Mesh-Free Numerical Approximation of the Biharmonic Equation via Optimized Kolmogorov--Arnold Neural Networks
Abstract
A mesh-free numerical framework based on \textit{Kolmogorov--Arnold Physics-Informed Neural Networks} (KAN-PINNs) is developed for the approximation of fourth-order elliptic boundary value problems, with specific application to the biharmonic equation governing thin plate deflection. Unlike conventional Multi-Layer Perceptrons relying on fixed nodal activations, learnable univariate functions parameterized via radial basis functions are deployed on network edges, while high-order differentiability is preserved through hyperbolic tangent activation mappings. The severe numerical stiffness inherent to fourth-order differential operators and fully clamped boundary conditions is addressed through a direct normalized residual formulation coupled with a hybrid, multi-stage AdamW-to-L-BFGS optimization pipeline. An automated \textit{24/7 hill-climbing search protocol} is implemented to systematically calibrate boundary penalty weights and optimization schedules. When evaluated on a smooth manufactured benchmark on the unit square, an error reduction factor exceeding 150× is attained over successive iterations, culminating in a final relative L2 error of 1.593 × 10−5 (0.00159%) and a training loss of 3.065 × 10−6 utilizing only $7,801$ trainable parameters. These results demonstrate that high-order PDE problems can be accurately resolved using compact, mesh-free KAN-PINNs architecture without requiring auxiliary variable transformations or discrete mesh generation.
Related
- An approximation for solutions to Helmholtz equation in polygons and its application in learning based numerical method
- Positive radial ground states for nonlinear biharmonic equations: maximum principles and expanding-domain approximation
- Existence of solutions for a nonlinear elliptic equation involving (p(x),q(x))-Laplacian and (p(x),q(x))-biharmonic Operator
- Existence Results for the Critical Biharmonic Elliptic Equation Involving Singular Nonlinearities
- Long-time reduction for the biharmonic nonlinear Schrödinger equation
BibTeX
@misc{rao2026meshfree,
title = {{Mesh-Free Numerical Approximation of the Biharmonic Equation via Optimized Kolmogorov--Arnold Neural Networks}},
author = {B. Veena S. N. Rao},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.39469},
eprint = {2609.39469},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.39469},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Mesh-Free Numerical Approximation of the Biharmonic Equation via Optimized Kolmogorov--Arnold Neural Networks
AU - B. Veena S. N. Rao
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.39469
UR - https://arxiv.org/abs/2609.39469
AB - A mesh-free numerical framework based on \textit{Kolmogorov--Arnold Physics-Informed Neural Networks} (KAN-PINNs) is developed for the approximation of fourth-order elliptic boundary value problems, with specific application to the biharmonic equation governing thin plate deflection. Unlike conventional Multi-Layer Perceptrons relying on fixed nodal activations, learnable univariate functions parameterized via radial basis functions are deployed on network edges, while high-order differentiability is preserved through hyperbolic tangent activation mappings. The severe numerical stiffness inherent to fourth-order differential operators and fully clamped boundary conditions is addressed through a direct normalized residual formulation coupled with a hybrid, multi-stage AdamW-to-L-BFGS optimization pipeline. An automated \textit{24/7 hill-climbing search protocol} is implemented to systematically calibrate boundary penalty weights and optimization schedules. When evaluated on a smooth manufactured benchmark on the unit square, an error reduction factor exceeding 150× is attained over successive iterations, culminating in a final relative L_2 error of 1.593 × 10^−5 (0.00159%) and a training loss of 3.065 × 10^−6 utilizing only $7,801$ trainable parameters. These results demonstrate that high-order PDE problems can be accurately resolved using compact, mesh-free KAN-PINNs architecture without requiring auxiliary variable transformations or discrete mesh generation.
ER -