Poisson Equation

The Poisson equation, a fundamental partial differential equation, describes the relationship between a source and its resulting potential field. Current research focuses on developing efficient numerical solvers, particularly for high-dimensional problems and those with complex geometries or boundary conditions, employing methods such as neural networks (including Physics-Informed Neural Networks and novel architectures like Neural Walk-on-Spheres), graph-based approaches, and Monte Carlo techniques. These advancements are improving the accuracy and speed of solving the Poisson equation across diverse fields, including computational fluid dynamics, molecular dynamics, and surface reconstruction from point clouds, leading to more efficient simulations and analyses.