Scalable nonlinear manifold reduced order model for dynamical systems

TL;DR

This paper presents a scalable nonlinear manifold reduced-order model that efficiently simulates large dynamical systems by training on smaller domains and deploying on larger ones, achieving high accuracy and significant speedup.

Contribution

It introduces a bottom-up training strategy for NM-ROMs that enhances scalability and demonstrates effectiveness on a 2D Burgers' equation example.

Findings

Achieves 1% relative error in simulations.

Provides nearly 700x speedup over traditional methods.

Demonstrates stable extrapolation from small to large domains.

Abstract

The domain decomposition (DD) nonlinear-manifold reduced-order model (NM-ROM) represents a computationally efficient method for integrating underlying physics principles into a neural network-based, data-driven approach. Compared to linear subspace methods, NM-ROMs offer superior expressivity and enhanced reconstruction capabilities, while DD enables cost-effective, parallel training of autoencoders by partitioning the domain into algebraic subdomains. In this work, we investigate the scalability of this approach by implementing a "bottom-up" strategy: training NM-ROMs on smaller domains and subsequently deploying them on larger, composable ones. The application of this method to the two-dimensional time-dependent Burgers' equation shows that extrapolating from smaller to larger domains is both stable and effective. This approach achieves an accuracy of 1% in relative error and provides a remarkable speedup of nearly 700 times.