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Learning of discrete models of variational PDEs from data

Christian Offen, Sina Ober-Blöbaum

TL;DR

This work presents a framework to learn discrete field theories directly on space-time lattices by training neural networks to represent a discrete Lagrangian density $L_d$ whose discrete Euler–Lagrange equations reproduce observed field data. A data-consistency loss combined with numerically informed regularisers guides the model toward nondegenerate, stable DEL systems, enabling accurate forward propagation and robust extrapolation. The approach preserves variational structure and, via Palais' principle of symmetric criticality, naturally captures highly symmetric solutions such as travelling waves—even when they are absent from the training data—contrasting with traditional model-order reduction that projects to latent variables. Demonstrations on the wave and Schrödinger equations show effective data fitting, accurate travelling-wave identification, and competitive performance against MOR baselines, with potential impact on structure-preserving surrogates for PDEs and conservation-law discovery.

Abstract

We show how to learn discrete field theories from observational data of fields on a space-time lattice. For this, we train a neural network model of a discrete Lagrangian density such that the discrete Euler--Lagrange equations are consistent with the given training data. We, thus, obtain a structure-preserving machine learning architecture. Lagrangian densities are not uniquely defined by the solutions of a field theory. We introduce a technique to derive regularisers for the training process which optimise numerical regularity of the discrete field theory. Minimisation of the regularisers guarantees that close to the training data the discrete field theory behaves robust and efficient when used in numerical simulations. Further, we show how to identify structurally simple solutions of the underlying continuous field theory such as travelling waves. This is possible even when travelling waves are not present in the training data. This is compared to data-driven model order reduction based approaches, which struggle to identify suitable latent spaces containing structurally simple solutions when these are not present in the training data. Ideas are demonstrated on examples based on the wave equation and the Schrödinger equation.

Learning of discrete models of variational PDEs from data

TL;DR

This work presents a framework to learn discrete field theories directly on space-time lattices by training neural networks to represent a discrete Lagrangian density whose discrete Euler–Lagrange equations reproduce observed field data. A data-consistency loss combined with numerically informed regularisers guides the model toward nondegenerate, stable DEL systems, enabling accurate forward propagation and robust extrapolation. The approach preserves variational structure and, via Palais' principle of symmetric criticality, naturally captures highly symmetric solutions such as travelling waves—even when they are absent from the training data—contrasting with traditional model-order reduction that projects to latent variables. Demonstrations on the wave and Schrödinger equations show effective data fitting, accurate travelling-wave identification, and competitive performance against MOR baselines, with potential impact on structure-preserving surrogates for PDEs and conservation-law discovery.

Abstract

We show how to learn discrete field theories from observational data of fields on a space-time lattice. For this, we train a neural network model of a discrete Lagrangian density such that the discrete Euler--Lagrange equations are consistent with the given training data. We, thus, obtain a structure-preserving machine learning architecture. Lagrangian densities are not uniquely defined by the solutions of a field theory. We introduce a technique to derive regularisers for the training process which optimise numerical regularity of the discrete field theory. Minimisation of the regularisers guarantees that close to the training data the discrete field theory behaves robust and efficient when used in numerical simulations. Further, we show how to identify structurally simple solutions of the underlying continuous field theory such as travelling waves. This is possible even when travelling waves are not present in the training data. This is compared to data-driven model order reduction based approaches, which struggle to identify suitable latent spaces containing structurally simple solutions when these are not present in the training data. Ideas are demonstrated on examples based on the wave equation and the Schrödinger equation.
Paper Structure (30 sections, 9 theorems, 80 equations, 13 figures)

This paper contains 30 sections, 9 theorems, 80 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Continuous and discrete variational principles and field theories
  3. Continuous variational principles
  4. Discrete variational principles
  5. Set-Up
  6. Examples
  7. Forward propagation
  8. Preparation of regularisation strategy
  9. Loss function and regulariser for training discrete Lagrangian densities
  10. Data consistency
  11. Regulariser
  12. Variational principles for travelling waves in discrete and continuous theories
  13. Numerical Experiments
  14. Wave equation
  15. Training data
  16. ...and 15 more sections

Key Result

Proposition 1

Let $u^{i}_{j}$, $u^{i+1}_{j}$, $u^{i}_{j+1}$, $u^{i-1}_{j}$, $u^{i-1}_{j+1}$, $u^{i}_{j-1}$, $u^{i+1}_{j-1}$ such that eq:DEL3pt holds. Let $\mathcal{O}\subset \mathbb{R}^d$ be a convex, neighbourhood of $u^\ast=u^{i+1}_{j}$, $\| \cdot \|$ a norm of $\mathbb{R}^d$ inducing an operator norm on $\mat and let $f(u^{(n)})$ denote the left hand side of eq:DEL3pt with $u_j^{i+1}$ replaced by $u^{(n)}$.

Figures (13)

  • Figure 1: Left: Motions of a data-driven model (solid lines) for the mathematical pendulum match motions of an analytic model (dotted). Centre: Numerically computed motions of the data-driven model are very inaccurate. Right: Applying the same numerical integrator to the analytic model yield much more accurate results. (Figure taken from LagrangianShadowIntegrators. See LagrangianShadowIntegrators for details.)
  • Figure 2: Visualisation of the 7 and 9 point stencil of \ref{['ex:3ptLd']}. Colours indicate that points are part of the same summand in \ref{['eq:DEL3pt']} or \ref{['eq:DEL4pt']}. Black vertices are present in several summands.
  • Figure 3: Behaviour of summands in regularisers $\ell_{\mathrm{reg}}$ of \ref{['eq:regLd3pt']}, \ref{['eq:regLdxdt']}, \ref{['eq:regLdxdtTamed']}, and $\ell_{\mathrm{reg}}^{\mathrm{illustration}}$.
  • Figure 4: Illustration of degeneracy of roots when learning without (left) or with (right) regularising terms. The orange curve refers to the trained model, the grey curve to the untrained model of $f$. The regulariser makes sure that the roots are non-degenerate and numerical root finding is well-conditioned.
  • Figure 5: As comparison to \ref{['fig:RegulariserIllustration']} (left), a posterior distribution for a Gaussian Processes for the same observations is computed without regularisation. The problem of degenerate roots of the mean is less pronounced but does occur for sparse training data.
  • ...and 8 more figures

Theorems & Definitions (41)

  • Example 1: Linear motions
  • Example 2: Wave equation
  • Example 3: Degenerate Lagrangian
  • Example 4: Schrödinger equation
  • Example 5: Linear motions
  • Example 6: 3 and 4 point discrete Lagrangian
  • Example 7: Discrete 2d wave equation
  • Example 8: Discrete Schrödinger equation
  • Remark 1
  • Proposition 1
  • ...and 31 more