An $r$-adaptive finite element method using neural networks for parametric self-adjoint elliptic problem

TL;DR

The paper presents an $r$-adaptive finite element framework that uses a neural network to optimize mesh node locations for parametric self-adjoint elliptic PDEs, solving each parametric instance with a standard FEM. By formulating the mesh adaptation as Ritz energy minimization, the approach preserves FEM robustness while allowing the NN to tailor the mesh density to solution features such as sharp gradients or singularities. The method extends to parametric PDEs by learning a parameter-dependent mesh through a neural network, with the objective averaged over parameter samples to achieve good generalization. Numerical experiments in 1D and 2D Poisson problems demonstrate significant accuracy gains over uniform meshes, especially in challenging regimes, while highlighting the need for exact or highly accurate integration to avoid spurious local minima and ensuring reliable convergence. The work is implemented in JAX, enabling automatic differentiation for mesh optimization while permitting the use of non-differentiable high-performance FE solvers, and points to future extensions to non-symmetric problems, higher-order elements, irregular geometries, and goal-oriented adaptivity.

Abstract

This work proposes an $r$-adaptive finite element method (FEM) using neural networks (NNs). The method employs the Ritz energy functional as the loss function, currently limiting its applicability to symmetric and coercive problems, such as those arising from self-adjoint elliptic problems. The objective of the NN optimization is to determine the mesh node locations. For simplicity in two-dimensional problems, these locations are assumed to form a tensor product structure. The method is designed to solve parametric partial differential equations (PDEs). For each PDE parameter instance, the optimal $r$-adapted mesh generated by the NN is then solved with a standard FEM. The construction of FEM matrices and load vectors is implemented such that their derivatives with respect to mesh node locations, required for NN training, can be efficiently computed using automatic differentiation. However, the linear equation solver does not need to be differentiable, enabling the use of efficient, readily available `out-of-the-box' solvers. Consequently, the proposed approach retains the robustness and reliability guarantees of the FEM for each parameter instance, while the NN optimization adaptively adjusts the mesh node locations. The method's performance is demonstrated on parametric Poisson problems using one- and two-dimensional tensor product meshes.

Figures (24)

• Figure 1: Piecewise-linear FEM basis functions in $\Omega=(a,b)$.
• Figure 2: Flowchart of the proposed $r$-adaptive method in the non-parametric case.
• Figure 3: Flowchart of the proposed $r$-adaptive method in the parametric case.
• Figure 4: For a Poisson problem $-u"(x)=f(x)$ with solution $u(x)=\arctan(\alpha(x-s))+\arctan(\alpha s)$ for $\alpha=50$ and $s=0.5$, as shown in the left panel, we illustrate the landscapes of the minimum Ritz energy over piecewise-linear functions as a single mesh-node position is varied. The minimal Ritz energy landscapes corresponding to the exact and inexact integration and their associated minimizers are shown in red and blue, respectively, in the central and right panels.
• Figure 5: Training histories during $r$-adaptive optimization for various mesh sizes $N$ (left panel) and relative errors of the uniform and $r$-adapted finite element solutions as a function of $N$ (right panel) for the non-parametric model problem \ref{['eq:model_problem1']} with manufactured solution $u(x) = \arctan(10(x-0.5)) + \arctan(5)$.

Theorems & Definitions (5)

• Remark 1
• Remark 2
• Remark 3: No need to compute derivatives of the solver of linear equations
• Remark 4
• Remark 5