Sparse Learning of Nonlinear PDE Dynamics using Kalman Smoothing

Abstract

Identifying differential equations from noisy measurement data requires fitting the underlying dynamics and assimilating the data, either implicitly or explicitly. The Sparse Identification of Nonlinear Dynamics (SINDy) method achieves this in two steps: a derivative estimation and smoothing step, followed by sparse regression over a library of candidate functions. Previous implementations of the derivative step in SINDy, including the Python package pysindy, relied on methods such as finite difference, L1 total variation minimization, or Savitzky-Golay filtering. Kalman smoothing, a classical approach for data assimilation with known noise statistics, has recently been incorporated into pysindy alongside hyperparameter optimization to enhance data denoising and improve differential equation discovery for several ordinary differential equation (ODE) systems. This thesis extends the use of Kalman smoothing in SINDy to partial differential equations (PDEs), enabling effective denoising and sparse identification of a range of PDE systems.