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arXiv2026

Physics-Informed Kolmogorov-Arnold Networks for Grad-Shafranov Tokamak Equilibria

We employ equation-driven, physics-constrained deep learning to solve the fixed-boundary Grad-Shafranov (GS) equilibrium problem, constructing axisymmetric magnetohydrodynamic equilibria with tokamak-relevant characteristics. Equilibria across linear (Solov'ev) and nonlinear profile functions are constructed using Physics-Informed Kolmogorov-Arnold Networks (KANs) that approximate GS solutions while satisfying appropriate boundary conditions. A highly nonlinear pressure profile recreating high-confinement mode phenomenology, such as pressure pedestals and significant bootstrap current components, is also considered. To enable efficient convergence, guided training schemes are employed, specifically homotopy-based continuation curriculum learning and transfer learning via pretrained networks. While computing nonlinear equilibria employing standard Multi-Layer Perceptrons under unguided physics-informed training remains an elusive or computationally inefficient task, our framework overcomes this limitation. Specifically, we demonstrate that the combination of three key elements, namely KAN architecture, guided training schemes, and the self-scaled Broyden optimization method, enables stable, efficient, and accurate equilibrium computation with simultaneous profile parameter identification in view of equilibrium constraints.

physics.plasm-ph↗

arXiv2025

Multi-soliton solutions and data-driven discovery of higher-order Burgers' hierarchy equations with physics informed neural networks

The Burgers hierarchy consists of nonlinear evolutionary partial differential equations (PDEs) with progressively higher-order dispersive and nonlinear terms. Notable members of this hierarchy are the Burgers equation and the Sharma-Tasso-Olver equation, which are widely applied in fields such as plasma physics, fluid mechanics, optics, and biophysics to describe nonlinear waves in inhomogeneous media. Various soliton and multi-soliton solutions to these equations have been identified and the fission and fusion of solitons have been studied using analytical and numerical techniques. Recently, deep learning methods, particularly Physics-Informed Neural Networks (PINNs), have emerged as a new approach for solving PDEs. These methods use deep neural networks to minimize PDE residuals while fitting relevant data. Although PINNs have been applied to equations like Burgers' and Korteweg-de Vries, higher-order members of the Burgers hierarchy remain unexplored in this context. In this study, we employ a PINN algorithm to approximate multi-soliton solutions of linear combinations of equations within the Burgers hierarchy. This semi-supervised approach encodes the PDE and relevant data, determining PDE parameters and resolving the linear combination to discover the PDE that describes the data. Additionally, we employ gradient-enhanced PINNs (gPINNs) and a conservation law, specific to the generic Burgers' hierarchy, to improve training accuracy. The results demonstrate the effectiveness of PINNs in describing multi-soliton solutions within the generic Burgers' hierarchy, their robustness to increased levels of data noise, and their limited yet measurable predictive capabilities. They also verify the potential for training refinement and accuracy improvement using enhanced approaches in certain cases, while enabling the discovery of the PDE model that describes the observed solitary structures.

physics.comp-ph↗