Enforcing boundary conditions for physics-informed neural operators

TL;DR

This paper introduces new methods for strongly enforcing boundary conditions in physics-informed neural operators, improving stability and accuracy especially on complex boundary geometries, and compares them with existing approaches.

Contribution

The paper generalizes existing boundary enforcement techniques and proposes a novel orthogonal projection method to handle piecewise $C^1$ boundaries effectively.

Findings

New methods outperform weak enforcement in accuracy.

Orthogonal projection approach enhances stability.

Techniques tested on Darcy flow and Navier-Stokes equations.

Abstract

Machine-learning based methods like physics-informed neural networks and physics-informed neural operators are becoming increasingly adept at solving even complex systems of partial differential equations. Boundary conditions can be enforced either weakly by penalizing deviations in the loss function or strongly by training a solution structure that inherently matches the prescribed values and derivatives. The former approach is easy to implement but the latter can provide benefits with respect to accuracy and training times. However, previous approaches to strongly enforcing Neumann or Robin boundary conditions require a domain with a fully $C^1$ boundary and, as we demonstrate, can lead to instability if those boundary conditions are posed on a segment of the boundary that is piecewise $C^1$ but only $C^0$ globally. We introduce a generalization of the approach by Sukumar \& Srivastava (doi: 10.1016/j.cma.2021.114333), and a new approach based on orthogonal projections that overcome this limitation. The performance of these new techniques is compared against weakly and semi-weakly enforced boundary conditions for the scalar Darcy flow equation and the stationary Navier-Stokes equations.