Scalable nonlinear manifold reduced order model for dynamical systems

TL;DR

This work addresses the computational bottleneck of simulating parameterized nonlinear systems by marrying nonlinear-manifold ROMs with algebraic domain decomposition, enabling parallel, localized autoencoder training. A bottom-up strategy trains NM-ROMs on small subdomains and deploys them on larger, composable domains, demonstrated on the 2D Burgers’ equation with impressive speedups (≈700x) and competitive accuracy (relative error around 1e-2). The method leverages per-subdomain encoders/decoders, SRPC constraints, and a Gauss-Newton–based inexact SQP solver to efficiently solve reduced-order problems without hyper-reduction in the reported setup. The results highlight scalable training and deployment, with potential improvements via adjoint-based optimization and applicability to broader PDE classes such as Kuramoto–Sivashinsky, KdV, and Navier–Stokes.

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.

Figures (2)

• Figure 1: Averaged speedups and errors from 50 distinct $2 \times 2$ test cases, evaluated across 25 different combinations of latent dimensions for vertical/horizontal port nodes ($\mathcal{P}$) and interior nodes ($\mathcal{I}$). The results illustrate the impact of varying latent space dimensions on performance metrics.
• Figure 2: The $u$ velocity predicted by the DD FOM and DD NM-ROM models, along with the absolute error between them, at various time instants. The initial condition was randomly sampled using (\ref{['eq:sampling']}) on a $10 \times 10$ configuration.