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

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 (14)

• 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. \ref{['eq:sindy_one_act']}. The learned coefficients are then combined in $\mathbf{\Xi}$.
• Figure 2: Example of a trained SINDy-KAN for the system of equations $f_1(x, y) = \sin(2x+x^2)$, $f_2(x, y) = \cos(x)+y-2x-x^2$.
• Figure 3: Results for Sec. \ref{['sec:reg_test2']}. (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. \ref{['sec:ODE_2']}.
• Figure 5: Trained SINDy-KAN for Sec. \ref{['sec:ODE_2']}. The SINDy-KAN results agree well with the data.