Higher-Order LaSDI: Reduced Order Modeling with Multiple Time Derivatives

2025-12-17

AI Summary

To address the severe degradation in prediction accuracy and stability of reduced-order models (ROMs) for long-time evolution of complex partial differential equations (PDEs), this work proposes a high-order low-dimensional modeling paradigm. The method explicitly incorporates multiple temporal derivatives to characterize system dynamics and introduces a novel Rollout loss function to enforce multi-step temporal consistency during optimization. We pioneer a joint training framework integrating high-order finite-difference discretizations with the Rollout loss, thereby overcoming the fundamental accuracy bottleneck of conventional ROMs in long-horizon prediction. The approach synergistically combines proper orthogonal decomposition (POD), supervised neural network modeling, and high-order temporal discretization. Evaluated on the 2D Burgers equation, the proposed method reduces long-time simulation error by over 60% and accelerates single-step inference by three orders of magnitude compared to baseline ROMs.

Technology Category

• Intelligent Robots: Learning & Optimization for ROB
• Computer Vision: Learning & Optimization for CV
• Search and Optimization: Learning to Search

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: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated 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

• Improves long-term predictive accuracy of reduced-order models for PDEs
• Introduces a flexible high-order finite-difference scheme for efficiency
• Proposes a Rollout loss to train models over arbitrary time horizons

Innovation

• Flexible high-order finite-difference scheme for efficiency
• Rollout loss trains models for long-term accuracy
• Demonstrated on 2D Burgers equation for validation