KAN/MultKAN with Physics-Informed Spline fitting (KAN-PISF) for ordinary/partial differential equation discovery of nonlinear dynamic systems

TL;DR

An equation discovery framework is proposed that includes a sequentially regularized derivatives for denoising algorithm to denoise the measure data to obtain accurate derivatives, KAN to identify the equation structure and suggest relevant nonlinear functions that are used to create a small overcomplete library of functions, and physics-informed spline fitting algorithm to filter the excess functions from the library and converge to the correct equation.

Abstract

Machine learning for scientific discovery is increasingly becoming popular because of its ability to extract and recognize the nonlinear characteristics from the data. The black-box nature of deep learning methods poses difficulties in interpreting the identified model. There is a dire need to interpret the machine learning models to develop a physical understanding of dynamic systems. An interpretable form of neural network called Kolmogorov-Arnold networks (KAN) or Multiplicative KAN (MultKAN) offers critical features that help recognize the nonlinearities in the governing ordinary/partial differential equations (ODE/PDE) of various dynamic systems and find their equation structures. In this study, an equation discovery framework is proposed that includes i) sequentially regularized derivatives for denoising (SRDD) algorithm to denoise the measure data to obtain accurate derivatives, ii) KAN to identify the equation structure and suggest relevant nonlinear functions that are used to create a small overcomplete library of functions, and iii) physics-informed spline fitting (PISF) algorithm to filter the excess functions from the library and converge to the correct equation. The framework was tested on the forced Duffing oscillator, Van der Pol oscillator (stiff ODE), Burger's equation, and Bouc-Wen model (coupled ODE). The proposed method converged to the true equation for the first three systems. It provided an approximate model for the Bouc-Wen model that could acceptably capture the hysteresis response. Using KAN maintains low complexity, which helps the user interpret the results throughout the process and avoid the black-box-type nature of machine learning methods.

Figures (20)

• Figure 1: Comparison between the MLP and KAN architecture
• Figure 2: Comparison between the KAN and MultKAN architecture
• Figure 3: Comparison between the interpretation of a nonlinear function from KAN and MultKAN.
• Figure 4: The proposed KAN-PISF equation discovery framework: a) The measured discrete noisy data, $u(x,t)$, and the underlying unknown governing equation, $h(x,t)$, for a two-dimensional dynamic system, b) The approximation of the measured data with a linear combination of B-spline basis functions and the derivatives obtained by analytical differentiation followed by SRDD denoising, c) MulKAN architecture to obtain the approximate structure of the underlying governing equation, $h(x,t)$, d) A dictionary of functions, $\Gamma$, is created based on the functions and equation structure suggested by KAN, whose unknown coefficients, $\theta$, are found by solving the linear regression problem to approximate $h(x,t)$, and e) PISF algorithm to sequentially eliminate the excess functions from the library to converge to the true equation.
• Figure 5: Comparing true and estimated displacement, velocity, and acceleration signals from KAN using automatic differentiation for noise-free data