Ashish Pal

Ashish Pal, Satish Nagarajaiah

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.

Ashish Pal, Sutanu Bhowmick, Satish Nagarajaiah

Sparse system identification of nonlinear dynamic systems is still challenging, especially for stiff and high-order differential equations for noisy measurement data. The use of highly correlated functions makes distinguishing between true and false functions difficult, which limits the choice of functions. In this study, an equation discovery method has been proposed to tackle these problems. The key elements include a) use of B-splines for data fitting to get analytical derivatives superior to numerical derivatives, b) sequentially regularized derivatives for denoising (SRDD) algorithm, highly effective in removing noise from signal without system information loss, c) uncorrelated component analysis (UCA) algorithm that identifies and eliminates highly correlated functions while retaining the true functions, and d) physics-informed spline fitting (PISF) where the spline fitting is updated gradually while satisfying the governing equation with a dictionary of candidate functions to converge to the correct equation sequentially. The complete framework is built on a unified deep-learning architecture that eases the optimization process. The proposed method is demonstrated to discover various differential equations at various noise levels, including three-dimensional, fourth-order, and stiff equations. The parameter estimation converges accurately to the true values with a small coefficient of variation, suggesting robustness to the noise.