# Robust and optimal sparse regression for nonlinear PDE models

## TL;DR

This paper presents a robust sparse regression method using weak formulation to accurately identify nonlinear PDE models from noisy data, outperforming existing techniques especially under varying noise conditions.

## Contribution

It introduces an optimized sparse regression approach with a scaling relation linking model accuracy to data resolution and noise levels, specifically for nonlinear PDEs.

## Key findings

- Achieves significantly better accuracy than existing methods.
- Derives a scaling relation between model accuracy, data resolution, and noise.
- Demonstrates effectiveness on the Kuramoto-Sivashinsky equation.

## Abstract

This paper investigates how models of spatiotemporal dynamics in the form of nonlinear partial differential equations can be identified directly from noisy data using a combination of sparse regression and weak formulation. Using the 4th-order Kuramoto-Sivashinsky equation for illustration, we show how this approach can be optimized in the limits of low and high noise, achieving accuracy that is orders of magnitude better than what existing techniques allow. In particular, we derive the scaling relation between the accuracy of the model, the parameters of the weak formulation, and the properties of the data, such as its spatial and temporal resolution and the level of noise.