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Learned Lagrangian Models of PDEs via Euler–Lagrange Residual Minimization

NeurIPS 2026 · Probabilistic and scientific modeling › Differential equations and PDEs · approach: Neural PDE solvers · 0 citations (OpenAlex)

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Proposes Euler–Lagrange Minimization (ELM), a mesh-free, near-symplectic optimization-based integrator that forward-simulates learned continuous Lagrangian densities for ODEs and 1D/2D wave PDEs.

Problem

Forward simulation of learned Lagrangian models of PDEs drifts in energy because standard integrators and learned operator/discrete-Lagrangian methods do not preserve conservative structure, and fixed discretizations couple all nodes into global solves.

Approach

Minimize the squared Euler–Lagrange residual of a learned continuous Lagrangian over Hermite-interpolated local space-time patches with damped Newton iteration swept by Jacobi, using oversampled Gauss–Legendre quadrature so the integrator is near-symplectic and boundary conditions enter as constraints on optimized variables.

Key results

• On the double pendulum over 100,000 s, ELM reaches 0.045% relative energy error versus 0.065% for GLRK and 77% for RK4.
• On the 1D wave, ELM with 21 nodes gives 21% energy error over 10,000 s versus 680% for LNN at dt=0.4; ELM with 51 nodes gives 1.9%.
• On the 2D wave over 1,000 s, ELM uses 3,521 parameters with 19% energy error, versus PDE-Refiner (~146M params) at 271% and FNO diverging by ~22 s.

Surprising findings

• ELM is the first to use a learned continuous Lagrangian density for forward simulation of a 2D PDE, reproducing double-slit interference from a model trained only on homogeneous domains.
• ELM with 3,521 parameters outperforms PDE-Refiner with about 146M parameters on the 2D wave, though it is roughly 37x slower per step (968 ms vs 26 ms).

Limitations

• ELM is more expensive per time step than standard integrators, especially for ODEs.
• Jacobi sweep converges slowly for strongly coupled systems and higher-dimensional PDEs with more patch overlap.
• Only Lagrangian densities with up to first derivatives are supported, excluding e.g. beam mechanics; dissipative systems not covered.

Experimental setup

Datasets: analytical double pendulum Lagrangian samples plane-wave superposition samples for 1D and 2D wave eigenmode-decomposed 2D wave trajectories (baselines)

Benchmarks: double pendulum energy drift 1D wave equation energy drift and L2 error 2D wave equation center amplitude, energy error, L2 error

Metrics: relative energy error relative L2 error ms/step parameter count

Baselines: LNN wave model RK4 Dormand-Prince Implicit Midpoint GLRK Fourier Neural Operator PDE-Refiner

Builds on: Lagrangian Neural Networks Gauss-Legendre Runge-Kutta classical variational integrators DeLaN

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