A Deep Learning approach for parametrized and time dependent Partial Differential Equations using Dimensionality Reduction and Neural ODEs

TL;DR

This paper introduces a novel deep learning framework combining dimensionality reduction and Neural ODEs to efficiently approximate solutions of time-dependent, parametric PDEs, significantly reducing computational costs while maintaining accuracy.

Contribution

It presents a new autoregressive, data-driven method that couples dimensionality reduction with Neural ODEs to learn PDE solution operators in a low-dimensional space, improving efficiency and accuracy.

Findings

DR coupled with Neural ODEs yields accurate PDE solutions.

The approach reduces computational load compared to traditional methods.

The method is faster and more accurate than existing deep learning models.

Abstract

Partial Differential Equations (PDEs) are central to science and engineering. Since solving them is computationally expensive, a lot of effort has been put into approximating their solution operator via both traditional and recently increasingly Deep Learning (DL) techniques. A conclusive methodology capable of accounting both for (continuous) time and parameter dependency in such DL models however is still lacking. In this paper, we propose an autoregressive and data-driven method using the analogy with classical numerical solvers for time-dependent, parametric and (typically) nonlinear PDEs. We present how Dimensionality Reduction (DR) can be coupled with Neural Ordinary Differential Equations (NODEs) in order to learn the solution operator of arbitrary PDEs. The idea of our work is that it is possible to map the high-fidelity (i.e., high-dimensional) PDE solution space into a reduced (low-dimensional) space, which subsequently exhibits dynamics governed by a (latent) Ordinary Differential Equation (ODE). Solving this (easier) ODE in the reduced space allows avoiding solving the PDE in the high-dimensional solution space, thus decreasing the computational burden for repeated calculations for e.g., uncertainty quantification or design optimization purposes. The main outcome of this work is the importance of exploiting DR as opposed to the recent trend of building large and complex architectures: we show that by leveraging DR we can deliver not only more accurate predictions, but also a considerably lighter and faster DL model compared to existing methodologies.