benchmark problems such as the double pendulum and one- and two-dimensional wave equations, and successfully generalizes to scenarios with spatially varying dynamics and complex boundary conditions.

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.

Problem

• Learned Lagrangian
• Partial Differential Equations
• Euler-Lagrange Residual
• Long-range Prediction
• Model Error

Innovation

• learned Lagrangian
• Euler-Lagrange residual minimization
• near-symplectic integrator
• mesh-free optimization
• physics-guided machine learning