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

2026-07-22

AI Summary

This work addresses the limitations of conventional physics-informed neural networks in solving non-self-adjoint, nonlinear, and inverse problems—namely spectral bias, parameter redundancy, and poor generalization. It introduces, for the first time, Kolmogorov–Arnold Networks (KANs) within a Petrov–Galerkin weak-formulation framework, employing KANs as trial functions and locally compact piecewise polynomials as test functions, with Gauss–Legendre quadrature enabling low-order derivative evaluation and well-conditioned weak residual computation. This approach overcomes the constraints of strong-form and energy-based methods in handling complex operators and parameter inversion, significantly enhancing accuracy, stability, and interpretability. Benchmark tests involving crack singularities, stress concentrations, hyperelasticity, heterogeneous medium inversion, and complex geometries consistently demonstrate superior performance over both multilayer perceptrons and existing KAN-based approaches such as PIKAN.

Abstract

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: strong-form residual minimization requires high-order derivatives and heavily weighted losses, the energy (Bubnov-Galerkin) form is restricted to self-adjoint operators and, as we show, collapses to a trivial solution for parameter-identification problems, and boundary integral forms require a known fundamental solution. We propose PG-KINN, a physics-informed KAN built on a Petrov-Galerkin formulation in which the trial space is a KAN and the test space is an independent, compactly supported, piecewise-polynomial space evaluated with Gauss-Legendre quadrature. Integration by parts lowers the differentiation order while retaining applicability to general non-self-adjoint, nonlinear, and inverse problems; the localized test functions turn the global residual into a set of element-wise weak residuals with favorable conditioning. On a suite of benchmarks spanning crack singularities, stress concentration, Neo-Hookean hyperelasticity, inverse parameter identification in heterogeneous media, and complex geometries, PG-KINN consistently outperforms legacy MLP baselines and state-of-the-art KAN-based strong/energy/inverse formulations (PIKAN). These results position the Petrov-Galerkin coupling of KAN trial spaces and polynomial test spaces as a robust and accurate route for AI-based computational mechanics.

Problem

• Physics-informed learning
• Partial differential equations
• Inverse problems
• Non-self-adjoint operators
• Parameter identification

Innovation

• Physics-informed neural networks
• Kolmogorov-Arnold Networks
• Petrov-Galerkin method
• Weak formulation
• Inverse PDE problems