Data-driven discovery of partial differential equations

TL;DR

This paper introduces a sparse regression approach for automatically discovering partial differential equations from time series data, effectively identifying governing physical laws in complex systems.

Contribution

It presents a novel, efficient, and robust method that uses sparsity and Pareto analysis to select the most accurate and parsimonious PDE models from data, capable of disambiguating similar dynamical terms.

Findings

Successfully applied to Navier-Stokes, quantum harmonic oscillator, and diffusion equations.

Capable of distinguishing between different PDEs with similar dynamics using multiple initial conditions.

Demonstrates robustness and computational efficiency in various canonical physics problems.

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 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 of mathematical physics including Navier-Stokes, the quantum harmonic oscillator, and the diffusion equation. Moreover, the method is capable of disambiguating between potentially non-unique 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 or the Korteweg-deVries equation, for instance. The method provides a promising new technique for discovering governing equations and physical laws in parametrized spatio-temporal systems where first-principles derivations are intractable.