Sequential decoder training for improved latent space dynamics identification

TL;DR

The paper addresses the interpretability-accuracy trade-off in latent-space ROMs for PDEs by introducing multi-stage LaSDI (mLaSDI), which sequentially adds residual-correcting decoders to refine reconstructions while preserving the first-stage interpretable latent dynamics learned via SINDy. The framework decomposes reconstruction into a primary latent-dynamics-driven term and subsequent residual terms, enabling improved accuracy without proportionally enlarging autoencoders. Applied to the 1D-1V Vlasov equation, mLaSDI consistently outperforms GPLaSDI in both accuracy and training efficiency across diverse architectures, achieving sub-percent errors in some cases and significantly reducing sensitivity to hyperparameters. The results suggest that multi-stage training can robustly enhance interpretability-accurate latent-space identification and may generalize to other latent-space methods and broader PDE families, offering a practical route to more efficient ROMs.

Abstract

Accurate numerical solutions of partial differential equations are essential in many scientific fields but often require computationally expensive solvers, motivating reduced-order models (ROMs). Latent Space Dynamics Identification (LaSDI) is a data-driven ROM framework that combines autoencoders with equation discovery to learn interpretable latent dynamics. However, enforcing latent dynamics during training can compromise reconstruction accuracy of the model for simulation data. We introduce multi-stage LaSDI (mLaSDI), a framework that improves reconstruction and prediction accuracy by sequentially learning additional decoders to correct residual errors from previous stages. Applied to the 1D-1V Vlasov equation, mLaSDI consistently outperforms standard LaSDI, achieving lower prediction errors and reduced training time across a wide range of architectures.

Figures (2)

• Figure 1: Schematic of mLaSDI. The first stage learns an autoencoder which reconstructs the training data $\mathbf{U}$, and latent space trajectories (solid black lines) are approximated using SINDy (dashed red lines). The second stage learns a new decoder which maps the SINDy trajectories in the latent space to the normalized residual error from the first stage.
• Figure 2: Applying GPlaSDI and mLaSDI with two stages to 1D-1V Vlasov equation using a wide range of model architectures in Table \ref{['tab:1d1v']}. Percentile errors for the first and second stages using errors from \ref{['eq:maxerrs']}.