Physics-informed graph Laplacian-augmented Kolmogorov–Arnold networks

Abstract

Physics-informed approximation of solutions to forward problems governed by partial differential equation on geometrically irregular domains become challenging when geometric representation, residual evaluation, and neural approximation are not consistently integrated. This study proposes a physics-informed graph Laplacian-augmented Kolmogorov–Arnold network (PI-GLaK) for approximating solutions to the steady incompressible Navier–Stokes equations in two-dimensional vascular geometries. The proposed framework establishes a coefficient-based neural parameterization of the velocity and pressure fields within a common mixed finite-element formulation. Mesh-induced graph connectivity, Laplacian-augmented Kolmogorov–Arnold network layers, and Galerkin residual evaluation are defined using the same Taylor–Hood finite-element representation. The proposed PI-GLaK was evaluated on aneurysm–bend, bifurcation, and stenosis–aneurysm configurations, and its performance was compared with that of a conventional physics-informed neural network (PINN) and a physics-informed graph Galerkin network (PI-GGN). The proposed PI-GLaK achieved total relative L2 errors of 0.048–0.069, compared with 0.017–8.98 for the conventional PINN and 1.03–3.24 for PI-GGN. The proposed PI-GLaK exhibited more consistent accuracy across the three configurations than the PINN baseline and substantially lower errors than the graph-based Galerkin baseline. A gradient-based analysis of the residual loss components characterized geometry-dependent interactions during training and their relationship to the optimization behavior of the proposed formulation. These results demonstrate that graph connectivity, neural coefficient parameterization, and Galerkin residual evaluation can be integrated within a common mixed finite-element representation for physics-informed approximation on geometrically irregular domains.