SINDy-KANs: Sparse identification of non-linear dynamics through Kolmogorov-Arnold networks

Amanda A. Howard; Nicholas Zolman; Bruno Jacob; Steven L. Brunton; Panos Stinis

cs.LGMar 19, 2026

SINDy-KANs: Sparse identification of non-linear dynamics through Kolmogorov-Arnold networks

Authors: Amanda A. Howard, Nicholas Zolman, Bruno Jacob, Steven L. Brunton, Panos Stinis

Organizations: Pacific Northwest National Laboratory, Richland, WA 99354 · University of Washington, Seattle, WA 98195

Abstract

Kolmogorov-Arnold networks (KANs) have arisen as a potential way to enhance the interpretability of machine learning. However, solutions learned by KANs are not necessarily interpretable, in the sense of being sparse or parsimonious. Sparse identification of nonlinear dynamics (SINDy) is a complementary approach that allows for learning sparse equations for dynamical systems from data; however, learned equations are limited by the library. In this work, we present SINDy-KANs, which simultaneously train a KAN and a SINDy-like representation to increase interpretability of KAN representations with SINDy applied at the level of each activation function, while maintaining the function compositions possible through deep KANs. We apply our method to a number of symbolic regression tasks, including dynamical systems, to show accurate equation discovery across a range of systems.

Figures & tables

Figure 1: Pictorial representation of a SINDy-KAN for a function $f(x,y)=\sin(x+y^{2})$ . Each KAN activation function has an associated least squares regression as given in Eq. 13 . The learned coefficients are then combined in $\mathbf{\Xi}$ .

Figure 2: Example of a trained SINDy-KAN for the system of equations $f_{1}(x,y)=\sin(2x+x^{2})$ , $f_{2}(x,y)=\cos(x)+y-2x-x^{2}$ .

Panel of Figure 3 — Figure 3: Results for Sec. 4.1 . (a) The final trained SINDy-KAN correctly learns the target equation. (b) The loss terms show two plateaus for the $L_{1}$ regularization term $\lambda_{1}||\mathbf{\Lambda}||_{1}$ .

Figure 4: Trained SINDy-KAN for Sec. 4.2.1 .

Figure 5: Trained SINDy-KAN for Sec. 4.2.1 . The SINDy-KAN results agree well with the data.

Panel of Figure 6 — Figure 6: Trained SINDy-KANs for the pendulum in Sec. 4.2.2 with one (a) and two (b) hidden layers.

Figure 7: SINDy-KAN results for Sec. 4.2.2 . (a-b) Results for the SINDy-KANs, compared with the numerical derivatives. (c) Loss for the SINDy-KAN.

Figure 8: The SINDy-KAN for ABC flow problem in Sec. 4.2.3 . For clarity, only the non-zero activation functions are shown.

Figure 9: The SINDy-KAN learns the dynamics of the ABC flow problem in Sec. 4.2.3 well.

Figure 10: The Lorenz system in Eqs. 28 – 30 and the system learned by the SINDy-KAN in Eqs. 32 – 34 for Sec. 4.2.4 . Both the true system and the learned system are evaluated numerically. We note that only data from $t\in[0,10]$ is used for training.

Figure 11: The trained SINDy-KAN for the Lorenz system in Sec. 4.2.4 . Multiplication nodes are denoted by “ $\times$ ” and addition nodes are denoted by “ $+$ ”.

Figure 12: The trained SINDy-KAN for the Kuramoto system in Sec. 4.2.5 . Multiplication nodes are denoted by “ $\times$ ” and addition nodes are denoted by “ $+$ ”. For clarity, only non-zero nodes are shown.

Appendix figures & tables7 assets

Supplementary material from the paper’s appendix.

Appendix

	$\mathcal{K}_{\Lambda}^{direct}$	$\mathcal{K}_{\Lambda}$ (From Sec. 4.1 )
Exact eq. $f(x,y)=\cos(x^{2}+y)$	$0.99986\cos(0.99996x^{2}+0.99995y)$	$0.99990\cos(1.00001x^{2}+0.99994y)$
Training time (s)	145	215

Appendix

Table 1: Comparison of SINDy-KANs and direct SINDy-KANs.

	$\mathcal{K}_{\Lambda}^{direct}$	$\mathcal{K}_{\Lambda}$ (From Sec. 4.2.1 )
$\mathbf{B}=\begin{bmatrix}-0.1&2&0\\ -2&-0.1&0\\ 0&0&-0.3\end{bmatrix}$	$\begin{bmatrix}-0.0989&1.9941&0\\ -1.9967&-0.0989&0\\ 0&0&-0.2964\end{bmatrix}$	$\begin{bmatrix}-0.0993&1.9946&0\\ -1.9972&-0.0995&0\\ 0&0&-0.2811\end{bmatrix}$

Appendix

Table 2: Comparison of SINDy-KANs and direct SINDy-KANs for the ODE problem in Sec. 4.2.1 .

Figure 13: An illustration of SINDy-KANs for dynamical systems. When training with data, the derivatives can either come from a pre-trained “forward KAN” or be found through finite differences or other methods.

Data $\mathbf{X}$ relative noise	$\mathcal{K}(\mathbf{X})$ error	$\dot{\mathcal{K}}(\mathbf{X})$ error	Learned equation
0%	5.688e-08	–	$\frac{dx}{dt}=-0.0997x+2.00044y$ $\frac{dy}{dt}=-2.00017x-0.09963y$ $\frac{dz}{dt}=-0.29567z$
6.27%	1.255e-05	0.00014	$\frac{dx}{dt}=-0.10575x+1.99087y$ $\frac{dy}{dt}=-2.00280x-0.10184y$ $\frac{dz}{dt}=-0.26376z$
31.37%	0.00043	0.0101	$\frac{dx}{dt}=-0.12326x+0.10216y^{3}+1.93752y+0.0897z^{3}$ $\frac{dy}{dt}=-2.00878x-0.12078y$ $\frac{dz}{dt}=-0.09579z^{3}-0.16220z$

Appendix

Table 3: Relative errors and learned equations for the ODE case from section 4.2.1 with noise added to the training data. Errors are computed by the relative mean squared error (MSE).

Panel of Figure 14 — Figure 14: Results for the test case from section 4.2.1 with 31.37% relative noise added to the training data (a-c). Although the derivatives used to train the SINDy-KAN are noisy (d-f), the SINDy-KAN is able to recover most of the equation.

Parameter	Sec. 4.1
KAN architecture	[2, 2, 1]
$g$	[3, 6]
Learning rate scales	[1, 0.6]
Boundaries	[0, 40k]
$k$	5
ADAM learning rate	1e-3

Appendix

Table 4: Hyperparameters used for training the results in section 4.1 .

g

is the number of grid points used in the KANs. The grid is refined at the schedule denoted by the boundaries, and the learning rate is scaled at the same boundaries.

k

is the polynomial degree for the B-splines.

Parameter	Sec. 4.2.1	Sec. 4.2.2	Sec. 4.2.3	Sec. 4.2.4	Sec. 4.2.5
KAN architecture	[3, 3]	[2, 2]	[3, 6, 3]	[3, 5, 3]	[3, 6, 3]
$g$	[3]	[5, 10]	[5, 7]	[5, 7, 10]	[5, 7, 10, 15]
Learning rate scales	[1]	[1, .6]	[1, .6]	[1, .6, .6]	[1, .6, .6, .1]
Boundaries	[0]	[0, 50k]	[0, 30k]	[0, 30k, 60k]	[0, 30k, 60k, 300k]
$k$	5	5	5	5	5
ADAM learning rate	1e-3	1e-3	5e-3	5e-3	1e-2

Appendix

Table 5: Hyperparameters used for training the results in section 4.2 .

g

is the number of grid points used. The grid is refined at the schedule denoted by the boundaries, and the learning rate is scaled at the same boundaries.

k

is the polynomial degree for the B-splines used.

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AC-SINDy: Compositional Sparse Identification of Nonlinear Dynamics

Peter Racioppo

Independent Researcher Los Angeles, CA 90731

Latent Dynamics Modeling Sparse Identification of Nonlinear Dynamics Nonlinear System Identification Symbolic Regression

Open paper AskPDF arXiv

We present AC-SINDy, a compositional extension of the Sparse Identification of Nonlinear Dynamics (SINDy) framework that replaces explicit feature libraries with a structured representation based on arithmetic circuits. Rather than enumerating candidate basis functions, the proposed approach constructs nonlinear features through compositions of linear functions and multiplicative interactions, yielding a compact and scalable parameterization and enabling sparsity to be enforced directly over the computational graph. We also introduce a formulation that separates state estimation from dynamics identification by combining latent state inference with shared dynamics and multi-step supervision, improving robustness to noise while preserving interpretability. Experiments on nonlinear and chaotic systems demonstrate that the method recovers accurate and interpretable governing equations while scaling more favorably than standard SINDy.

Abstract

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