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Orthogonal Polynomial Kolmogorov–Arnold Networks (OP-KANs) blend the compactness of polynomial bases with neural-network flexibility. We study replacing fully connected networks (FCNs) in Finite-Basis Physics-Informed Networks (FB-PINNs) with OP-KANs for solving PDEs. We use a generalized OP-KAN layer supporting Chebyshev, Legendre, Jacobi, Hermite and any recurrence-defined basis. Our JAX implementation fuses basis evaluation and contraction into one Accelerated Linear Algebra (XLA) kernel, reducing backward floating-point operations (FLOPs) by up to 80% and training time by 50%. Across a ten-problem benchmark, from 1D harmonic oscillators to the 3D Taylor Green vortex, OP-KANs match or surpass FCN accuracy with roughly 10 fewer parameters. On discontinuous or high-frequency tasks, a two-layer Jacobi-KAN reduces mean relative absolute error by 30% and variance by up to 75% with comparable training time. Scheduled training further lowers peak memory by 65%, enabling single-GPU solutions for problems that exceed memory budget when using FCNs. OP-KANs thus emerge as an efficient, interpretable alternative to FCNs in scientific machine learning, especially when memory or data are scarce.

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Orthogonal Polynomial Kolmogorov–Arnold Networks for Physics–Informed PDE Solving

  • 会议论文

  • 发表日期：

    2026年

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  • Sumith Salluri,
  • Pancham Shukla

摘要

Orthogonal Polynomial Kolmogorov–Arnold Networks (OP-KANs) blend the compactness of polynomial bases with neural-network flexibility. We study replacing fully connected networks (FCNs) in Finite-Basis Physics-Informed Networks (FB-PINNs) with OP-KANs for solving PDEs. We use a generalized OP-KAN layer supporting Chebyshev, Legendre, Jacobi, Hermite and any recurrence-defined basis. Our JAX implementation fuses basis evaluation and contraction into one Accelerated Linear Algebra (XLA) kernel, reducing backward floating-point operations (FLOPs) by up to 80% and training time by 50%. Across a ten-problem benchmark, from 1D harmonic oscillators to the 3D Taylor Green vortex, OP-KANs match or surpass FCN accuracy with roughly 10 fewer parameters. On discontinuous or high-frequency tasks, a two-layer Jacobi-KAN reduces mean relative absolute error by 30% and variance by up to 75% with comparable training time. Scheduled training further lowers peak memory by 65%, enabling single-GPU solutions for problems that exceed memory budget when using FCNs. OP-KANs thus emerge as an efficient, interpretable alternative to FCNs in scientific machine learning, especially when memory or data are scarce.