Using Noisy or Incomplete Data to Discover Models of Spatiotemporal Dynamics

TL;DR

This paper introduces a weak formulation approach for sparse regression that accurately discovers complex spatiotemporal PDE models from noisy data, including models with latent variables, overcoming derivative evaluation challenges.

Contribution

It presents a novel weak formulation method enabling robust PDE discovery from noisy data, including high-order derivatives and unmeasured variables.

Findings

Successfully reconstructs high-order PDEs from noisy data.

Accurately models systems with unmeasured latent variables.

Demonstrates effectiveness on turbulent flow data.

Abstract

Sparse regression has recently emerged as an attractive approach for discovering models of spatiotemporally complex dynamics directly from data. In many instances, such models are in the form of nonlinear partial differential equations (PDEs); hence sparse regression typically requires evaluation of various partial derivatives. However, accurate evaluation of derivatives, especially of high order, is infeasible when the data are noisy, which has a dramatic negative effect on the result of regression. We present a novel and rather general approach that addresses this difficulty by using a weak formulation of the problem. For instance, it allows accurate reconstruction of PDEs involving high-order derivatives, such as the Kuramoto-Sivashinsky equation, from data with a considerable amount of noise. The flexibility of our approach also allows reconstruction of PDE models that involve latent variables which cannot be measured directly with acceptable accuracy. This is illustrated by reconstructing a model for a weakly turbulent flow in a thin fluid layer, where neither the forcing nor the pressure field is known.