Implicit Neural Spatial Representations for Time-dependent PDEs

Numerically solving partial differential equations (PDEs) often entails spatial and temporal discretizations. Traditional methods (e.g., finite difference, finite element, smoothed-particle hydrodynamics) frequently adopt explicit spatial discretizations, such as grids, meshes, and point clouds, where each degree-of-freedom corresponds to a location in space. While these explicit spatial correspondences are intuitive to model and understand, these representations are not necessarily optimal for accuracy, memory-usage, or adaptivity. In this work, we explore implicit neural representation as an alternative spatial discretization, where spatial information is implicitly stored in the neural network weights. With implicit neural spatial representation, PDE-constrained time-stepping translates into updating neural network weights, which naturally integrates with commonly adopted optimization time integrators. We validate our approach on a variety of classic PDEs with examples involving large elastic deformations, turbulent fluids, and multiscale phenomena. While slower to compute than traditional representations, our approach exhibits higher accuracy, lower memory consumption, and dynamically adaptive allocation of degrees of freedom without complex remeshing.