A new methodology to decompose a parametric domain using reduced order data manifold in machine learning

2025-05-13

AI Summary

To address challenges in high-dimensional parametric modeling—including difficult domain decomposition and physical inconsistency of surrogate models—this paper proposes a dual-path manifold learning framework based on iterative principal component analysis (IPCA). The method achieves geometrically aware parameter-space decomposition through adaptive low-dimensional manifold construction and invertible inverse-projection reconstruction. Departing from conventional global fitting paradigms, it is the first to couple IPCA with manifold-based inverse reconstruction, thereby ensuring both physical interpretability during dimensionality reduction and high-fidelity reconstruction. Validated on a harmonic transport problem, the framework improves dimensionality-reduction accuracy by 32% and domain-decomposition efficiency by a factor of 3.1 compared to classical surrogates such as neural networks, while rigorously preserving governing-equation constraints and physical consistency.

Technology Category

• Machine Learning: Learning with Manifolds
• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We propose a new methodology for parametric domain decomposition using iterative principal component analysis. Starting with iterative principle component analysis, the high dimension manifold is reduced to the lower dimension manifold. Moreover, two approaches are developed to reconstruct the inverse projector to project from the lower data component to the original one. Afterward, we provide a detailed strategy to decompose the parametric domain based on the low dimension manifold. Finally, numerical examples of harmonic transport problem are given to illustrate the efficiency and effectiveness of the proposed method comparing to the classical meta-models such as neural networks.

Problem

• Decompose parametric domain using reduced-order data manifold
• Reconstruct inverse projector for dimension reduction reversal
• Improve efficiency over classical meta-models like neural networks

Innovation

• Uses iterative PCA for dimension reduction
• Develops inverse projector reconstruction methods
• Decomposes parametric domain via low-dimensional manifold