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

Chetra Mang, Axel TahmasebiMoradi, Mouadh Yagoubi

TL;DR

The paper tackles the challenge of high‑dimensional parametric domains in machine learning by proposing a domain‑decomposition framework built on iterative principal component analysis (iPCA). It develops two inverse projection strategies to recover original data from reduced coordinates, constructs a stretched 1‑manifold from the reduced data, and introduces a line‑similarity based decomposition (LISSDA) to partition the parametric space. Learning and prediction are performed in the latent space, either through interpolation in the stretched manifold or via per‑domain MLPs, with validation on a 1D harmonic transport problem showing competitive accuracy and improved data efficiency. The work demonstrates that exploiting low‑dimensional manifold structure can yield accurate, scalable predictions and offers a path for extending local, domain‑dependent models to complex physical problems.

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.

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

TL;DR

The paper tackles the challenge of high‑dimensional parametric domains in machine learning by proposing a domain‑decomposition framework built on iterative principal component analysis (iPCA). It develops two inverse projection strategies to recover original data from reduced coordinates, constructs a stretched 1‑manifold from the reduced data, and introduces a line‑similarity based decomposition (LISSDA) to partition the parametric space. Learning and prediction are performed in the latent space, either through interpolation in the stretched manifold or via per‑domain MLPs, with validation on a 1D harmonic transport problem showing competitive accuracy and improved data efficiency. The work demonstrates that exploiting low‑dimensional manifold structure can yield accurate, scalable predictions and offers a path for extending local, domain‑dependent models to complex physical problems.

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.
Paper Structure (20 sections, 4 theorems, 5 equations, 5 figures, 4 tables, 2 algorithms)

This paper contains 20 sections, 4 theorems, 5 equations, 5 figures, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related work
  3. Main contribution
  4. METHODOLOGY
  5. Reduced dimension component using iterative principal component analysis
  6. First approach - pseudo-inverse
  7. Second approach - component complement inverse
  8. Stretched reduced order data manifold
  9. 1-manifold data
  10. 1-manifold stretching
  11. Decomposition of parametric domain using reduced order data manifold
  12. Learning in latent space
  13. Prediction in latent space
  14. RESULTS AND DISCUSSIONS
  15. Harmonic transport problem
  16. ...and 5 more sections

Key Result

Theorem 2.1

Given the original data $X$ of dimension $p\in \mathbb N$, there exists $k\in \mathbb N$ such that a reduced components $X_k$ of dimension $p_k\in \mathbb N$ and $p_k<p$ is expressed as $X_k=\pi_k(X)=X\pi_k$ where $\pi_k = \prod\limits_{j\in \{1...k\}}V_j$ and $V_j$ is the singular vector of the SVD

Figures (5)

  • Figure 1: (a) Definition of turning point and curve and (b) turning curve is stretched by a mirror function.
  • Figure 2: Triangulation of 1-manifold by ball pivoting algorithm.
  • Figure 3: Connected curve of the 1-manifold.
  • Figure 4: Stretched manifold and parametric domain decomposition for the first case with the threshold constant $\gamma=4$.
  • Figure 5: References and predictions of the test dataset and their absolute errors.

Theorems & Definitions (8)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • proof : Proof of Theorem 2.1
  • proof : Proof of Theorem 2.2
  • proof : Proof of Theorem 2.3
  • proof : Proof of Theorem 2.4