Revealing Low-Dimensional Structure in 2D Richtmyer-Meshkov Instabilities via Parametric Reduced-Order Modeling

TL;DR

This work develops a parametric reduced-order modeling framework for 2D Richtmyer-Meshkov instability using Latent Space Dynamics Identification (LaSDI) with autoencoder compression. The method reveals that, after a nonlinear embedding of the material interface, RMI dynamics can be represented by a low-dimensional ($n_z=3$) linear dynamical system in latent space, even as parameters such as EOS and initial perturbations vary. Training proceeds in two phases to learn both the latent dynamics and a neural parameter-to-coefficients map, enabling accurate forward predictions of late-time interface evolution from limited bitmap observables. The approach offers a practical, non-intrusive surrogate for engineering tasks like design optimization and parameter inference in RMI-dominated systems, and suggests a new direction for theory by exposing a low-dimensional structure in strongly nonlinear multi-mode RMI dynamics.

Abstract

Efficient modeling of the Richtmyer-Meshkov instability (RMI) is essential to many engineering tasks, including high-speed combustion and drive and capsule geometry optimization in Inertial Confinement Fusion (ICF). In the latter, RMI causes the ablator and fuel to mix, introducing cold spots into the fuel and lowering performance; controlling RMI is thus a core ICF design concern. In this work, we introduce a reduced-order model for two-dimensional RMI based on the Latent Space Dynamics Identification (LaSDI) algorithm. We demonstrate the efficacy of the proposed methodology in efficiently parametrizing the solution space over a high-dimensional parameter vector consisting of material EOS parameters and initial conditions known to affect RMI growth rates. Using only late-time partial observations of the dynamics, we use our framework to not only provide a highly efficient dynamic surrogate model, but to reveal that the RMI exhibits the structure of a surprisingly low-dimensional and linear dynamical system, into the nonlinear growth regime, after a suitable nonlinear transformation is applied to the material interface, which we approximate as a trained autoencoder. Our use of practical observables and fundamental parameters suggests that such ROMs may be useful for downstream engineering tasks which confront the RMI, while the low-dimensional representation suggests a new direction for theoretical work.

Figures (10)

• Figure 1: Diagram of parametrized reduced-order model, consisting of a shallow autoencoder network with two hidden layers in the encoder $\phi$ and decoder $\psi$, an expansive parameters-to-coefficients network ${\mathcal{M}}$ with four hidden layers, and a parametrized linear latent-space dynamical system. The layer widths are indicated in each layer, and $n_z \in \{2,3,4\}$ is examined.
• Figure 2: ROM performance over the weak coupling regime: profiles 1-26 over parameters $[v_{0},s_1,c_s]$ (see Tables \ref{['tab:initialcoeffs']}, \ref{['tab:matparams']}). Tables include 95th percentile mean pixel error (MPE) and Jaccard loss (JL) over the training (tr.) and testing (te.) sets for models with latent space dimension $n_z\in \{2,3,4\}$. Images depict final-time model outputs ($t=40$ indicating the final snapshot) for testing simulations at the 95th percentile in JL using (top to bottom) $n_z = \{2,3,4\}$ (i.e. the model performed better than the depicted case on 95% test cases). Left to right: ground truth, ROM prediction, difference between predicted and truth, and latent space dynamics over the time interval. The latent space trajectories are plotted for the AE, V-ROM, and F-ROM models in dashed, dot-dashed, and dotted lines, respectively, with colors representing different latent space coordinates over time.
• Figure 3: Performance of F-ROMs in modeling RMI under weak mode coupling according to the minimum, median, and maximum JL (top) and MPE (bottom) metrics \ref{['eq:metrics']} over the test set as a function of time.
• Figure 4: 95th percentile scores for modeling strong coupling between $k=2$ and $k=4$ modes, with variable $v_{0}$ and EOS parameters $[s_1,c_s]$. Figures are analogous to those in \ref{['fig:weakcoupling']}.
• Figure 5: Performance of F-ROMs in modeling RMI under strong mode coupling according to the minimum, median, and maximum JL (top) and MPE (bottom) metrics \ref{['eq:metrics']} over the test set as a function of time.