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arxiv: 2606.19984 · v4 · pith:3UKF6R2Xnew · submitted 2026-06-18 · 💻 cs.LG

Kolmogorov-Arnold Reservoir Computing

Juntian Huang , J\"urgen Kurths , Ying Tang This is my paper

Pith reviewed 2026-07-03 23:37 UTC · model grok-4.3

classification 💻 cs.LG
keywords reservoir computingKolmogorov-Arnold networksdynamical systemspartial differential equationsbasis function expansionsclosed-form trainingforecasting
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The pith

Kolmogorov-Arnold Reservoir Computing replaces fixed reservoirs with explicit basis expansions to enable closed-form training while preserving expressive capacity for dynamical systems.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces KARC by substituting traditional reservoirs with basis-function expansions drawn from the Kolmogorov-Arnold representation theorem. This yields a lightweight variant of Kolmogorov-Arnold networks that supports the efficient closed-form training typical of reservoir computing. The design targets long-range dependencies in dynamical systems without recurrence or rapid feature expansion. A sympathetic reader would care if this delivers stronger forecasting on partial differential equations at costs comparable to existing methods while opening links to generative modeling.

Core claim

KARC is a lightweight design of Kolmogorov-Arnold networks that preserves their potential expressive capacity while admitting efficient closed-form training of reservoir computing and outperforming existing methods on PDE benchmarks at comparable cost.

What carries the argument

Explicit basis-function expansions inspired by the Kolmogorov-Arnold representation theorem, which replace reservoirs to support closed-form training and maintain expressive power.

If this is right

  • Outperforms existing reservoir computing methods on challenging benchmarks including partial differential equations at comparable cost.
  • Can be integrated with generative diffusion models for facilitating text-to-image generation.
  • Establishes a principled bridge between reservoir computing and Kolmogorov-Arnold networks.
  • Yields a unified framework for efficient dynamical forecasting and generative modeling.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The closed-form training could simplify scaling to longer time horizons compared with methods that require recurrence or feature explosion.
  • Integration with diffusion models may support hybrid models that enforce physical constraints in generative tasks beyond images.
  • The approach might reduce hyperparameter sensitivity that affects conventional reservoir computing.
  • Application to real-world time series from climate or fluid dynamics could test whether the basis expansions generalize beyond the reported PDE benchmarks.

Load-bearing premise

The selected basis-function expansions will capture long-range dependencies in the target dynamical systems without recurrence or rapid feature growth.

What would settle it

Showing that KARC fails to outperform standard reservoir computing or next-generation reservoir computing on PDE forecasting tasks at comparable computational cost would falsify the central performance claim.

Figures

Figures reproduced from arXiv: 2606.19984 by Juntian Huang, J\"urgen Kurths, Ying Tang.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 1
Figure 1. Figure 1: Effect of low-rank readout factorization on KARC forecasting for the Kuramoto-Sivashinsky equa￾tion. Forecasting results are shown for the Kuramoto-Sivashinsky equation with domain size L=22, using KARC with Fourier basis functions. The top row shows the ground-truth spatiotemporal field. The following rows show KARC forecasts with different low-rank readout ratios, where ratio=dl/dh, dl is the low-rank re… view at source ↗
Figure 1
Figure 1. Figure 1: Effect of low-rank readout factorization on KARC forecasting for the Kuramoto-Sivashinsky equa￾tion. Forecasting results are shown for the Kuramoto-Sivashinsky equation with domain size L=22, using KARC with Fourier basis functions. The top row shows the ground-truth spatiotemporal field. The following rows show KARC forecasts with different low-rank readout ratios, where ratio=dl/dh, dl is the low-rank re… view at source ↗
Figure 2
Figure 2. Figure 2: Forecasting performance of RC with different reservoir sizes on the double-scroll system. Each row corresponds to a different reservoir dimension dh, and each column represents one state variable of the double-scroll system. Blue and gray curves denote the predicted and reference trajectories, respectively. 4 2 0 order = 2 1e6 V1 0 1 2 1e6 V2 0 2 4 1e6 I 2 0 2 order = 3 1 0 1 2 0 2 2 0 2 order = 4 1 0 1 2 … view at source ↗
Figure 3
Figure 3. Figure 3: Forecasting performance of NG-RC with different orders on the double-scroll system. Rows correspond to different orders and columns represent the three state variables V1, V2 and I. Yellow and orange curves denote the predicted and reference trajectories, respectively. without providing a clear improvement over the third-order model, we adopt the third-order NG-RC model in the main-text comparison as a mor… view at source ↗
Figure 3
Figure 3. Figure 3: Forecasting performance of random MLP features with different feature dimensions on the Kuramoto-Sivashinsky equation. The top row shows the reference spatiotemporal field for the Kuramoto-Sivashinsky equation with domain size L = 22. The remaining rows show forecasts generated using random MLP features with different feature dimensions dh, together with the corresponding pointwise prediction errors. Dimen… view at source ↗
Figure 4
Figure 4. Figure 4: Forecasting performance of KARC with different bases on the double-scroll system. Rows correspond to different basis functions, including Fourier, B-spline and Chebyshev bases, and columns represent the three state variables V1, V2 and I. Blue and orange curves denote the predicted and reference trajectories, respectively. indicate that the choice of basis function plays an important role in KARC, and that… view at source ↗
Figure 4
Figure 4. Figure 4: Forecasting performance of RC with different reservoir dimensions on the double-scroll system. Each row corresponds to a different reservoir dimension dh, and each column represents one state variable of the double-scroll system. The trajectory colors follow the same convention as in the main-text double-scroll experiment. We first examine the sensitivity of RC to reservoir dimension on the double-scroll s… view at source ↗
Figure 5
Figure 5. Figure 5: Forecasting performance of RC with different reservoir sizes on the Kuramoto-Sivashinsky equa￾tion. The top-left panel shows the reference spatiotemporal field, and the subsequent rows show RC forecasts obtained with different reservoir dimensions dh = 1000, 3000, 8000 and 12000. The right column reports the corresponding pointwise prediction errors. We next investigate whether increasing the reservoir dim… view at source ↗
Figure 5
Figure 5. Figure 5: Forecasting performance of NG-RC with different orders on the double-scroll system. Rows correspond to different orders and columns represent the three state variables V1, V2 and I. The trajectory colors follow the same convention as in the main-text double-scroll experiment. 2 0 2 Fourier V1 1 0 1 V2 2 0 2 I 2 0 2 B-spline 1 0 1 2 0 2 0 5 10 15 20 25 max t 2 0 2 Chebyshev 0 5 10 15 20 25 max t 1 0 1 0 5 1… view at source ↗
Figure 6
Figure 6. Figure 6: Forecasting performance of VolterraRC with different kernel parameters λ on the Kuramoto￾Sivashinsky equation. The top row shows the reference spatiotemporal field, and the subsequent rows show the forecasts and corresponding pointwise errors for different λ. in the figure, however, increasing λ does not monotonically improve the forecasting performance. For relatively small or moderate values of λ, Volter… view at source ↗
Figure 6
Figure 6. Figure 6: Forecasting performance of KARC with different bases on the double-scroll system. Rows correspond to different basis functions, including Fourier, B-spline and Chebyshev bases, and columns represent the three state variables V1, V2 and I. The trajectory colors follow the same convention as in the main-text double-scroll experiment. to the oscillatory nature of the double-scroll dynamics [PITH_FULL_IMAGE:f… view at source ↗
Figure 7
Figure 7. Figure 7: Forecasting performance of KARC with different bases on the Kuramoto-Sivashinsky equation. The top-left panel shows the reference spatiotemporal field, and the subsequent rows show KARC forecasts obtained using Fourier, B-spline and Chebyshev bases. The right column presents the corresponding pointwise prediction errors. The results compare how the choice of basis function affects the spatiotemporal foreca… view at source ↗
Figure 7
Figure 7. Figure 7: Forecasting performance of RC with different reservoir dimensions on the Kuramoto-Sivashinsky equation. The top-left panel shows the reference spatiotemporal field, and the subsequent rows show RC forecasts obtained with different reservoir dimensions dh = 1000, 3000, 8000 and 12000. The right column reports the corresponding pointwise prediction errors. We next extend the sensitivity analysis to the Kuram… view at source ↗
Figure 8
Figure 8. Figure 8: Forecasting performance of RC with different reservoir sizes on the shallow water equations. The first row shows the reference solution fields from step 10 to step 100, and the remaining rows show the corresponding prediction errors of RC with reservoir dimensions dh = 3000, dh = 5000, and dh = 8000. Truth step=10 step=20 step=30 step=40 step=50 step=60 step=70 step=80 step=90 step=100 = 0.1 0 = 0.5 0 = 0.… view at source ↗
Figure 8
Figure 8. Figure 8: Forecasting performance of VolterraRC with different kernel parameters α on the Kuramoto￾Sivashinsky equation. The top row shows the reference spatiotemporal field, and the subsequent rows show the forecasts and corresponding pointwise errors for different α. Finally, [PITH_FULL_IMAGE:figures/full_fig_p035_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Forecasting performance of VolterraRC with different kernel parameters λ on the shallow water equations. The first row shows the reference solution fields from step 10 to step 100. The remaining rows show the corresponding pointwise prediction errors of VolterraRC with different values of the kernel memory parameter λ. We also evaluate RC with different reservoir dimensions on the shallow water equations. … view at source ↗
Figure 9
Figure 9. Figure 9: Forecasting performance of KARC with different bases on the Kuramoto-Sivashinsky equation. The top-left panel shows the reference spatiotemporal field, and the subsequent rows show KARC forecasts obtained using Fourier, B-spline and Chebyshev bases. The right column presents the corresponding pointwise prediction errors. The results compare how the choice of basis function affects the spatiotemporal foreca… view at source ↗
Figure 10
Figure 10. Figure 10: Forecasting performance of KARC with different bases on the shallow water equation. The first row shows the reference solution fields from step 10 to step 100, and the remaining rows show the corresponding prediction errors of KARC using Fourier, B-spline, and Chebyshev basis functions. equations. Different from the results on the Kuramoto-Sivashinsky equation, increasing λ improves the forecasting perfor… view at source ↗
Figure 10
Figure 10. Figure 10: Forecasting performance of RC with different reservoir dimensions on the shallow water equations. The first row shows the reference solution fields from step 10 to step 100, and the remaining rows show the corresponding prediction errors of RC with reservoir dimensions dh = 3000, dh = 5000, and dh = 8000. Truth step=10 step=20 step=30 step=40 step=50 step=60 step=70 step=80 step=90 step=100 = 0.1 0 = 0.5 … view at source ↗
Figure 11
Figure 11. Figure 11: Additional text-to-image generation examples. Each row corresponds to a text prompt. Each column shows the results produced by FLUX.1-dev (baseline), Spectrum, KARC with Fourier bases, and KARC with B-spline bases, respectively. We further provide additional text-to-image generation examples using FLUX.1-dev [6] to evaluate the visual quality of KARC-based acceleration. We compare the original spectrum [7… view at source ↗
Figure 11
Figure 11. Figure 11: Forecasting performance of VolterraRC with different kernel parameters α on the shallow water equations. The first row shows the reference solution fields from step 10 to step 100. The remaining rows show the corresponding pointwise prediction errors of VolterraRC with different values of the kernel memory parameter α. We next turn to the shallow water equations, a benchmark that embodies additional physi… view at source ↗
Figure 12
Figure 12. Figure 12: Forecasting performance of RC, NG-RC, and KARC on the lorenz63 system. Rows correspond to different models, and columns correspond to the three state variables of the lorenz63 system. Λmax denotes the largest Lyapunov exponent, and one unit on the horizontal axis represents one Lyapunov time. In the main text, the double-scroll system was used as a representative low-dimensional chaotic ODE benchmark to a… view at source ↗
Figure 12
Figure 12. Figure 12: Forecasting performance of KARC with different bases on the shallow water equation. The first row shows the reference solution fields from step 10 to step 100, and the remaining rows show the corresponding prediction errors of KARC using Fourier, B-spline, and Chebyshev basis functions. and 0.90, the accumulated error is noticeably reduced, and the model better preserves the spatiotemporal evolution of th… view at source ↗
Figure 13
Figure 13. Figure 13: (a) Schematic illustration of the hybrid KARC+FNO-2D architecture. (b) Qualitative comparison between KARC-only and KARC+FNO-2D predictions at selected forecasting times. The rows show the 64 × 64 ground truth, the KARC-only prediction computed at 32 × 32 resolution and upsampled to 64 × 64, and the final 64 × 64 KARC+FNO-2D prediction, respectively. (c) Forecasting errors on the Navier-Stokes equation.Th… view at source ↗
Figure 14
Figure 14. Figure 14: Forecasting performance of RC, NG-RC, and KARC on the Lorenz63 system. Rows correspond to different models, and columns correspond to the three state variables of the Lorenz63 system. Λmax denotes the largest Lyapunov exponent, and one unit on the horizontal axis represents one Lyapunov time. In the main text, the double-scroll system was used as a representative low-dimensional chaotic ODE benchmark to a… view at source ↗
Figure 15
Figure 15. Figure 15: (a) Schematic illustration of the hybrid KARC+FNO-2D architecture. (b) Qualitative comparison between KARC-only and KARC+FNO-2D predictions at selected forecasting times. The rows show the 64 × 64 ground truth, the KARC-only prediction computed at 32 × 32 resolution and upsampled to 64 × 64, and the final 64 × 64 KARC+FNO-2D prediction, respectively. (c) Forecasting errors on the Navier-Stokes equation.Th… view at source ↗
read the original abstract

Reservoir computing offers a lightweight framework for forecasting dynamical systems but may struggle to capture long-range dependencies due to limited representational capacity. Conventional reservoir computing recurrently uses fixed reservoirs with hyperparameter sensitivity, while the next generation reservoir computing removes recurrence at the cost of rapidly growing feature dimensions. Here, we develop Kolmogorov-Arnold Reservoir Computing (KARC), which replaces reservoirs with explicit basis-function expansions inspired by the Kolmogorov-Arnold representation theorem. We rigorously show that KARC is a lightweight design of Kolmogorov-Arnold networks (KANs), preserving the potential expressive capacity of KANs while admitting efficient closed-form training of reservoir computing. At comparable cost, KARC outperforms existing reservoir computing methods on challenging benchmarks including partial differential equations. It can also be integrated with generative diffusion models for facilitating text-to-image generation. This work thus establishes a principled bridge between reservoir computing and KANs, yielding a unified framework for efficient dynamical forecasting and generative modeling.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces Kolmogorov-Arnold Reservoir Computing (KARC), which replaces recurrent reservoirs with explicit univariate basis-function expansions drawn from the Kolmogorov-Arnold representation theorem. It claims to rigorously establish that KARC is a lightweight realization of Kolmogorov-Arnold networks (KANs) that retains their expressive capacity while permitting closed-form linear training, and that it outperforms standard and next-generation reservoir computing on PDE forecasting benchmarks at comparable cost; an additional integration with diffusion models for text-to-image generation is mentioned.

Significance. If the central claims hold, the work supplies a principled bridge between reservoir computing and KANs, yielding a framework that avoids both recurrence and rapid feature-dimension growth while preserving the potential for high expressivity in dynamical forecasting and generative tasks.

major comments (2)
  1. [Abstract] The central claim that fixed explicit basis expansions capture long-range dependencies in PDEs without recurrence or dimension growth is load-bearing for the outperformance assertion, yet the manuscript supplies no scaling analysis versus prediction horizon, versus basis size, or versus system stiffness to test this assumption (see skeptic note on the weakest assumption).
  2. [Abstract] The abstract asserts a 'rigorous' proof that KARC is a lightweight KAN design admitting closed-form training, but provides neither the relevant equations, the explicit form of the basis expansions, nor the regularization procedure; without these the soundness of the closed-form claim cannot be assessed.
minor comments (2)
  1. Dataset details, error bars, and hyperparameter selection protocol for the PDE benchmarks are absent from the abstract and should be supplied with the experimental results.
  2. The integration with generative diffusion models is stated without any description of how the KARC component is inserted or evaluated; this section requires at least a schematic and quantitative results.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments, which help improve the clarity and rigor of our work. We address each major comment in detail below.

read point-by-point responses

  1. Referee: [Abstract] The central claim that fixed explicit basis expansions capture long-range dependencies in PDEs without recurrence or dimension growth is load-bearing for the outperformance assertion, yet the manuscript supplies no scaling analysis versus prediction horizon, versus basis size, or versus system stiffness to test this assumption (see skeptic note on the weakest assumption).

    Authors: We agree that explicit scaling analyses would strengthen the manuscript. Our current experiments demonstrate outperformance on PDE benchmarks with different prediction horizons and system types, but we did not include dedicated scaling studies. In the revised version, we will add scaling plots versus prediction horizon, basis size, and stiffness parameters to directly test the assumption. revision: yes

  2. Referee: [Abstract] The abstract asserts a 'rigorous' proof that KARC is a lightweight KAN design admitting closed-form training, but provides neither the relevant equations, the explicit form of the basis expansions, nor the regularization procedure; without these the soundness of the closed-form claim cannot be assessed.

    Authors: The abstract is a high-level summary. The rigorous proof that KARC is a lightweight KAN, the explicit univariate basis expansions (based on the Kolmogorov-Arnold theorem), and the closed-form training via regularized linear regression are provided in detail in Sections 2-4 of the manuscript, including the relevant equations for the basis functions and the training procedure. To address the concern, we will revise the abstract to include a brief reference to these sections and key elements of the formulation. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation is self-contained via explicit basis replacement and linear readout.

full rationale

The paper defines KARC by replacing the reservoir with fixed univariate basis expansions drawn from the Kolmogorov-Arnold theorem, then shows that the resulting model admits a closed-form linear solution for the readout weights exactly as in standard reservoir computing. This reduction follows directly from the linearity of the final layer with respect to the chosen basis coefficients and does not rely on fitting any target quantity to itself or on self-citation chains that presuppose the result. The expressive-capacity claim is an existence statement inherited from the representation theorem rather than a fitted prediction, and no step renames an empirical pattern or imports a uniqueness result from the authors' prior work as an external fact. The derivation therefore remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The abstract invokes the Kolmogorov-Arnold representation theorem as the source of the basis expansions and assumes that a finite truncation of those expansions suffices for the target tasks; no free parameters, additional axioms, or invented entities are named.

axioms (1)
  • standard math Kolmogorov-Arnold representation theorem guarantees that the required multivariate functions can be expressed as finite sums of univariate functions.
    Invoked to justify the replacement of random reservoirs by explicit basis expansions.

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