Unfitted finite element interpolated neural networks

TL;DR

The paper introduces unfitted finite element interpolated neural networks (unfitted feinn) to solve PDEs on complex geometries by interpolating neural networks onto unfitted FE spaces and minimizing residual-based losses with Nitsche-based weak boundary conditions. It develops continuous and discretised frameworks, analyzes loss formulations (including discrete dual norms), and extends to inverse problems. Through extensive 2D and 3D experiments, it shows that NN surrogates can outperform FE interpolations in accuracy, rival or exceed PINNs in efficiency, and remain robust to small cut cells and hyperparameters. The approach enables accurate forward and inverse solving on geometries defined by level-set functions or STL meshes, with notable improvements in convergence and training speed. Overall, unfitted feinn offers a scalable, geometry-agnostic, data-efficient pathway for PDE surrogates on complex domains, with clear avenues for adaptive refinement and time-dependent extensions.

Abstract

We present a novel approach that integrates unfitted finite element methods and neural networks to approximate partial differential equations on complex geometries. Easy-to-generate background meshes (e.g., a simple Cartesian mesh) that cut the domain boundary (i.e., they do not conform to it) are used to build suitable trial and test finite element spaces. The method seeks a neural network that, when interpolated onto the trial space, minimises a discrete norm of the weak residual functional on the test space associated to the equation. As with unfitted finite elements, essential boundary conditions are weakly imposed by Nitsche's method. The method is robust to variations in Nitsche coefficient values, and to small cut cells. We experimentally demonstrate the method's effectiveness in solving both forward and inverse problems across various 2D and 3D complex geometries, including those defined by implicit level-set functions and explicit stereolithography meshes. For forward problems with smooth analytical solutions, the trained neural networks achieve several orders of magnitude smaller $H^1$ errors compared to their interpolation counterparts. These interpolations also maintain expected $h$- and $p$-convergence rates. Using the same amount of training points, the method is faster than standard PINNs (on both GPU and CPU architectures) while achieving similar or superior accuracy. Moreover, using a discrete dual norm of the residual (achieved by cut cell stabilisation) remarkably accelerates neural network training and further enhances robustness to the choice of Nitsche coefficient values. The experiments also show the method's high accuracy and reliability in solving inverse problems, even with incomplete observations.

Figures (16)

• Figure 1: (a) Background box and physical domain, (b) trial and (c) test fe spaces.
• Figure 2: Analytic geometries used in the 2D experiments.
• Figure 3: Convergence history of $u^{id}$ in $L^2$ and $H^1$ errors for the 2D Poisson problem, using different Nitsche coefficients during training. The top row shows $L^2$ errors and the bottom row represents $H^1$ errors. The first column displays errors of the interpolated nn and the second column corresponds to errors of the nn themselves. The red dashed lines represent the fem errors.
• Figure 4: Convergence history of the nn $H^1$ error over training iterations and wall-clock time in seconds for the 2D Poisson problem using pinn and $\ell^2$-feinn. Both methods are implemented in TensorFlow, employ identical nn architectures with the same initialisation, and are trained on GPU using TensorFlow's BFGS with an equal number of training points.
• Figure 5: $L^2$ and $H^1$ errors of the trained nn for the 2D Poisson problem plotted against the centre position of the moving disk.

Theorems & Definitions (1)

• Remark 3.1