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arxiv: 2606.19984 · v3 · pith:3UKF6R2Xnew · submitted 2026-06-18 · 💻 cs.LG

Kolmogorov-Arnold Reservoir Computing

Juntian Huang , Jurgen Kurths , Ying Tang This is my paper

Pith reviewed 2026-06-30 10:49 UTC · model grok-4.3

classification 💻 cs.LG
keywords reservoir computingKolmogorov-Arnold networksdynamical systemsbasis function expansionsforecastingpartial differential equationsmachine learning
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The pith

Kolmogorov-Arnold Reservoir Computing replaces reservoirs with basis-function expansions to combine KAN expressivity with closed-form reservoir training.

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

The paper develops Kolmogorov-Arnold Reservoir Computing by substituting conventional reservoirs with explicit basis-function expansions drawn from the Kolmogorov-Arnold representation theorem. This substitution is shown to preserve the expressive capacity claimed for Kolmogorov-Arnold networks while enabling the efficient closed-form training that defines reservoir computing. The result addresses limitations of standard reservoir methods in long-range dependencies and of next-generation variants in exploding feature dimensions. If the substitution holds, high-fidelity forecasting of dynamical systems becomes possible at lower cost, with demonstrated gains on partial differential equation benchmarks and compatibility with diffusion models for generation tasks.

Core claim

KARC 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 text-to-image generation.

What carries the argument

Explicit basis-function expansions inspired by the Kolmogorov-Arnold representation theorem, used in place of recurrent reservoirs to carry the network's representational capacity.

If this is right

  • At comparable cost, KARC outperforms existing reservoir computing methods on challenging benchmarks including partial differential equations.
  • It can be integrated with generative diffusion models for text-to-image generation.
  • This establishes a principled bridge between reservoir computing and KANs, enabling efficient and high-fidelity dynamical system forecasting.

Where Pith is reading between the lines

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

  • The same basis expansions could be tested as drop-in replacements in other recurrent architectures that currently rely on random or trained reservoirs.
  • Because training remains closed-form, KARC may scale more readily to very long time series than gradient-based KAN variants.
  • Integration with diffusion models opens the possibility of using KARC as a temporal prior inside generative pipelines for physics-informed image or video synthesis.

Load-bearing premise

The basis-function expansions inspired by the Kolmogorov-Arnold representation theorem can be substituted for conventional reservoirs without loss of the expressive capacity that KANs are claimed to possess.

What would settle it

A benchmark dynamical system on which KARC matches or exceeds full KAN performance in accuracy yet requires substantially more parameters or training time than standard reservoir methods would falsify the efficiency-plus-capacity claim.

Figures

Figures reproduced from arXiv: 2606.19984 by Juntian Huang, Jurgen 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 trainable 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 text-to-image generation. This work thus establishes a principled bridge between reservoir computing and KANs, enabling efficient and high-fidelity dynamical system forecasting.

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

1 major / 0 minor

Summary. The manuscript introduces Kolmogorov-Arnold Reservoir Computing (KARC), which substitutes explicit basis-function expansions inspired by the Kolmogorov-Arnold representation theorem for conventional reservoirs in reservoir computing. It claims KARC is a lightweight design of Kolmogorov-Arnold networks (KANs) that preserves their expressive capacity while enabling efficient closed-form training, outperforms existing RC methods on dynamical-system benchmarks including PDEs at comparable cost, and integrates with generative diffusion models.

Significance. If the central claims of preserved expressive capacity and rigorous closed-form training hold, the work would establish a useful bridge between reservoir computing and KANs, offering a principled route to efficient, high-fidelity forecasting of dynamical systems with potential impact in scientific machine learning.

major comments (1)
  1. Abstract: the claim of a 'rigorous' demonstration that KARC preserves the expressive capacity of KANs is load-bearing for the central contribution, yet the provided material supplies no derivation, theorem statement, or proof outline; without explicit verification of this step the superiority and preservation assertions cannot be assessed.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their review and for highlighting this critical point about the central claim. We address the comment below and commit to a revision that supplies the missing supporting material.

read point-by-point responses

  1. Referee: Abstract: the claim of a 'rigorous' demonstration that KARC preserves the expressive capacity of KANs is load-bearing for the central contribution, yet the provided material supplies no derivation, theorem statement, or proof outline; without explicit verification of this step the superiority and preservation assertions cannot be assessed.

    Authors: We agree the abstract's assertion requires explicit verification. The current manuscript states the claim at a high level in Section 3 but does not include a formal theorem or proof outline. In the revised manuscript we will insert a new subsection (Section 3.2) containing: (i) the theorem statement that KARC constitutes a lightweight KAN design preserving expressive capacity via finite Kolmogorov-Arnold basis expansions in the reservoir, and (ii) a concise proof sketch relying on the universal approximation property of KANs together with the closed-form linear readout. This addition will allow direct assessment of the preservation argument and the closed-form training claim. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's central claim is that KARC replaces conventional reservoirs with explicit basis-function expansions drawn from the Kolmogorov-Arnold theorem, yielding a lightweight KAN variant that supports closed-form training while preserving expressive capacity. This construction is presented as an independent design choice whose equivalence to KANs is asserted via a rigorous demonstration rather than by redefining the target quantity in terms of itself or by fitting parameters to the very quantities being predicted. No self-citation chain is invoked as the sole justification for a uniqueness theorem, no fitted input is relabeled as a prediction, and no ansatz is smuggled through prior self-work. The derivation therefore remains self-contained against external benchmarks and does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The design rests on the Kolmogorov-Arnold representation theorem as a standard mathematical fact and introduces the KARC architecture itself; no free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • standard math The Kolmogorov-Arnold representation theorem applies to the explicit basis-function expansions used in place of reservoirs.
    Invoked to justify the substitution that preserves expressive capacity.

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