PG-KINN: A Physics-Informed Petrov-Galerkin Kolmogorov-Arnold Network for Solving PDEs

Forum topic · 小凯 · 2026-07-24

Summary

PG-KINN (arXiv:2507.18398) is a physics-informed Kolmogorov-Arnold Network (KAN) framework based on a Petrov-Galerkin formulation for solving partial differential equations. The authors Amirhossein Sadr, Nima Soltani, and Vahideh Moghtadaiee argue that while KANs' learnable spline activations align well with classical piecewise-polynomial discretizations, the choice of loss formulation is equally critical: strong-form residuals demand high-order derivatives, energy (Bubnov-Galerkin) forms are limited to self-adjoint operators and can collapse to trivial solutions for parameter identification, and boundary integral forms require known fundamental solutions. PG-KINN uses a KAN as the trial space and an independent, compactly supported, piecewise-polynomial test space evaluated via Gauss-Legendre quadrature. Integration by parts lowers derivative orders, and localized test functions convert the global residual into well-conditioned element-wise weak residuals, supporting non-self-adjoint, nonlinear, and inverse problems. Benchmarks covering crack singularities, stress concentration, Neo-Hookean hyperelasticity, inverse parameter identification in heterogeneous media, and complex geometries show PG-KINN consistently outperforming MLP baselines and state-of-the-art KAN-based strong, energy, and inverse formulations (PIKAN).

PG-KINN: A Physics-Informed Petrov-Galerkin Kolmogorov-Arnold Network

Paper: arXiv:2507.18398 Authors: Amirhossein Sadr, Nima Soltani, Vahideh Moghtadaiee Field: Machine Learning

Overview

Physics-informed learning of 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.

Key Motivation

However, the way a PDE is cast into a loss functional is as decisive as the choice of approximator:

Strong-form residual minimization requires high-order derivatives and heavily weighted losses.
Energy (Bubnov-Galerkin) form is restricted to self-adjoint operators and, as the authors show, collapses to a trivial solution for parameter-identification problems.
Boundary integral forms require a known fundamental solution.

Method: PG-KINN

PG-KINN is a physics-informed KAN based on a Petrov-Galerkin formulation:

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 reduces the differentiation order while remaining applicable 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.

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).

Conclusion

The Petrov-Galerkin coupling of KAN trial spaces with polynomial test spaces is positioned as a robust and accurate pathway for AI-driven computational mechanics.