Deep-learning of Parametric Partial Differential Equations from Sparse and Noisy Data

by   Hao Xu, et al.

Data-driven methods have recently made great progress in the discovery of partial differential equations (PDEs) from spatial-temporal data. However, several challenges remain to be solved, including sparse noisy data, incomplete candidate library, and spatially- or temporally-varying coefficients. In this work, a new framework, which combines neural network, genetic algorithm and adaptive methods, is put forward to address all of these challenges simultaneously. In the framework, a trained neural network is utilized to calculate derivatives and generate a large amount of meta-data, which solves the problem of sparse noisy data. Next, genetic algorithm is utilized to discover the form of PDEs and corresponding coefficients with an incomplete candidate library. Finally, a two-step adaptive method is introduced to discover parametric PDEs with spatially- or temporally-varying coefficients. In this method, the structure of a parametric PDE is first discovered, and then the general form of varying coefficients is identified. The proposed algorithm is tested on the Burgers equation, the convection-diffusion equation, the wave equation, and the KdV equation. The results demonstrate that this method is robust to sparse and noisy data, and is able to discover parametric PDEs with an incomplete candidate library.



There are no comments yet.


page 1

page 2

page 3

page 4


DLGA-PDE: Discovery of PDEs with incomplete candidate library via combination of deep learning and genetic algorithm

Data-driven methods have recently been developed to discover underlying ...

Deep-learning based discovery of partial differential equations in integral form from sparse and noisy data

Data-driven discovery of partial differential equations (PDEs) has attra...

Online Weak-form Sparse Identification of Partial Differential Equations

This paper presents an online algorithm for identification of partial di...

Convergent numerical method for a linearized travel time tomography problem with incomplete data

We propose a new numerical method to solve the linearized problem of tra...

DeepMoD: Deep learning for Model Discovery in noisy data

We introduce DeepMoD, a deep learning based model discovery algorithm wh...

Sparsely constrained neural networks for model discovery of PDEs

Sparse regression on a library of candidate features has developed as th...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.