Enforcing boundary conditions for physics-informed neural operators

Niklas Göschel, Sebastian Götschel, Daniel Ruprecht

Keywords: physics-informed neural operators, physics-informed neural networks, boundary conditions, approximate distance function, hard constraints

Abstract:

Machine-learning based techniques like physics-informed neural networks (PINNs) and physics-informed neural operators (PINO) are becoming increasingly adept at solving even complex systems of partial differential equations (PDEs).
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, N. & Srivastava, A., 2022, https://doi.org/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.

Primary Area: Machine learning for sciences (e.g. climate, health, life sciences, physics, social sciences)

Submission Number: 13031