AKAPINN: Adaptive Kolmogorov-Arnold Physics-Informed Neural Networks for approximating solutions to quasilinear partial differential equations
Abstract
A controlled comparative study of a mesh-free Kolmogorov--Arnold Physics-Informed Neural Network (KAN-PINN) applied to nonlinear strain-limiting partial differential equations is presented. Three distinct training runs across varying material-parameter pairs $(α, β)$ are evaluated within a fixed computational pipeline comprising an exact hard boundary ansatz, a robust pseudo-Huber residual loss, and a multi-stage Adam-to-L-BFGS optimizer schedule. A fundamental metric inversion is revealed by the analysis: while training loss reflects operator-specific residual scaling and constitutive stiffness, the operator-independent relative L2 error demonstrates that intermediate parameter regimes achieve superior solution accuracy and enhanced generalization.
Related
- Rigidity results via a Poincaré type formula for stable solutions of quasilinear anisotropic elliptic equations
- Liouville Theorems for Stable Solutions to Anisotropic Quasilinear Equations
- Fundamental solutions to Kolmogorov-Fokker-Planck equations with unbounded lower order terms
- Kolmogorov-Arnold Networks for Free-Boundary Partial Differential Equations
- Asymptotic Formula of the Lifespan of Classical Solutions to 1D Quasilinear Wave Equations
BibTeX
@misc{rao2026akapinn,
title = {{AKAPINN: Adaptive Kolmogorov-Arnold Physics-Informed Neural Networks for approximating solutions to quasilinear partial differential equations}},
author = {B. Veena S. N. Rao},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.25507},
eprint = {2609.25507},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25507},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - AKAPINN: Adaptive Kolmogorov-Arnold Physics-Informed Neural Networks for approximating solutions to quasilinear partial differential equations
AU - B. Veena S. N. Rao
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.25507
UR - https://arxiv.org/abs/2609.25507
AB - A controlled comparative study of a mesh-free Kolmogorov--Arnold Physics-Informed Neural Network (KAN-PINN) applied to nonlinear strain-limiting partial differential equations is presented. Three distinct training runs across varying material-parameter pairs $(α, β)$ are evaluated within a fixed computational pipeline comprising an exact hard boundary ansatz, a robust pseudo-Huber residual loss, and a multi-stage Adam-to-L-BFGS optimizer schedule. A fundamental metric inversion is revealed by the analysis: while training loss reflects operator-specific residual scaling and constitutive stiffness, the operator-independent relative L_2 error demonstrates that intermediate parameter regimes achieve superior solution accuracy and enhanced generalization.
ER -