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

NeurIPS 2026 · Probabilistic and scientific modeling › Dynamical systems · approach: State-space inference for stochastic systems · 0 citations (OpenAlex)

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Replaces encoder/decoder of a Deep-Koopman operator with two-layer KANs and uses a post-hoc level-set/gradient-identity analysis to read out a symbolic observable dictionary in a basis chosen after training.

Problem

Koopman/dictionary selection for data-driven dynamics requires committing to a function library before training (EDMD, SINDy, KANDy), while Deep-Koopman autoencoders avoid this but yield uninterpretable latent observables.

Approach

Two-layer KAN encoder/decoder in a Deep-Koopman model with a stable generator K = Omega - L^T L; post-hoc level-set decomposition f = h∘g with h' recovered by the gradient identity h'(g) = ∇f·∇g / ||∇g||^2, so the readout basis is chosen after training.

Key results

• On Lorenz, pruned Deep-Koopman-KANDy reaches per-seed Jaccard 0.79 ± 0.06 with union recall 1.0 over 5 seeds, versus 0.60 for EDMD-DL and SINDy-AE.
• On Ikeda with a misspecified degree-3 polynomial readout, four latent coordinates reach R^2(h∘g) in [0.92, 0.99].
• On the Arnold cat map (negative control), one-step MSE is about 3e-5 but the prediction horizon extends only about 2-4 steps, consistent with the Lyapunov limit.

Surprising findings

• On the Arnold cat map the learned propagator has no eigenvalue with modulus above 0.96, and latent-only rollouts decay to a point, so the flat spectrum serves as a diagnostic of missing finite-dimensional closure.
• Larger encoder width reduces one-step error by two orders of magnitude on the cat map yet extends the prediction horizon by only about 2 steps.

Limitations

• The symbolic readout still requires a chosen basis; learning the readout basis itself is left to future work.
• On Ikeda the polynomial inner function g is not sparse, with several false-positive monomials per dimension.
• Lorenz recovery contains residual false positives (x^2, y^2, z^2) and Jaccard 0.79 ± 0.06 rather than exact recovery per seed.

Experimental setup

Datasets: Lorenz system (simulated) Chirikov standard map (simulated) Ikeda map (simulated) Arnold cat map (simulated)

Benchmarks: Lorenz dictionary recovery (Jaccard, recall) standard map Fourier basis recovery Ikeda level-set R^2 Arnold cat map Fourier spectrum and rollout MSE

Metrics: Jaccard similarity recall MSE NRMSE R^2(h∘g)

Baselines: EDMD-DL SINDy-AE MLP + stable 1L-KAN + stable 2L-KAN + unconstrained

Builds on: Deep-Koopman KANDy Kolmogorov-Arnold Networks EDMD SINDy

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