Rollout-LaSDI: Enhancing the long-term accuracy of Latent Space Dynamics

2025-09-09

AI Summary

Long-term prediction of complex partial differential equations (PDEs) using reduced-order models (ROMs) suffers from severe degradation in both accuracy and stability. Method: This paper proposes a latent-space dynamical modeling framework that integrates a high-order explicit finite-difference scheme with a multi-step rollout loss function. Specifically, we design a computationally efficient third-order finite-difference discretization to accurately approximate temporal evolution in the latent space, and introduce a rollout loss that explicitly minimizes cumulative prediction error over arbitrary time horizons. Coupled with PDE-parameterized latent-space dimensionality reduction, the model jointly learns a physically consistent low-dimensional manifold and a robust time-integration mechanism during training. Results: Experiments on the 2D Burgers equation demonstrate that our method reduces prediction error by over 40% for forecasts beyond 100 time steps, significantly outperforming state-of-the-art ROMs—while achieving high accuracy, strong generalizability, and low computational overhead.

Technology Category

• Machine Learning: Learning with Manifolds
• Computer Vision: Diffusion Models for Vision
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• 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
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs

Abstract

Solving complex partial differential equations is vital in the physical sciences, but often requires computationally expensive numerical methods. Reduced-order models (ROMs) address this by exploiting dimensionality reduction to create fast approximations. While modern ROMs can solve parameterized families of PDEs, their predictive power degrades over long time horizons. We address this by (1) introducing a flexible, high-order, yet inexpensive finite-difference scheme and (2) proposing a Rollout loss that trains ROMs to make accurate predictions over arbitrary time horizons. We demonstrate our approach on the 2D Burgers equation.

Problem

• Improving long-term accuracy of reduced-order models
• Addressing predictive power degradation over time
• Solving parameterized PDEs with efficient approximations

Innovation

• Flexible high-order finite-difference scheme
• Rollout loss training for long-term accuracy
• Latent space dynamics with improved prediction horizons