Data-driven discovery of partial differential equations

Abstract

Researchers propose sparse regression for identifying governing partial differential equations for spatiotemporal systems.

Sparse model selection via integral terms

“…In practice Eq. [5] does not hold because of the noise and we are left with the more difficult inverse problem, i.e. X(x) ≈ D(x)c. Instead, we determine the coefficient vector c that satisfies the least-squares constraint…”

“…In both plots, two terms are chosen from the 15 term trial set. The first plot is shows the result of a sparse optimization method for fitting the trial functions directly to the ODE, inspired by the trial functions used in [3][4][5][6] but using a similar model to Eq. [7].…”

“…In [3], a sequential thresholded least-squares algorithm is used to fit a set of candidate polynomials to computed velocity data. Sparse optimization for learning partial differential equations from spatio-temporal data is detailed in [4,5]. In [6], a sparse optimization problem is proposed for joint model selection and outlier detection, which allows for learning in the presence of time-intervals with large corruption.…”

Methods for data-driven multiscale model discovery for materials

“…A major extension of the SINDy modeling framework generalized the library to include partial derivatives, enabling the identification of partial differential equations (PDEs) [58,59]. The resulting algorithm, called the PDE functional identification of nonlinear dynamics (FIND), which is especially relevant for materials discovery, has been demonstrated to successfully identify several canonical PDEs from classical physics, purely from noisy data.…”

“…The resulting algorithm, called the PDE functional identification of nonlinear dynamics (FIND), which is especially relevant for materials discovery, has been demonstrated to successfully identify several canonical PDEs from classical physics, purely from noisy data. These PDEs include Navier-Stokes, Kuramoto-Sivashinsky, Schrödinger, reaction diffusion, Burgers, Korteweg-de Vries, and the diffusion equation for Brownian motion [58]. The PDE-FIND algorithm is particularly promising for identifying macroscopic models for nonlinear, anisotropic, and heterogeneous materials that are not amenable to linear and random homogenization techniques [3].…”