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Learning the Basis: A Kolmogorov-Arnold Network Approach Embedding Green's Function Priors

Rui Zhu, Yuexing Peng, George C. Alexandropoulos, Wenbo Wang, Wei Xiang

TL;DR

Classical MoM relies on a fixed RWG basis to solve the EFIE, limiting adaptivity of current distributions. PhyKAN introduces a Kolmogorov–Arnold Network (KAN) based learnable basis within a dual-branch architecture that embeds Green's function priors to enforce the EFIE residual during training. The approach achieves sub-0.01 reconstruction error across canonical geometries and accurate unsupervised radar cross section (RCS) predictions, with ablations showing the beneficial role of physical priors. This framework provides a physics-consistent, geometry-adaptive bridge between traditional EM solvers and data-driven models for electromagnetic modeling.

Abstract

The Method of Moments (MoM) is constrained by the usage of static, geometry-defined basis functions, such as the Rao-Wilton-Glisson (RWG) basis. This letter reframes electromagnetic modeling around a learnable basis representation rather than solving for the coefficients over a fixed basis. We first show that the RWG basis is essentially a static and piecewise-linear realization of the Kolmogorov-Arnold representation theorem. Inspired by this insight, we propose PhyKAN, a physics-informed Kolmogorov-Arnold Network (KAN) that generalizes RWG into a learnable and adaptive basis family. Derived from the EFIE, PhyKAN integrates a local KAN branch with a global branch embedded with Green's function priors to preserve physical consistency. It is demonstrated that, across canonical geometries, PhyKAN achieves sub-0.01 reconstruction errors as well as accurate, unsupervised radar cross section predictions, offering an interpretable, physics-consistent bridge between classical solvers and modern neural network models for electromagnetic modeling.

Learning the Basis: A Kolmogorov-Arnold Network Approach Embedding Green's Function Priors

TL;DR

Classical MoM relies on a fixed RWG basis to solve the EFIE, limiting adaptivity of current distributions. PhyKAN introduces a Kolmogorov–Arnold Network (KAN) based learnable basis within a dual-branch architecture that embeds Green's function priors to enforce the EFIE residual during training. The approach achieves sub-0.01 reconstruction error across canonical geometries and accurate unsupervised radar cross section (RCS) predictions, with ablations showing the beneficial role of physical priors. This framework provides a physics-consistent, geometry-adaptive bridge between traditional EM solvers and data-driven models for electromagnetic modeling.

Abstract

The Method of Moments (MoM) is constrained by the usage of static, geometry-defined basis functions, such as the Rao-Wilton-Glisson (RWG) basis. This letter reframes electromagnetic modeling around a learnable basis representation rather than solving for the coefficients over a fixed basis. We first show that the RWG basis is essentially a static and piecewise-linear realization of the Kolmogorov-Arnold representation theorem. Inspired by this insight, we propose PhyKAN, a physics-informed Kolmogorov-Arnold Network (KAN) that generalizes RWG into a learnable and adaptive basis family. Derived from the EFIE, PhyKAN integrates a local KAN branch with a global branch embedded with Green's function priors to preserve physical consistency. It is demonstrated that, across canonical geometries, PhyKAN achieves sub-0.01 reconstruction errors as well as accurate, unsupervised radar cross section predictions, offering an interpretable, physics-consistent bridge between classical solvers and modern neural network models for electromagnetic modeling.

Paper Structure

This paper contains 7 sections, 14 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Surface Current as a Learning Problem
  3. Proposed PhyKAN Framework
  4. Neural Network Architecture
  5. Physics-Informed Loss Function
  6. Numerical Results and Discussion
  7. Conclusion

Figures (3)

  • Figure 1: (a) The architecture of the proposed PhyKAN comprising a local branch that encodes geometric features using a conventional KAN and global branch that incorporates Green’s function priors for physically consistent prediction of the induced surface current. (b) The structure of the conventional KAN ref11, where learnable activation functions enable adaptive basis modeling.
  • Figure 2: Reconstruction results for the surfaces of two objects with different variants of the proposed two-branch PhyKAN architecture.
  • Figure 3: Bistatic RCS at $\phi = 0^{\circ}$ for (a) the cube and (b) the cone objects, comparing different PhyKAN variants against the MoM reference.