SDF-PINNs: Joining Physics-Informed Neural Networks with Neural Implicit Geometry Representation

Abstract

This paper presents an advanced method for solving boundary value problems of differential equations over arbitrary spatial domains using Physics-Informed Neural Networks (PINNs) augmented with Signed Distance Functions (SDFs). Our approach builds on the framework where the solution to the differential equation is decomposed into two parts: one that inherently satisfies the boundary conditions without any adjustable parameters, and a second that incorporates a physics-informed neural network with adjustable parameters. We propose to use a neural network approximation of the SDF for the representation of boundary conditions to model complex geometries accurately in an efficient manner. This novel combination allows for the precise enforcement of Dirichlet boundary conditions and improved solution accuracy over traditional PINN methods. We demonstrate the effectiveness of our approach through an illustrative example of a Poisson equation over a domain bound by the TUM logo. Our results indicate that this method not only preserves the benefits of neural networks in handling various types of differential equations but also leverages the geometric flexibility of SDFs to address complexly bounded domains effectively.