Operator Inference Aware Quadratic Manifolds with Isotropic Reduced Coordinates for Nonintrusive Model Reduction

2025-07-27

AI Summary

In non-intrusive model order reduction (MOR), quadratic manifold embeddings often exhibit oscillatory and non-smooth behavior, leading to an imbalance between reconstruction and long-term prediction errors. Method: This paper proposes an operator-inference-oriented joint optimization framework. It constructs a quadratic manifold parameterized by isotropic reduced coordinates and simultaneously minimizes both snapshot reconstruction error and long-term prediction error of the reduced-order model (ROM). A greedy training strategy is introduced to enforce embedding smoothness and learnability. Contribution/Results: To our knowledge, this is the first work to explicitly incorporate downstream prediction performance—rather than solely reconstruction fidelity—into the manifold learning objective, thereby mitigating geometric distortion inherent in reconstruction-only approaches. Evaluated on convection- and turbulence-dominated benchmark problems, the proposed method improves ROM prediction accuracy by up to two orders of magnitude over baseline methods, significantly enhancing robustness and generalization capability.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Learning to Search
• Intelligent Robots: Learning & Optimization for ROB

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

Abstract

Quadratic manifolds for nonintrusive reduced modeling are typically trained to minimize the reconstruction error on snapshot data, which means that the error of models fitted to the embedded data in downstream learning steps is ignored. In contrast, we propose a greedy training procedure that takes into account both the reconstruction error on the snapshot data and the prediction error of reduced models fitted to the data. Because our procedure learns quadratic manifolds with the objective of achieving accurate reduced models, it avoids oscillatory and other non-smooth embeddings that can hinder learning accurate reduced models. Numerical experiments on transport and turbulent flow problems show that quadratic manifolds trained with the proposed greedy approach lead to reduced models with up to two orders of magnitude higher accuracy than quadratic manifolds trained with respect to the reconstruction error alone.

Problem

• Minimizes reconstruction and prediction errors in reduced modeling
• Avoids oscillatory embeddings for accurate reduced models
• Improves accuracy by two orders in transport and flow problems

Innovation

• Greedy training for reconstruction and prediction errors
• Quadratic manifolds avoiding non-smooth embeddings
• Two orders accuracy gain in reduced models