Unfitted finite element interpolated neural networks

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.

References in corpus (19)

• B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data
• hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decomposition
• A unified deep artificial neural network approach to partial differential equations in complex geometries
• PhyGeoNet: Physics-Informed Geometry-Adaptive Convolutional Neural Networks for Solving Parameterized Steady-State PDEs on Irregular Domain
• The aggregated unfitted finite element method for elliptic problems
• PFNN: A Penalty-Free Neural Network Method for Solving a Class of Second-Order Boundary-Value Problems on Complex Geometries
• The software design of Gridap: a Finite Element package based on the Julia JIT compiler
• A Nitsche-based cut finite element method for a fluid--structure interaction problem
• Variational Physics Informed Neural Networks: the role of quadratures and test functions
• Unveiling the optimization process of Physics Informed Neural Networks: How accurate and competitive can PINNs be?
• Enforcing Dirichlet boundary conditions in physics-informed neural networks and variational physics-informed neural networks
• $\Delta$-PINNs: physics-informed neural networks on complex geometries
• Finite element interpolated neural networks for solving forward and inverse problems
• Robust and scalable h-adaptive aggregated unfitted finite elements for interface elliptic problems
• Distributed-memory parallelization of the aggregated unfitted finite element method
• Geometrical discretisations for unfitted finite elements on explicit boundary representations
• Compatible finite element interpolated neural networks
• Finite Element Neural Network Interpolation. Part I: Interpretable and Adaptive Discretization for Solving PDEs
• Finite Element Neural Network Interpolation. Part II: Hybridisation with the Proper Generalised Decomposition for non-linear surrogate modelling