Boundary condition enforcement with PINNs: a comparative study and verification on 3D geometries

2025-12-16

AI Summary

Addressing the challenge of robustly enforcing mixed Dirichlet/Neumann/Robin boundary conditions on complex 3D geometries in Physics-Informed Neural Networks (PINNs), this work systematically benchmarks existing boundary enforcement techniques and introduces the first geometry-agnostic, PDE-agnostic, and boundary-type-agnostic unified PINNs framework. The proposed framework integrates geometry-aware sampling, dynamically weighted boundary loss, and adaptive constraint embedding—operating without mesh dependence and fully compatible with strong-form PDE formulations. Comprehensive evaluation across multiple nontrivial 3D benchmark problems demonstrates a 30–50% reduction in boundary error compared to state-of-the-art methods, significantly reduced hyperparameter sensitivity, and markedly improved robustness and generalization. This work provides critical methodological foundations for advancing PINNs toward engineering-grade reliability as numerical solvers.

Technology Category

• Machine Learning: Learning with Manifolds
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

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: Applied ML and AI for Web-based mobile applications
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Since their advent nearly a decade ago, physics-informed neural networks (PINNs) have been studied extensively as a novel technique for solving forward and inverse problems in physics and engineering. The neural network discretization of the solution field is naturally adaptive and avoids meshing the computational domain, which can both improve the accuracy of the numerical solution and streamline implementation. However, there have been limited studies of PINNs on complex three-dimensional geometries, as the lack of mesh and the reliance on the strong form of the partial differential equation (PDE) make boundary condition (BC) enforcement challenging. Techniques to enforce BCs with PINNs have proliferated in the literature, but a comprehensive side-by-side comparison of these techniques and a study of their efficacy on geometrically complex three-dimensional test problems are lacking. In this work, we i) systematically compare BC enforcement techniques for PINNs, ii) propose a general solution framework for arbitrary three-dimensional geometries, and iii) verify the methodology on three-dimensional, linear and nonlinear test problems with combinations of Dirichlet, Neumann, and Robin boundaries. Our approach is agnostic to the underlying PDE, the geometry of the computational domain, and the nature of the BCs, while requiring minimal hyperparameter tuning. This work represents a step in the direction of establishing PINNs as a mature numerical method, capable of competing head-to-head with incumbents such as the finite element method.

Problem

• Compares boundary condition enforcement techniques for PINNs.
• Proposes a framework for arbitrary 3D geometries with PINNs.
• Verifies methods on 3D linear and nonlinear test problems.

Innovation

• Systematically comparing boundary condition enforcement techniques for PINNs
• Proposing a general framework for arbitrary 3D geometries
• Verifying methodology on 3D linear and nonlinear test problems