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

TL;DR

This paper introduces a systematic method to exactly enforce Dirichlet, Neumann, and Robin boundary conditions on curved domain boundaries in physics-informed machine learning, ensuring high accuracy even on complex geometries.

Contribution

The paper develops a new approach combining domain mappings, TFC constrained expressions, and transfinite interpolations for exact boundary condition enforcement in complex domains.

Findings

Boundary conditions enforced with machine accuracy.

Applicable to complex quadrilateral domains with curved boundaries.

Effective for both linear and nonlinear problems.

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.