arxiv: 2605.06000 · v1 · submitted 2026-05-07 · 🧮 math.DS

Recognition: unknown

Deep-Koopman-KANDy: Dictionary Discovery for Deep-Koopman Operators with Kolmogorov-Arnold Networks for Dynamics

Erik Bollt, Jeremie Fish, Kevin Slote

Pith reviewed 2026-05-08 04:30 UTC · model grok-4.3

classification 🧮 math.DS

keywords deep koopman operatorskolmogorov-arnold networksdictionary discoverydynamical systemssymbolic regressiondata-driven modelinglorenz systemkoopman theory

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The pith

Deep-Koopman operators with two-layer KAN encoders and decoders allow post-training recovery of symbolic dictionaries via level-set and chain-rule identities.

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

The paper develops Deep-Koopman-KANDy to address the need to precommit to a function library when using methods like EDMD or SINDy for learning dynamical systems. It replaces the encoder and decoder networks in a standard Deep-Koopman model with two-layer Kolmogorov-Arnold Networks, then applies a level-set construction together with a chain-rule gradient identity to reveal the compositional form of the learned observables in any chosen basis after training is complete. The method is demonstrated on the Lorenz system, where it recovers the target polynomial terms, the standard map where it finds a matching Fourier basis, the Ikeda map where it identifies the correct foliation coordinate, and the Arnold cat map where it correctly detects the lack of a sparse finite-dimensional representation. A reader would care because this removes the upfront library choice while still producing interpretable symbolic output from flexible deep models.

Core claim

By training a Deep-Koopman Operator whose encoder and decoder are two-layer KANs, a level-set construction combined with the chain-rule gradient identity exposes the compositional structure of the learned latent observables as a sparse symbolic dictionary in a post-hoc basis. On the Lorenz system this recovers the dictionary {x, y, z, xy, xz} with perfect recall and Jaccard score 0.79 plus or minus 0.06; on the standard map it recovers a low-order Fourier basis; on the Ikeda map a misspecified polynomial readout still recovers the foliation coordinate approximately x squared plus y squared together with a nontrivial outer function; and on the Arnold cat map the method fails to produce a spur

What carries the argument

Two-layer Kolmogorov-Arnold Networks as the encoder and decoder of a Deep-Koopman Operator, together with a level-set construction and chain-rule gradient identity that extracts the compositional structure of the learned observables.

If this is right

On the Lorenz system the method recovers the target dictionary {x, y, z, xy, xz} with perfect recall.
On the Chirikov standard map the method recovers a low-order Fourier basis that matches the known analytical structure.
On the Ikeda map a misspecified polynomial readout still recovers the correct foliation coordinate approximately equal to x squared plus y squared along with a nontrivial outer function.
On the Arnold cat map the method fails to find a sparse closure, consistent with the theoretical impossibility of finite-dimensional Koopman closure.
The approach permits dictionary discovery without requiring the practitioner to select a function library before training.

Where Pith is reading between the lines

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

The post-training readout could be applied to already-trained Deep-Koopman models provided their encoder and decoder admit similar compositional exposure.
The technique offers a route to validate whether a deep Koopman model has captured essential dynamics in an interpretable symbolic form rather than an opaque latent space.
Success on the Ikeda map despite a misspecified readout basis suggests the method may work for systems whose Koopman representations are structured but not polynomial.

Load-bearing premise

The observables learned by the two-layer KAN encoder and decoder possess a compositional structure that the level-set construction and chain-rule gradient identity can reliably extract.

What would settle it

Applying the full pipeline to the Lorenz system and finding that the recovered dictionary misses the terms xy and xz or yields a Jaccard score well below 0.79 would falsify the recovery claim.

Figures

Figures reproduced from arXiv: 2605.06000 by Erik Bollt, Jeremie Fish, Kevin Slote.

**Figure 1.** Figure 1: Deep-Koopman-KANDy architecture. A two-layer KAN encoder ( view at source ↗

**Figure 2.** Figure 2: The hidden activation layer of the KAN defines a manifold in latent space. The level view at source ↗

**Figure 3.** Figure 3: Lorenz attractor and learned Koopman observable. (left) Attractor colored by the learned observable g(x, y, z). (center) Ground-truth and predicted trajectory over ≈ 2τΛ. (right) Level-set bands g(x, y, z) ≈ ci induce a structured foliation of the attractor. Headline result. Deep-Koopman-KANDy with architecture [3, 5, 5] recovers T exactly at the union level: every term in T appears in at least one latent … view at source ↗

**Figure 4.** Figure 4: Deep-Koopman-KANDy on the standard map at view at source ↗

**Figure 5.** Figure 5: Arnold cat map: ground truth, full model, and two ablations. view at source ↗

**Figure 6.** Figure 6: Ikeda map: level-set decomposition with a misspecified polynomial readout dictionary. The Ikeda map admits no sparse polynomial representation, so the degree-3 polynomial Lasso used to recover the inner function g is misspecified by construction. We use it anyway, treating the polynomial as a flexible interpolant rather than a structural prior. (left) Attractor colored by the reconstruction h(g(x)) for ea… view at source ↗

read the original abstract

Symbolic library -- or Koopman dictionary -- selection is a fundamental challenge in data-driven dynamical systems. Extended Dynamic Mode Decomposition (EDMD), Sparse Identification of Nonlinear Dynamics (SINDy), and Kolmogorov--Arnold Networks for Dynamics (KANDy) all require the practitioner to commit to a function library at training time; Deep-Koopman Operators avoid this commitment but produce uninterpretable latent observables. We propose Deep-Koopman-KANDy, a structured approach to post-hoc symbolic dictionary readout that combines Deep-Koopman modeling with Kolmogorov-Arnold Networks for Dynamics (KANDy). The encoder and decoder of a Deep-Koopman Operator are replaced with two-layer Kolmogorov--Arnold Networks (KANs), and a level-set construction together with a chain-rule gradient identity exposes the compositional structure of the learned observables in a basis chosen \emph{after} training. We evaluate the method on the Lorenz system, the Chirikov standard map, the Ikeda map, and the Arnold cat map. On Lorenz it recovers the target dictionary $\{x,y,z,xy,xz\}$ with perfect recall and Jaccard score $0.79\pm0.06$; on the standard map it recovers a low-order Fourier basis matching the analytical structure; on Ikeda -- which has no sparse polynomial representation -- a misspecified polynomial readout still recovers the correct foliation coordinate $g\approx x^2+y^2$ together with a nontrivial outer function; and on the Arnold cat map -- used as a negative control because finite-dimensional Koopman closure is provably impossible -- the method fails to find a sparse closure, as expected.

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.

Desk Editor's Note private letter to a colleague

This paper gives a workable post-training way to pull symbolic dictionaries out of Deep-Koopman models by swapping in two-layer KANs and using level-set gradients, and the benchmarks back the claim on standard systems.

read the letter

The new element is the post-hoc readout: train a Deep-Koopman model with KAN encoders and decoders, then apply a level-set construction and chain-rule identity to expose the learned observables in a user-chosen basis after the fact. This avoids committing to a library at training time, unlike EDMD or SINDy, and differs from plain Deep-Koopman by making the latent functions more readable. The experiments test exactly that on four systems and report clean recoveries where expected: the target polynomial set on Lorenz, a low-order Fourier basis on the standard map, the foliation coordinate on Ikeda even under polynomial misspecification, and failure on the Arnold cat map as a negative control. Those results line up with the central claim and include a sensible check against a case where finite-dimensional closure is impossible. The approach is coherent and the empirical controls do not contradict the method on the reported data. The soft spot is the lack of implementation specifics. The abstract gives no hyperparameter sweeps for KAN widths or activations, no noise robustness checks, and no ablation on how the gradient identity behaves under different random seeds or data lengths. That leaves open whether the recoveries hold up outside these clean benchmarks or depend on careful tuning. The paper is aimed at researchers who already work with Koopman operators or data-driven symbolic methods and want an interpretable lift without upfront library selection. A reader focused on nonlinear dynamics or control would get practical value from the method and the targeted tests. It deserves a serious referee because the idea is self-contained, the experiments are well-chosen, and the reported outcomes show no internal contradictions. I would send it to review.

Referee Report

1 major / 2 minor

Summary. The paper proposes Deep-Koopman-KANDy, which augments Deep-Koopman operators by replacing the encoder/decoder with two-layer Kolmogorov-Arnold Networks (KANs). A level-set construction and chain-rule gradient identity are then applied to expose the compositional structure of the learned observables, enabling post-hoc recovery of a sparse symbolic dictionary in a user-chosen basis. Experiments on the Lorenz system recover the target dictionary {x,y,z,xy,xz} with perfect recall and Jaccard score 0.79±0.06; the standard map yields a low-order Fourier basis; the Ikeda map recovers the foliation coordinate g≈x²+y² even under polynomial misspecification; and the Arnold cat map (negative control) fails to produce a sparse closure, as expected.

Significance. If the recoveries hold under fuller scrutiny, the work offers a practical bridge between the flexibility of latent Deep-Koopman models and the interpretability of dictionary-based methods such as EDMD, SINDy, and KANDy. The benchmark results, including exact recall on Lorenz, appropriate Fourier recovery on the standard map, recovery of the correct foliation coordinate on Ikeda despite basis misspecification, and expected failure on the Arnold cat map, provide concrete evidence that the KAN-based compositional readout can succeed where purely data-driven latent spaces remain opaque. The explicit negative control and the post-training nature of the dictionary selection are particular strengths.

major comments (1)

[Abstract and Experiments] The abstract and experimental sections report quantitative recovery metrics (perfect recall, Jaccard 0.79±0.06 on Lorenz; correct foliation coordinate on Ikeda) but omit full implementation details, hyperparameter sensitivity studies, and error analysis. These omissions are load-bearing for the central claim that the level-set/chain-rule construction reliably exposes the target dictionary across systems.

minor comments (2)

[Methods] The precise mathematical form of the level-set construction and the chain-rule gradient identity should be stated explicitly (with equation numbers) in the methods section to allow independent verification.
[Experiments] Clarify the exact user-chosen basis and the stopping criterion for the post-hoc readout in each experiment; the current description leaves the selection procedure somewhat implicit.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their positive summary, recognition of the method's strengths, and recommendation for minor revision. We address the major comment below and will make the requested additions.

read point-by-point responses

Referee: [Abstract and Experiments] The abstract and experimental sections report quantitative recovery metrics (perfect recall, Jaccard 0.79±0.06 on Lorenz; correct foliation coordinate on Ikeda) but omit full implementation details, hyperparameter sensitivity studies, and error analysis. These omissions are load-bearing for the central claim that the level-set/chain-rule construction reliably exposes the target dictionary across systems.

Authors: We agree that the current presentation would benefit from expanded details to support the reliability claim. In the revised version we will add: complete implementation specifications (network widths, activation choices, training schedules, and basis libraries); hyperparameter sensitivity results across key parameters such as regularization strength, number of KAN grid points, and dictionary size; and error analysis including run-to-run variance, failure-mode identification, and quantitative metrics on all four systems. These will appear in an expanded Experiments section together with a new supplementary note on reproducibility. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper introduces a post-training readout procedure that replaces the encoder/decoder of a Deep-Koopman model with two-layer KANs and then applies a level-set construction plus chain-rule identity to recover a symbolic dictionary in a user-chosen basis. All reported results consist of empirical recovery of externally known analytical dictionaries (Lorenz polynomial set, low-order Fourier basis on the standard map, foliation coordinate on Ikeda, and expected failure on the Arnold cat map). These recoveries are measured against independent benchmark systems and known closed-form structures rather than against quantities fitted inside the same optimization; no step reduces a claimed prediction to a fitted input by construction, and no load-bearing uniqueness theorem or ansatz is imported solely via self-citation. The derivation therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The claim rests on standard neural network differentiability and dynamical systems theory with no new postulated entities. Architecture choices act as free parameters but are not central to the recovery claim.

free parameters (1)

KAN layer widths and activation choices
Two-layer KAN architecture hyperparameters selected by the user that determine the expressivity of the readout.

axioms (1)

standard math The chain rule applies to the composed functions realized by the KAN encoder/decoder.
Invoked to obtain the gradient identity that exposes the compositional structure of the observables.

pith-pipeline@v0.9.0 · 5605 in / 1273 out tokens · 49475 ms · 2026-05-08T04:30:05.131497+00:00 · methodology

discussion (0)

Reference graph

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This yields a one-dimensional dataset{(g j, q j)}N j=1 whereg j =g k(xj)andq j ≈h ′ k(gj)

points with∥∇g k(xj)∥2 below the 5th percentile are discarded (ill-conditioned denomina- tors), and outliers inh ′ k beyond3×IQRare removed. This yields a one-dimensional dataset{(g j, q j)}N j=1 whereg j =g k(xj)andq j ≈h ′ k(gj). Step 3: Integration and reconstruction.We fit a polynomial ˆh′ k(ζ) = Pp l=0 bl (ζ−¯gk)l to the binned medians of{(g j, qj)}(...