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Compatible finite element interpolated neural networks

Santiago Badia, Wei Li, Alberto F. Martín

TL;DR

This work extends the FEINN framework to PDEs with weak solutions in $H(\textbf{curl})$ and $H(\textbf{div})$ by integrating curl/div-conforming discrete de Rham spaces into neural interpolations, and it introduces TraceFEINN for surface Darcy problems. The authors demonstrate that interpolated neural networks on compatible finite element spaces, combined with residual minimisation in a discrete dual norm, yield stable, structure-preserving discretisations for forward and inverse problems, including Maxwell-type and surface Darcy models. Across extensive numerical experiments, compatible FEINNs outperform standard FEM for smooth solutions, accurately resolve surface PDEs on spheres, and compete with or surpass adjoint neural methods for inverse problems, especially when aided by adaptivity and preconditioning. The framework provides a scalable, mathematically grounded approach for combining neural approximations with curl/div-conforming finite element spaces, with broad implications for electromagnetism, geophysics, and surface PDEs.

Abstract

We extend the finite element interpolated neural network (FEINN) framework from partial differential equations (PDEs) with weak solutions in $H^1$ to PDEs with weak solutions in $H(\textbf{curl})$ or $H(\textbf{div})$. To this end, we consider interpolation trial spaces that satisfy the de Rham Hilbert subcomplex, providing stable and structure-preserving neural network discretisations for a wide variety of PDEs. This approach, coined compatible FEINNs, has been used to accurately approximate the $H(\textbf{curl})$ inner product. We numerically observe that the trained network outperforms finite element solutions by several orders of magnitude for smooth analytical solutions. Furthermore, to showcase the versatility of the method, we demonstrate that compatible FEINNs achieve high accuracy in solving surface PDEs such as the Darcy equation on a sphere. Additionally, the framework can integrate adaptive mesh refinements to effectively solve problems with localised features. We use an adaptive training strategy to train the network on a sequence of progressively adapted meshes. Finally, we compare compatible FEINNs with the adjoint neural network method for solving inverse problems. We consider a one-loop algorithm that trains the neural networks for unknowns and missing parameters using a loss function that includes PDE residual and data misfit terms. The algorithm is applied to identify space-varying physical parameters for the $H(\textbf{curl})$ model problem from partial or noisy observations. We find that compatible FEINNs achieve accuracy and robustness comparable to, if not exceeding, the adjoint method in these scenarios.

Compatible finite element interpolated neural networks

TL;DR

This work extends the FEINN framework to PDEs with weak solutions in and by integrating curl/div-conforming discrete de Rham spaces into neural interpolations, and it introduces TraceFEINN for surface Darcy problems. The authors demonstrate that interpolated neural networks on compatible finite element spaces, combined with residual minimisation in a discrete dual norm, yield stable, structure-preserving discretisations for forward and inverse problems, including Maxwell-type and surface Darcy models. Across extensive numerical experiments, compatible FEINNs outperform standard FEM for smooth solutions, accurately resolve surface PDEs on spheres, and compete with or surpass adjoint neural methods for inverse problems, especially when aided by adaptivity and preconditioning. The framework provides a scalable, mathematically grounded approach for combining neural approximations with curl/div-conforming finite element spaces, with broad implications for electromagnetism, geophysics, and surface PDEs.

Abstract

We extend the finite element interpolated neural network (FEINN) framework from partial differential equations (PDEs) with weak solutions in to PDEs with weak solutions in or . To this end, we consider interpolation trial spaces that satisfy the de Rham Hilbert subcomplex, providing stable and structure-preserving neural network discretisations for a wide variety of PDEs. This approach, coined compatible FEINNs, has been used to accurately approximate the inner product. We numerically observe that the trained network outperforms finite element solutions by several orders of magnitude for smooth analytical solutions. Furthermore, to showcase the versatility of the method, we demonstrate that compatible FEINNs achieve high accuracy in solving surface PDEs such as the Darcy equation on a sphere. Additionally, the framework can integrate adaptive mesh refinements to effectively solve problems with localised features. We use an adaptive training strategy to train the network on a sequence of progressively adapted meshes. Finally, we compare compatible FEINNs with the adjoint neural network method for solving inverse problems. We consider a one-loop algorithm that trains the neural networks for unknowns and missing parameters using a loss function that includes PDE residual and data misfit terms. The algorithm is applied to identify space-varying physical parameters for the model problem from partial or noisy observations. We find that compatible FEINNs achieve accuracy and robustness comparable to, if not exceeding, the adjoint method in these scenarios.

Paper Structure

This paper contains 30 sections, 3 theorems, 41 equations, 16 figures.

Table of Contents

  1. Introduction
  2. Methodology
  3. Notation
  4. Continuous problems
  5. Maxwell's equations
  6. Darcy equation on a closed surface
  7. Finite element discretisation
  8. Discrete de Rham complexes
  9. Edge and face finite elements
  10. Linearised test spaces
  11. Maxwell problem finite element discretisation
  12. Surface Darcy problem finite element discretisation
  13. Neural networks
  14. Compatible finite element interpolated neural networks
  15. Trace finite element interpolated neural networks
  16. ...and 15 more sections

Key Result

Proposition 2.1

The trial space $U_h^1$ and the linearised test space ${V_h^1}$ for quadrilateral and hexahedral Nédélec elements of the first kind have the same number of dof per geometrical entity (edge, face, cell) on the mesh $\mathcal{T}_h$ and as a result the same dimensions.

Figures (16)

  • Figure 1: Error convergence of feinn and nn solutions with respect to the mesh size of the trial fe space for the forward Maxwell problem with a smooth solution. Different colours represent different orders of trial bases.
  • Figure 2: $H(\textbf{curl})$ error history of and feinn and nn during training using different dual norms in the preconditioned loss for the forward Maxwell problem with a smooth solution; "N/A" indicates no preconditioning. The number in the legends indicates iterations in the initialisation step. Mesh size $h=2^{-4}$ and order $k_U=4$ were used.
  • Figure 3: Error convergence of feinn and nn solutions with respect to the order of trial bases for the forward Maxwell problem with a smooth solution. Loss function subject to minimisation is equipped with preconditioner $\mathbf{B}_{\mathrm{lin}}$ and the $H(\textbf{curl})$ norm. Different colours represent different mesh sizes.
  • Figure 4: Interpolated nn solutions on a $6\times50\times50$ grid for the forward Darcy problem on a sphere.
  • Figure 5: Error convergence of feinn and nn solutions with respect to the mesh size of the trial fe space for the forward Darcy problem on the unit sphere. Colours distinguish between flux and pressure.
  • ...and 11 more figures

Theorems & Definitions (5)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Theorem 2.3