Data-Driven Deep Learning of Partial Differential Equations in Modal Space

TL;DR

This paper introduces a data-driven deep learning framework to approximate the evolution operator of unknown PDEs in modal space, enabling solution prediction without explicit PDE term identification.

Contribution

It proposes a novel approach to approximate PDE evolution operators in modal space using neural networks, bypassing the need for explicit PDE term recovery.

Findings

Effective approximation of PDE evolution operators demonstrated on various PDEs.

High predictive accuracy shown even with discontinuous solutions.

Framework applicable to complex PDEs like Burgers' equation.

Abstract

We present a framework for recovering/approximating unknown time-dependent partial differential equation (PDE) using its solution data. Instead of identifying the terms in the underlying PDE, we seek to approximate the evolution operator of the underlying PDE numerically. The evolution operator of the PDE, defined in infinite-dimensional space, maps the solution from a current time to a future time and completely characterizes the solution evolution of the underlying unknown PDE. Our recovery strategy relies on approximation of the evolution operator in a properly defined modal space, i.e., generalized Fourier space, in order to reduce the problem to finite dimensions. The finite dimensional approximation is then accomplished by training a deep neural network structure, which is based on residual network (ResNet), using the given data. Error analysis is provided to illustrate the predictive accuracy of the proposed method. A set of examples of different types of PDEs, including inviscid Burgers' equation that develops discontinuity in its solution, are presented to demonstrate the effectiveness of the proposed method.