Learning Partial Differential Equations from Data Using Neural Networks
This work addresses the challenge of PDE discovery from data for researchers in computational science, though it is incremental as it builds on prior methods by extending to more general PDE types.
The authors tackled the problem of estimating unknown partial differential equations from noisy data by developing a neural network framework that interpolates samples and extracts PDEs, showing it outperforms other methods, particularly in low signal-to-noise regimes.
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.