Neural network-augmented differentiable finite element method for boundary value problems

Classical numerical methods such as finite element method (FEM) face limitations due to their low efficiency when addressing large-scale problems. As a novel paradigm, the physics-informed neural network (PINN) has demonstrated significant potential to solve partial differential equations. However, conventional PINNs utilize meshless control at discrete sampling points, which limits their ability to effectively handle complex boundaries. Moreover, catastrophic failure may occur in the deep energy method (DEM, a specific type of PINN). To handle these challenges, this study proposes a Neural Network-augmented Differentiable Finite Element Method (NNDFEM) by combining PINN and finite element approximation. In NNDFEM, the neural network backend solely predicts nodal variables. Derivatives and complex boundary conditions can be well handled by the finite element frontend. The governing equation over the domain, Dirichlet, and Neumann boundary conditions are directly enforced on the finite element frontend. Thus, losses of boundary conditions in PINN are rendered unnecessary. The overfitting problem in DEM is also significantly mitigated. Fully connected neural network (FCNN), modified FCNN, and graph-convolutional network are tested as backends. NNDFEM circumvents nodal force calculation and matrix assembly in FEM. Functional losses of linear elasticity, finite strain nonlinear elasticity, heat conduction, and flow in porous media are validated. A systematic exploration unveils the role of 3D finite element mesh. For large-scale problems, a multi-fidelity learning strategy is employed. Thus, the three-dimensional case with over three million degrees of freedom trains well in two minutes. Benefiting from the fast inference of the neural network backend, the forward pass is 8,550 times faster than FEM.