  With the rapid advancement of graphical processing units, Physics-Informed
Neural Networks (PINNs) are emerging as a promising tool for solving partial
differential equations (PDEs). However, PINNs are not well suited for solving
PDEs with multiscale features, particularly suffering from slow convergence and
poor accuracy. To address this limitation of PINNs, this article proposes
physics-informed cell representations for resolving multiscale Poisson problems
using a model architecture consisting of multilevel multiresolution grids
coupled with a multilayer perceptron (MLP). The grid 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. The
variational (weak) form based loss function accelerates computation by allowing
the linear interpolation of feature vectors within grid cells. This cell-based
MLP model also facilitates the use of a decoupled training scheme for Dirichlet
boundary conditions and a parameter-sharing scheme for periodic boundary
conditions, delivering superior accuracy compared to conventional PINNs.
Furthermore, the numerical examples highlight improved speed and accuracy in
solving PDEs with nonlinear or high-frequency boundary conditions and provide
insights into hyperparameter selection. In essence, by cell-based MLP model
along with the parallel tiny-cuda-nn library, our implementation improves
convergence speed and numerical accuracy.
