Source-linked AI summary
Data-driven discovery of partial differential equations
Samuel H. Rudy, Steven L. Brunton, Joshua L. Proctor, J. Nathan Kutz
TL;DR
The paper addresses how to uncover governing PDEs from spatio-temporal data when first-principles derivations are difficult. PDE-FIND uses sparse regression over candidate terms, including spatial derivatives, and identifies canonical PDEs while handling sampling and ambiguities across solutions.
Problem
Underlying physical laws remain difficult to uncover from time-series data with spatio-temporal activity, especially for complex systems that have eluded first-principles derivation.
Method
PDE-FIND constructs a library of linear, nonlinear, and derivative terms, then uses sparse regression, sequential threshold ridge regression, and Pareto selection to identify parsimonious PDEs.
Results
PDE-FIND identifies governing PDEs across canonical mathematical-physics systems, including strongly nonlinear models, even with significant spatial subsampling.
Takeaways & Limitations
The method provides a data-driven tool for discovering physical laws in spatial-domain applications where first-principles derivations may be intractable.
Takeaways & Limitations
The method assumes the unknown PDE right-hand side contains only a few active terms, making it sparse in the space of possible functions.
Abstract
from arXiv · showhide
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.