Date of Award
2026-05-01
Degree Name
Master of Science
Department
Computational Science
Advisor(s)
Peimeng Yin
Abstract
This thesis develops and analyzes neural-network-based solvers for the Poisson equation on polygonal domains, where re-entrant corners induce reduced solution regularity and challenge standard numerical methods.
Three neural formulations are investigated: Physics-Informed Neural Networks (PINNs), Physics-Informed Extreme Learning Machines (PIELMs), and Rank-Inspired Neural Networks (RINNs). PINNs rely on gradient-based optimization with automatic differentiation, while PIELMs employ randomly initialized hidden features with least-squares training, achieving significantly lower computational cost. RINNs extend this framework through covariance-driven orthogonalization to improve numerical conditioning and stability.
On convex polygonal domains, all three methods are benchmarked against analytical solutions. Both PIELM and RINN consistently achieve higher accuracy at lower computational cost than PINNs. On non-convex domains, singular solutions of the form r^alpha sin(alpha theta) are examined, where the dominant exponent alpha depends on the re-entrant corner angle.
A singularity-enriched neural ansatz is proposed, decomposing the solution into smooth and singular components using a compactly supported cutoff function. This formulation, termed SEPIELM, enables accurate recovery of the singular coefficient gamma with high numerical precision.
Two extensions are introduced: RBF-SEPIELM, which uses Gaussian radial basis functions for improved spatial localization, and Graded RBF-SEPIELM, which employs graded center distributions and spatially varying shape parameters to concentrate resolution near the singularity.
In benchmark problems, SEPIELM achieves a relative L2 error of 8.49 x 10^-4, improving upon previously reported SEPINN results, while being over two orders of magnitude faster than standard PINNs. The graded RBF-SEPIELM achieves the best performance, with a relative L2 error of 3.48 x 10^-4 in 5.18 seconds, compared to 1540 seconds for PINNs.
These results demonstrate that explicitly incorporating singular and geometric structure into neural approximations is essential for accurate and efficient solution of elliptic problems on non-smooth domains.
Language
en
Provenance
Received from ProQuest
Copyright Date
2026-05
File Size
98 p.
File Format
application/pdf
Rights Holder
Harshini Reddy Kodiganti
Recommended Citation
Kodiganti, Harshini Reddy, "Singularity-Enriched Neural Networks For Elliptic Problems In Polygonal Domains" (2026). Open Access Theses & Dissertations. 4709.
https://scholarworks.utep.edu/open_etd/4709