Higher-Order-KANDy recovers higher-order dynamics without a candidate library and selects the exact active set at 7.5% noise
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Synopsis
The authors propose Higher-Order-KANDy, which combines delay and derivative (jet) embeddings with derivative-ordered KANDy layers stacked into a machine learning architecture and learns a delay-to-jet map as a Kolmogorov-Arnold network, handling higher-order systems natively without a candidate library; on synthetic benchmarks it recovers equations under coarse sampling and noise, and at noise levels up to 7.5% it selects the exact active set on every seed, whereas an oracle weak-form regression admits a spurious term at that noise level, though the oracle remains the more accurate coefficient estimator.
Figure 1: Overview of the Higher-Order-KANDy framework. (a) A delay embedding y ( t ) = ( x ( t ) , x ( t − τ ) , … , x ( t − ( m − 1 ) τ ) ) ∈ ℝ m y(t)=(x(t),x(t-\tau),\dots,x(t-(m-1)\tau))\in\mathbb{R}^{m} ( m m delays, step τ \tau ) is passed through (b) a 1-KANDy lifting layer that estimates the latent velocity x ^ ′ \hat{x}^{\prime} . (c) Repeated differentiation builds the derivative stack 𝒟 k = { x , x ^ ′ , … , x ^ ( k − 1 ) } \mathcal{D}_{k}=\{x,\hat{x}^{\prime},\ldots,\hat{x}^{(k-1)}\} , from which (d) the feature set Θ k ( 𝒟 k , t ) \Theta_{k}(\mathcal{D}_{k},t) is assembled. (e) A k k -KANDy equation layer learns univariate edge functions φ k , j \varphi_{k,j} whose sum predicts x ^ ( k ) \hat{x}^{(k)} , so the edges read out directly as the additive terms of the governing equation.
arXivInterpretation
Introduces Higher-Order-KANDy, which uses delay and derivative (jet) embeddings together with derivative-ordered KANDy layers stacked into a machine learning architecture to learn governing equations of higher-order dynamical systems directly from data. Compared with SINDy and WSINDy, the method needs no candidate library, for example recovering the full-period -sin(u) with nothing named in advance, and it works natively for higher-order systems rather than first reducing them to first-order form. The abstract reports equation recovery on synthetic benchmarks under coarse sampling and noise, with comparisons to SINDy and WSINDy on noisy data.
Maintains the correct term structure as noise grows, selecting the exact active set on every seed at noise levels up to 7.5%. At the same noise level, an oracle weak-form regression admits a spurious term, which Higher-Order-KANDy does not. The abstract states the 7.5% noise level, the per-seed selection result, and the comparison with the oracle weak-form regression, while noting the oracle remains the more accurate coefficient estimator.
Leverages the relationship between delay and differential embeddings and the Kolmogorov-Arnold representation theorem to construct a delay-to-jet map learned as a Kolmogorov-Arnold network. This construction preserves derivative structure and yields meaningful terms that appear in the governing equation, addressing both the expansion of candidate terms from reducing higher-order equations and noise from estimating derivatives. The abstract grounds this in the method construction and synthetic benchmark results, without reporting network sizes or dataset counts.
Perspective
The work targets researchers and practitioners who need to derive governing equations of higher-order dynamical systems from data, in synthetic benchmark settings with coarse sampling and noise; its value lies in producing physically meaningful terms without a pre-specified candidate library and in offering a concrete realization of the relationship between delay and differential embeddings. The results described in the abstract are limited to synthetic benchmarks, so the scope should be read as equation recovery and term-structure selection within that setting.
The abstract does not report network sizes, training details, the number of benchmark systems, or perturbation types beyond noise, nor does it describe performance on real measurement data; the oracle weak-form regression remains more accurate in coefficients, suggesting a trade-off between term-structure selection and coefficient accuracy worth watching. In addition, the available content here is the abstract, without figures or experimental details, which limits how specifically the numerical and ablation conclusions can be restated.