PG-KINN: A Physics-Informed Petrov-Galerkin Kolmogorov-Arnold Network for Solving Forward and Inverse PDEs
2026-07-22 • Machine Learning
Machine Learning
AI summaryⓘ
The authors explain that common neural networks used to solve physics problems with partial differential equations (PDEs) have some drawbacks in accuracy and interpretation. They introduce a new method called PG-KINN, which combines a special type of network (KAN) with a mathematical technique called the Petrov-Galerkin formulation. This approach reduces complexity by using simpler test functions and helps solve more general and difficult PDE problems, including those with complex shapes and unknown parameters. Their tests show PG-KINN works better than older methods on various challenging science and engineering problems.
Partial Differential EquationsMultilayer PerceptronsKolmogorov Arnold NetworksPetrov-Galerkin FormulationPhysics-Informed Neural NetworksStrong-form Residual MinimizationEnergy Method (Bubnov-Galerkin)Inverse ProblemsGauss-Legendre QuadratureComputational Mechanics
Authors
Amirhossein Sadr, Nima Soltani, Vahideh Moghtadaiee, Aida Pakniyat, Dara Rahmati, Saeid Gorgin
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.