Learning partial differential equations via data discovery and sparse optimization

Abstract

We investigate the problem of learning an evolution equation directly from some given data. This work develops a learning algorithm to identify the terms in the underlying partial differential equations and to approximate the coefficients of the terms only using data. The algorithm uses sparse optimization in order to perform feature selection and parameter estimation. The features are data driven in the sense that they are constructed using nonlinear algebraic equations on the spatial derivatives of the data. Several numerical experiments show the proposed method's robustness to data noise and size, its ability to capture the true features of the data, and its capability of performing additional analytics. Examples include shock equations, pattern formation, fluid flow and turbulence, and oscillatory convection.

Sparse model selection via integral terms

“…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.…”

Data-Driven Partial Derivative Equations Discovery with Evolutionary Approach

“…features are written back in the form (2) and regression is used to find the coefficients. Usually, in regression all possible combinations [12] of the feature vectors Eq. 4 are chosen for minimization problem Eq.…”

“…7 is the parameter of the evolutionary algorithm. The second remark is that, in contrast to the existing algorithms [11,12,7], the target feature is chosen randomly, whereas in the sparse-regression only cases time-derivative is used.…”