Sparsistent Model Discovery

TL;DR

This paper introduces the randomized adaptive Lasso within the DeepMod framework, enabling more robust discovery of nonlinear and chaotic PDEs from noisy, limited data, surpassing previous methods in noise tolerance and automation.

Contribution

It demonstrates that the randomized adaptive Lasso can recover PDEs under violated irrepresentability conditions, improving robustness and automation in deep learning-based model discovery.

Findings

Recovered PDEs at twice the noise-to-sample ratio of state-of-the-art methods.

Achieved accurate model discovery with a single hyperparameter set.

Successfully identified nonlinear and chaotic PDEs from limited noisy data.

Abstract

Discovering the partial differential equations underlying spatio-temporal datasets from very limited and highly noisy observations is of paramount interest in many scientific fields. However, it remains an open question to know when model discovery algorithms based on sparse regression can actually recover the underlying physical processes. In this work, we show the design matrices used to infer the equations by sparse regression can violate the irrepresentability condition (IRC) of the Lasso, even when derived from analytical PDE solutions (i.e. without additional noise). Sparse regression techniques which can recover the true underlying model under violated IRC conditions are therefore required, leading to the introduction of the randomised adaptive Lasso. We show once the latter is integrated within the deep learning model discovery framework DeepMod, a wide variety of nonlinear and chaotic canonical PDEs can be recovered: (1) up to $\mathcal{O}(2)$ higher noise-to-sample ratios than state-of-the-art algorithms, (2) with a single set of hyperparameters, which paves the road towards truly automated model discovery.