Exact and soft boundary conditions in Physics-Informed Neural Networks for the Variable Coefficient Poisson equation

TL;DR

This paper compares soft loss-based and exact boundary condition approaches in Physics-Informed Neural Networks for solving the variable coefficient Poisson equation, providing practical implementation resources.

Contribution

It offers a detailed comparison of BC imposition methods in PINNs and provides practical code resources for implementation.

Findings

Exact BC imposition improves solution accuracy.

Soft BC imposition is easier to implement but less precise.

Resources include code examples and step-by-step guides.

Abstract

Boundary conditions (BCs) are a key component in every Physics-Informed Neural Network (PINN). By defining the solution to partial differential equations (PDEs) along domain boundaries, BCs constrain the underlying boundary value problem (BVP) that a PINN tries to approximate. Without them, unique PDE solutions may not exist and finding approximations with PINNs would be a challenging, if not impossible task. This study examines how soft loss-based and exact distance function-based BC imposition approaches differ when applied in PINNs. The well known variable coefficient Poisson equation serves as the target PDE for all PINN models trained in this work. Besides comparing BC imposition approaches, the goal of this work is to also provide resources on how to implement these PINNs in practice. To this end, Keras models with Tensorflow backend as well as a Python notebook with code examples and step-by-step explanations on how to build soft/exact BC PINNs are published alongside this review.