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KANO: Kolmogorov-Arnold Neural Operator

Jin Lee, Ziming Liu, Xinling Yu, Yixuan Wang, Haewon Jeong, Murphy Yuezhen Niu, Zheng Zhang

TL;DR

The paper tackles learning operators for position‑dependent dynamics (variable‑coefficient PDEs) with interpretability. It proposes the Kolmogorov‑Arnold Neural Operator (KANO), a dual‑domain operator that jointly parameterizes in spatial and spectral bases via a learnable symbol $\boldsymbol{p}(\mathbf{x},\bm{\xi})$ and Kolmogorov–Arnold Networks to form a pseudo‑differential operator, enabling symbolic interpretability and robust generalization. Theoretical analysis shows FNO’s pure‑spectral bottleneck and curse of dimensionality for spectrally dense operators, while KANO’s dual‑domain expressivity yields input‑independent, compact representations and provable approximation guarantees. Empirically, KANO outperforms FNO across synthetic position‑dependent benchmarks and a long‑horizon quantum dynamics task, recovering ground‑truth coefficients to four decimals and achieving state fidelities on the order of $10^{-6}$ with orders of magnitude fewer parameters, while providing symbolic recovery of the learned operators. This work shifts operator learning toward interpretable, symbolically retrievable scientific models with strong out‑of‑distribution generalization.

Abstract

We introduce Kolmogorov--Arnold Neural Operator (KANO), a dual-domain neural operator jointly parameterized by both spectral and spatial bases with intrinsic symbolic interpretability. We theoretically demonstrate that KANO overcomes the pure-spectral bottleneck of Fourier Neural Operator (FNO): KANO remains expressive over generic position-dependent dynamics (variable coefficient PDEs) for any physical input, whereas FNO stays practical only for spectrally sparse operators and strictly imposes a fast-decaying input Fourier tail. We verify our claims empirically on position-dependent differential operators, for which KANO robustly generalizes but FNO fails to. In the quantum Hamiltonian learning benchmark, KANO reconstructs ground-truth Hamiltonians in closed-form symbolic representations accurate to the fourth decimal place in coefficients and attains $\approx 6\times10^{-6}$ state infidelity from projective measurement data, substantially outperforming that of the FNO trained with ideal full wave function data, $\approx 1.5\times10^{-2}$, by orders of magnitude.

KANO: Kolmogorov-Arnold Neural Operator

TL;DR

The paper tackles learning operators for position‑dependent dynamics (variable‑coefficient PDEs) with interpretability. It proposes the Kolmogorov‑Arnold Neural Operator (KANO), a dual‑domain operator that jointly parameterizes in spatial and spectral bases via a learnable symbol and Kolmogorov–Arnold Networks to form a pseudo‑differential operator, enabling symbolic interpretability and robust generalization. Theoretical analysis shows FNO’s pure‑spectral bottleneck and curse of dimensionality for spectrally dense operators, while KANO’s dual‑domain expressivity yields input‑independent, compact representations and provable approximation guarantees. Empirically, KANO outperforms FNO across synthetic position‑dependent benchmarks and a long‑horizon quantum dynamics task, recovering ground‑truth coefficients to four decimals and achieving state fidelities on the order of with orders of magnitude fewer parameters, while providing symbolic recovery of the learned operators. This work shifts operator learning toward interpretable, symbolically retrievable scientific models with strong out‑of‑distribution generalization.

Abstract

We introduce Kolmogorov--Arnold Neural Operator (KANO), a dual-domain neural operator jointly parameterized by both spectral and spatial bases with intrinsic symbolic interpretability. We theoretically demonstrate that KANO overcomes the pure-spectral bottleneck of Fourier Neural Operator (FNO): KANO remains expressive over generic position-dependent dynamics (variable coefficient PDEs) for any physical input, whereas FNO stays practical only for spectrally sparse operators and strictly imposes a fast-decaying input Fourier tail. We verify our claims empirically on position-dependent differential operators, for which KANO robustly generalizes but FNO fails to. In the quantum Hamiltonian learning benchmark, KANO reconstructs ground-truth Hamiltonians in closed-form symbolic representations accurate to the fourth decimal place in coefficients and attains state infidelity from projective measurement data, substantially outperforming that of the FNO trained with ideal full wave function data, , by orders of magnitude.

Paper Structure

This paper contains 62 sections, 5 theorems, 96 equations, 6 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Operator Learning and Fourier Neural Operator
  4. Fourier Neural Operator (FNO).
  5. Kolmogorov--Arnold Network
  6. Theoretical Analysis on FNO's Pure-Spectral Bottleneck
  7. The Pure-Spectral Bottleneck of FNO
  8. FNO suffers Curse of Dimensionality on Position-Dependent Dynamics
  9. Reviewing Remark 21 & 22 of uatfno
  10. Kolmogorov–Arnold Neural Operator
  11. KANO Architecture
  12. Symbolic Interpretability of KANO.
  13. KANO's Dual-Domain Expressivity
  14. KANO practically has no input constraint.
  15. Experimental Results
  16. ...and 47 more sections

Key Result

Lemma 1

A single position operator, spatial multiplier by $x$, induces algebraic decay in output Fourier tail when the input is band-limited.

Figures (6)

  • Figure 1: (a)$\mathbfcal{L}_{\tiny{\text{FNO}}}$ architecture. (b)$\mathbfcal{L}_{\tiny{\text{KANO}}}$ architecture.
  • Figure 2: Loss test results. (a)$\mathbfcal{G}_1$(b)$\mathbfcal{G}_2$(c)$\mathbfcal{G}_3$. Note the logarithmic scale.
  • Figure 3: Interpolation test results. (a)$\mathbfcal{G}_1$(b)$\mathbfcal{G}_2$(c)$\mathbfcal{G}_3$.
  • Figure 4: (a)$\boldsymbol{p}(x,\xi)$ of $\mathbfcal{G}_1$. The middle edge does not contribute to the output. (b)$\boldsymbol{p}(x,\xi)$ of $\mathbfcal{G}_2$. (c)$\boldsymbol{p}(x,\xi)$ of $\mathbfcal{G}_3$. (d)$\bm{\Phi}$ of $\mathbfcal{G}_3$. Edge of the residual in (d) looks linear, so we compared two scenarios, linear and cubic, which the latter achieved lower loss and better generalization.
  • Figure 5: pos&mom type training results. (a) Structure of the potential $w(x)$(b)$\boldsymbol{p}(x,\xi)$ of DW. (c)$\boldsymbol{p}(x,\xi)$ of NLSE. (d)$\varphi(\lvert\cdot\rvert,\angle\cdot)$ of NLSE. Potential $w(x)$ structure is clearly reconstructed.
  • ...and 1 more figures

Theorems & Definitions (16)

  • Lemma 1: Position operator elongates Fourier tail
  • proof : Sketch of proof.
  • Theorem 1: Curse of dimensionality on position operators
  • proof : Sketch of proof.
  • Remark 1
  • Remark 2: Complexity analyses of KANO
  • Theorem 2: KANO stays practical for smooth symbol
  • proof : Sketch of proof.
  • Corollary 1: KANO is practical for generic position-dependent dynamics
  • Remark 3: Scope of KANO
  • ...and 6 more