Energy loss informed cell-based multilayer perceptron for solving elliptic PDEs with multiscale features

Physics-Informed Neural Networks (PINNs) are emerging as a promising tool for solving partial differential equations (PDEs). However, standard PINNs can suffer from slow convergence and poor accuracy while solving elliptic PDEs with multiscale features. To address this limitation, an improved physics-informed cell representation is developed using a model architecture consisting of multilevel multiresolution grids coupled with a multilayer perceptron (MLP). The grid nodal parameters (i.e., the level-dependent feature vectors) and the MLP parameters (i.e., the weights and biases) are determined using gradient-descent based optimization along with auto-differentiation. The variational or energy loss function accelerates computation by allowing the linear interpolation of feature vectors within grid cells. The concatenated feature vectors from the cells at multiple levels are passed to the MLP to predict the PDE solution variable. This cell-based MLP model facilitates the use of a decoupled training scheme for applying Dirichlet boundary conditions and a parameter-sharing scheme for enforcing periodic boundary conditions. The numerical examples presented highlight improved speed and accuracy compared to standard PINNs for solving elliptic PDEs with nonlinear, high-contrast, or high-frequency features and provide insights into hyperparameter selection. The proposed cell-based MLP model along with the parallel tiny-cuda-nn library can be a viable approach for solving geomechanics problems involving Poisson and Helmholtz equations describing time-independent reactive-diffusive and wave phenomena.