Online Weak-form Sparse Identification of Partial Differential Equations
This provides a streaming alternative for system identification in higher dimensions, which is incremental as it builds on existing weak-form sparse identification methods.
The paper tackled the problem of identifying partial differential equations (PDEs) from noisy data in an online setting, and the result was a method that successfully identified and tracked systems with time-varying coefficients, demonstrated on equations in up to three spatial dimensions.
This paper presents an online algorithm for identification of partial differential equations (PDEs) based on the weak-form sparse identification of nonlinear dynamics algorithm (WSINDy). The algorithm is online in a sense that if performs the identification task by processing solution snapshots that arrive sequentially. The core of the method combines a weak-form discretization of candidate PDEs with an online proximal gradient descent approach to the sparse regression problem. In particular, we do not regularize the $\ell_0$-pseudo-norm, instead finding that directly applying its proximal operator (which corresponds to a hard thresholding) leads to efficient online system identification from noisy data. We demonstrate the success of the method on the Kuramoto-Sivashinsky equation, the nonlinear wave equation with time-varying wavespeed, and the linear wave equation, in one, two, and three spatial dimensions, respectively. In particular, our examples show that the method is capable of identifying and tracking systems with coefficients that vary abruptly in time, and offers a streaming alternative to problems in higher dimensions.