A Comprehensive Review of Latent Space Dynamics Identification Algorithms for Intrusive and Non-Intrusive Reduced-Order-Modeling

TL;DR

This chapter presents strategies to enforce the laws of thermodynamics into LaSDI models (tLaSDI), enhance robustness in the presence of noise through the weak form (WLaSDI), select high-fidelity training data efficiently through active learning (gLaSDI, GPLaSDI), and quantify the ROM prediction uncertainty through Gaussian processes (GPLaSDI).

Abstract

Numerical solvers of partial differential equations (PDEs) have been widely employed for simulating physical systems. However, the computational cost remains a major bottleneck in various scientific and engineering applications, which has motivated the development of reduced-order models (ROMs). Recently, machine-learning-based ROMs have gained significant popularity and are promising for addressing some limitations of traditional ROM methods, especially for advection dominated systems. In this chapter, we focus on a particular framework known as Latent Space Dynamics Identification (LaSDI), which transforms the high-fidelity data, governed by a PDE, to simpler and low-dimensional latent-space data, governed by ordinary differential equations (ODEs). These ODEs can be learned and subsequently interpolated to make ROM predictions. Each building block of LaSDI can be easily modulated depending on the application, which makes the LaSDI framework highly flexible. In particular, we present strategies to enforce the laws of thermodynamics into LaSDI models (tLaSDI), enhance robustness in the presence of noise through the weak form (WLaSDI), select high-fidelity training data efficiently through active learning (gLaSDI, GPLaSDI), and quantify the ROM prediction uncertainty through Gaussian processes (GPLaSDI). We demonstrate the performance of different LaSDI approaches on Burgers equation, a non-linear heat conduction problem, and a plasma physics problem, showing that LaSDI algorithms can achieve relative errors of less than a few percent and up to thousands of times speed-ups.

Figures (11)

• Figure 1: 1D Burgers Equation -- Maximum relative error ($\%$) using LaSDI on a pre-selected uniform training grid, and using gLaSDI (residual-based) active learning. The black boxes represent the parameters corresponding to the training FOM datapoints (and the red boxes represent the four initialization training points, in the gLaSDI case).
• Figure 2: 1D Burgers Equation -- Maximum relative error ($\%$) using GPLaSDI and using gLaSDI
• Figure 3: 1D Burgers Equation -- Maximum Predictive Variance
• Figure 4: 1D Burgers Equation -- Maximum relative error ($\%$) using LaSDI and using WLaSDI (no noise case). Note that this error is bounded below by the autoencoder projection error, which is about $3\%$ across the parameter space in this case.
• Figure 5: 1D Burgers Equation -- Maximum relative error ($\%$) using LaSDI and using WLaSDI with $10\%$ added noise. The relative error is bounded below by the autoencoder projection error, which is about $5\%$ across the parameter space.