Paper Overview
- Field: Machine Learning
- Authors: Amirhossein Sadr, Nima Soltani, Vahideh Moghtadaiee
- arXiv: 2507.18398
- Strong-form residual minimization requires high-order derivatives and heavily weighted losses.
- Energy (Bubnov-Galerkin) formulations are restricted to self-adjoint operators and, as the authors show, collapse to a trivial solution for parameter-identification problems.
- Boundary integral forms require known fundamental solutions.
- The trial space is a KAN.
- The test space is an independent, compactly supported, piecewise-polynomial space evaluated with Gauss-Legendre quadrature.
- Integration by parts lowers the differentiation order while preserving applicability to general non-self-adjoint, nonlinear, and inverse problems.
- Localized test functions transform the global residual into a set of well-conditioned element-wise weak residuals.
Summary
Physics-informed learning of partial differential equations (PDEs) has been dominated by multilayer perceptrons (MLPs), whose spectral bias and dense parameterization limit both accuracy and interpretability. Kolmogorov Arnold Networks (KANs) mitigate these limitations because their learnable spline activations are structurally aligned with the piecewise-polynomial bases of classical discretizations.
However, the way a PDE is cast into a loss functional is as decisive as the choice of approximator:
Method
The authors propose PG-KINN, a physics-informed KAN based on a Petrov-Galerkin formulation, where:
Results
Across benchmarks covering crack singularities, stress concentration, Neo-Hookean hyperelasticity, inverse parameter identification in heterogeneous media, and complex geometries, PG-KINN consistently outperforms conventional MLP baselines and state-of-the-art KAN-based strong/energy/inverse formulations (PIKAN). The results position the Petrov-Galerkin coupling of KAN trial spaces with polynomial test spaces as a robust and accurate path for AI-driven computational mechanics.
Source: arXiv:2507.18398