Reduced Order Modeling for Parameterized Time-Dependent PDEs using Spatially and Memory Aware Deep Learning

TL;DR

This paper introduces a nonintrusive reduced order modeling approach for parameterized time-dependent PDEs using convolutional autoencoders and memory-aware neural networks, enabling efficient multi-query solutions.

Contribution

It combines nonlinear spatial reduction with memory-aware neural networks for time-stepping, offering a novel, stable, and generalizable ROM framework for complex PDEs.

Findings

Successfully applied to heat, advection, and Navier-Stokes equations.

Demonstrates improved efficiency for multi-query PDE problems.

Ensures stability and generalization in neural network-based ROMs.

Abstract

We present a novel reduced order model (ROM) approach for parameterized time-dependent PDEs based on modern learning. The ROM is suitable for multi-query problems and is nonintrusive. It is divided into two distinct stages: A nonlinear dimensionality reduction stage that handles the spatially distributed degrees of freedom based on convolutional autoencoders, and a parameterized time-stepping stage based on memory aware neural networks (NNs), specifically causal convolutional and long short-term memory NNs. Strategies to ensure generalization and stability are discussed. The methodology is tested on the heat equation, advection equation, and the incompressible Navier-Stokes equations, to show the variety of problems the ROM can handle.