Learning Partial Differential Equations from Data Using Neural Networks

TL;DR

This paper presents a neural network-based framework for estimating unknown partial differential equations from noisy data, capable of handling linear combinations of user-defined functions and outperforming existing methods especially with noisy data.

Contribution

The authors introduce a novel deep learning approach with regularization for PDE discovery from noisy data, extending previous methods to more general PDE forms.

Findings

Outperforms existing methods in noisy environments

Effective in low signal-to-noise ratio regimes

Validated on simulated data with known PDEs

Abstract

We develop a framework for estimating unknown partial differential equations from noisy data, using a deep learning approach. Given noisy samples of a solution to an unknown PDE, our method interpolates the samples using a neural network, and extracts the PDE by equating derivatives of the neural network approximation. Our method applies to PDEs which are linear combinations of user-defined dictionary functions, and generalizes previous methods that only consider parabolic PDEs. We introduce a regularization scheme that prevents the function approximation from overfitting the data and forces it to be a solution of the underlying PDE. We validate the model on simulated data generated by the known PDEs and added Gaussian noise, and we study our method under different levels of noise. We also compare the error of our method with a Cramer-Rao lower bound for an ordinary differential equation. Our results indicate that our method outperforms other methods in estimating PDEs, especially in the low signal-to-noise regime.