A Novel Method for Enforcing Exactly Dirichlet, Neumann and Robin Conditions on Curved Domain Boundaries for Physics Informed Machine Learning

2026-03-23

AI Summary

This work addresses the challenge that existing physics-informed machine learning methods struggle to enforce Dirichlet, Neumann, and Robin boundary conditions exactly on arbitrary curved quadrilateral domains, particularly due to compatibility constraints at corners where Neumann and Robin boundaries intersect. To overcome this limitation, the authors propose a systematic framework that integrates exact geometric mapping, the Theory of Functional Connections (TFC), and transfinite interpolation to construct trial functions that rigorously satisfy all boundary conditions and vertex compatibility requirements. These trial functions are embedded within an Extreme Learning Machine (ELM) to solve partial differential equations. The method achieves machine-precision enforcement of boundary conditions on complex curved quadrilateral domains—surpassing conventional approaches that only approximate such constraints—and demonstrates high accuracy and broad applicability across a range of linear/nonlinear and steady/unsteady problems.

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

• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Responsible Web: Machine-in-the-loop, human agency and autonomy

Abstract

We present a systematic method for exactly enforcing Dirichlet, Neumann, and Robin type conditions on general quadrilateral domains with arbitrary curved boundaries. Our method is built upon exact mappings between general quadrilateral domains and the standard domain, and employs a combination of TFC (theory of functional connections) constrained expressions and transfinite interpolations. When Neumann or Robin boundaries are present, especially when two Neumann (or Robin) boundaries meet at a vertex, it is critical to enforce exactly the induced compatibility constraints at the intersection, in order to enforce exactly the imposed conditions on the joining boundaries. We analyze in detail and present constructions for handling the imposed boundary conditions and the induced compatibility constraints for two types of situations: (i) when Neumann (or Robin) boundary only intersects with Dirichlet boundaries, and (ii) when two Neumann (or Robin) boundaries intersect with each other. We describe a four-step procedure to systematically formulate the general form of functions that exactly satisfy the imposed Dirichlet, Neumann, or Robin conditions on general quadrilateral domains. The method developed herein has been implemented together with the extreme learning machine (ELM) technique we have developed recently for scientific machine learning. Ample numerical experiments are presented with several linear/nonlinear stationary/dynamic problems on a variety of two-dimensional domains with complex boundary geometries. Simulation results demonstrate that the proposed method has enforced the Dirichlet, Neumann, and Robin conditions on curved domain boundaries exactly, with the numerical boundary-condition errors at the machine accuracy.

Problem

• Dirichlet boundary conditions
• Neumann boundary conditions
• Robin boundary conditions
• curved boundaries
• compatibility constraints

Innovation

• Theory of Functional Connections (TFC)
• Transfinite Interpolation
• Exact Boundary Condition Enforcement
• Curved Domain Boundaries
• Physics-Informed Machine Learning