mLaSDI: Multi-stage latent space dynamics identification

2025-06-10

AI Summary

Reduced-order models (ROMs) for efficient partial differential equation (PDE) solving suffer from reconstruction distortion and dynamical inconsistency in complex or high-frequency regimes. Method: This paper proposes Multi-Stage Latent Space Dynamics Identification (MS-LaSDI), a non-intrusive, multi-stage latent-space dynamics learning framework. It cascades lightweight autoencoders and employs residual learning to progressively correct both latent-space reconstruction errors and discrepancies in the fitted ordinary differential equation (ODE) dynamics—thereby overcoming the inherent fidelity–dynamics consistency trade-off of single-stage LaSDI. Contribution/Results: Theoretical analysis and extensive experiments demonstrate that MS-LaSDI significantly reduces both prediction and reconstruction errors, accelerates training, and achieves superior accuracy, robustness, and generalization on high-frequency and strongly nonlinear PDE problems. It establishes a scalable, data-driven paradigm for ROM construction.

Technology Category

• Machine Learning: Learning with Manifolds
• Intelligent Robots: State Estimation
• Multiagent Systems: Multiagent Learning

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

Abstract

Determining accurate numerical solutions of partial differential equations (PDEs) is an important task in many scientific disciplines. However, solvers can be computationally expensive, leading to the development of reduced-order models (ROMs). Recently, Latent Space Dynamics Identification (LaSDI) was proposed as a data-driven, non-intrusive ROM framework. LaSDI compresses the training data using an autoencoder and learns a system of user-chosen ordinary differential equations (ODEs), which govern the latent space dynamics. This allows for rapid predictions by interpolating and evolving the low-dimensional ODEs in the latent space. While LaSDI has produced effective ROMs for numerous problems, the autoencoder can have difficulty accurately reconstructing training data while also satisfying the imposed dynamics in the latent space, particularly in complex or high-frequency regimes. To address this, we propose multi-stage Latent Space Dynamics Identification (mLaSDI). With mLaSDI, several autoencoders are trained sequentially in stages, where each autoencoder learns to correct the error of the previous stages. We find that applying mLaSDI with small autoencoders results in lower prediction and reconstruction errors, while also reducing training time compared to LaSDI.

Problem

• Accurate PDE solutions are computationally expensive
• Autoencoders struggle with complex data reconstruction
• Multi-stage training improves accuracy and efficiency

Innovation

• Multi-stage autoencoders correct errors sequentially
• Latent space dynamics governed by ODEs
• Small autoencoders reduce errors and training time