A Method for Exact Enforcement of Boundary Conditions on General Curved Triangular Domains: Algorithm and Application to Scientific Machine Learning
Abstract
We present a method for enforcing exactly Dirichlet, Neumann, and Robin conditions on general triangular domains with arbitrarily curved boundaries. The algorithmic constructions developed here complements those in a previous work for general quadrilateral domains. Our method combines (i) the exact mapping between a general curved triangular domain and the standard triangle, (ii) TFC constrained expressions, and (iii) Barnhill transfinite interpolations, to provide explicit closed-form trial ansatze that mathematically satisfy the imposed boundary conditions exactly. Corner compatibility induced by the boundary conditions is pivotal, and enforcing exactly the imposed boundary conditions hinges on the exact enforcement of the induced compatibility constraints. The corner compatibility associated with the acute vertices of the standard triangle and the coupling between gradient components when the hypotenuse corresponds to a Neumann/Robin boundary pose particular challenges. Two strategies are critical to resolving these issues, by reformulating conditional corner constraints into equivalent unconditional forms and by introducing a separate TFC-style auxiliary edge construction together with an associated linear solve on the hypotenuse. We analyze the corner compatibility constraints and present formulations for the general form of functions that exactly satisfy the imposed Dirichlet, Neumann, or Robin conditions on curved triangular domains. The proposed method has been implemented using extreme learning machine type randomized neural networks for solving PDE problems. We present ample numerical experiments with linear/nonlinear PDE problems on a variety of domains with complex boundary geometries. Numerical results show that the current method has enforced the Dirichlet, Neumann, and Robin conditions on curved domain boundaries to the machine precision.
Related
- Concurrent enforcement of polyconvexity and true-stress-true-strain monotonicity in incompressible isotropic hyperelasticity: application to neural network constitutive models
- Optimal enforcement of natural conditions on smooth boundaries with piecewise polynomials upon fitted straight-edged meshes
- Necessary and sufficient conditions for the self-normalized central limit theorem for independent triangular arrays
- Mediation and moderation in distance-based regression with triangular fuzzy data: simulation evidence and an application to academic performance
- An Efficient iLQR Algorithm for Optimal Control with Bounded States and Inputs: Application to Spacecraft Maneuvers
BibTeX
@misc{dong2026method,
title = {{A Method for Exact Enforcement of Boundary Conditions on General Curved Triangular Domains: Algorithm and Application to Scientific Machine Learning}},
author = {Suchuan Dong and Yuchuan Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.17392},
eprint = {2610.17392},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.17392},
}
RIS
TY - UNPB
TI - A Method for Exact Enforcement of Boundary Conditions on General Curved Triangular Domains: Algorithm and Application to Scientific Machine Learning
AU - Suchuan Dong
AU - Yuchuan Zhang
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.17392
UR - https://arxiv.org/abs/2610.17392
AB - We present a method for enforcing exactly Dirichlet, Neumann, and Robin conditions on general triangular domains with arbitrarily curved boundaries. The algorithmic constructions developed here complements those in a previous work for general quadrilateral domains. Our method combines (i) the exact mapping between a general curved triangular domain and the standard triangle, (ii) TFC constrained expressions, and (iii) Barnhill transfinite interpolations, to provide explicit closed-form trial ansatze that mathematically satisfy the imposed boundary conditions exactly. Corner compatibility induced by the boundary conditions is pivotal, and enforcing exactly the imposed boundary conditions hinges on the exact enforcement of the induced compatibility constraints. The corner compatibility associated with the acute vertices of the standard triangle and the coupling between gradient components when the hypotenuse corresponds to a Neumann/Robin boundary pose particular challenges. Two strategies are critical to resolving these issues, by reformulating conditional corner constraints into equivalent unconditional forms and by introducing a separate TFC-style auxiliary edge construction together with an associated linear solve on the hypotenuse. We analyze the corner compatibility constraints and present formulations for the general form of functions that exactly satisfy the imposed Dirichlet, Neumann, or Robin conditions on curved triangular domains. The proposed method has been implemented using extreme learning machine type randomized neural networks for solving PDE problems. We present ample numerical experiments with linear/nonlinear PDE problems on a variety of domains with complex boundary geometries. Numerical results show that the current method has enforced the Dirichlet, Neumann, and Robin conditions on curved domain boundaries to the machine precision.
ER -