Data-driven discovery of partial differential equations

Abstract

We propose a sparse regression method capable of discovering the governing partial differential equation(s) of a given system by time series measurements in the spatial domain. The regression framework relies on sparsity-promoting techniques to select the nonlinear and partial derivative terms of the governing equations that most accurately represent the data, bypassing a combinatorially large search through all possible candidate models. The method balances model complexity and regression accuracy by selecting a parsimonious model via Pareto analysis. Time series measurements can be made in an Eulerian framework, where the sensors are fixed spatially, or in a Lagrangian framework, where the sensors move with the dynamics. The method is computationally efficient, robust, and demonstrated to work on a variety of canonical problems spanning a number of scientific domains including Navier-Stokes, the quantum harmonic oscillator, and the diffusion equation. Moreover, the method is capable of disambiguating between potentially nonunique dynamical terms by using multiple time series taken with different initial data. Thus, for a traveling wave, the method can distinguish between a linear wave equation and the Korteweg-de Vries equation, for instance. The method provides a promising new technique for discovering governing equations and physical laws in parameterized spatiotemporal systems, where first-principles derivations are intractable.

Keywords: data-driven discovery; dynamical systems; partial differential equations; sparse regression.

Fig. 1. Steps in the PDE functional identification of nonlinear dynamics (PDE-FIND) algorithm, applied to infer the Navier-Stokes equations from data.

(1a) Data are collected as snapshots of a solution to a PDE. (1b) Numerical derivatives are taken, and data are compiled into a large matrix Θ, incorporating candidate terms for the PDE. (1c) Sparse regressions are used to identify active terms in the PDE. (2a) For large data sets, sparse sampling may be used to reduce the size of the problem. (2b) Subsampling the data set is equivalent to taking a subset of rows from the linear system in Eq. 2. (2c) An identical sparse regression problem is formed but with fewer rows. (d) Active terms in ξ are synthesized into a PDE.

Fig. 2. Inferring the diffusion equation from a single Brownian motion.

(A) Time series is broken into many short random walks that are used to construct histograms of the displacement. (B) Brownian motion trajectory following the diffusion equation. (C) Parameter error $({||\widehat{\mathrm{\xi }}-\mathrm{\xi }*||}_1)$ versus length of known time series. Blue symbols correspond to correct identification of the structure of the diffusion model u_t = cu_xx.

Fig. 3. Inferring nonlinearity via observing solutions at multiple amplitudes.

(A) Example two-soliton solution to the KdV equation. (B) Applying our method to a single soliton solution determines that it solves the standard advection equation. (C) Looking at two completely separate solutions reveals nonlinearity.