Enforcing boundary conditions for physics-informed neural operators

2025-10-28

AI Summary

This work addresses the instability arising when enforcing Neumann and Robin boundary conditions on piecewise $C^1$ (globally only $C^0$) boundaries in Physics-Informed Neural Operators (PINOs). We propose a novel strong-constraint method based on orthogonal projection, extending the Sukumar & Srivastava framework to relax the conventional requirement of globally $C^1$ boundaries. By constructing admissible trial functions and applying orthogonal projection onto the constrained function space, our approach ensures exact and numerically stable satisfaction of boundary conditions. The method unifies weak, semi-weak, and strong formulations for boundary treatment. We validate it on scalar Darcy flow and steady-state Navier–Stokes equations: results demonstrate significantly improved training stability and numerical accuracy, enhanced robustness on complex geometries, faster convergence, and lower approximation errors compared to standard approaches.

Technology Category

• Constraint Satisfaction and Optimization: Constraint Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks

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.

Problem

• Enforcing boundary conditions in physics-informed neural operators
• Overcoming instability from piecewise smooth boundaries
• Comparing strong enforcement methods for PDE boundary conditions

Innovation

• Generalizing existing strong boundary enforcement approach
• Introducing orthogonal projection method for boundary conditions
• Overcoming limitations of piecewise C1 boundary instability