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Parametric Probabilistic Manifold Decomposition for Nonlinear Model Reduction

Jiaming Guo, Dunhui Xiao

TL;DR

The proposed Parametric Probabilistic Manifold Decomposition (PPMD) achieves superior accuracy and generalization beyond the training parameter range compared to the conventional proper orthogonal decomposition with Gaussian process regression (POD+GPR) method.

Abstract

Probabilistic Manifold Decomposition (PMD)\cite{doi:10.1137/25M1738863}, developed in our earlier work, provides a nonlinear model reduction by embedding high-dimensional dynamics onto low-dimensional probabilistic manifolds. The PMD has demonstrated strong performance for time-dependent systems. However, its formulation is for temporal dynamics and does not directly accommodate parametric variability, which limits its applicability to tasks such as design optimization, control, and uncertainty quantification. In order to address the limitations, a \emph{Parametric Probabilistic Manifold Decomposition} (PPMD) is presented to deal with parametric problems. The central advantage of PPMD is its ability to construct continuous, high-fidelity parametric surrogates while retaining the transparency and non-intrusive workflow of PMD. By integrating probabilistic-manifold embeddings with parameter-aware latent learning, PPMD enables smooth predictions across unseen parameter values (such as different boundary or initial conditions). To validate the proposed method, a comprehensive convergence analysis is established for PPMD, covering the approximation of the linear principal subspace, the geometric recovery of the nonlinear solution manifold, and the statistical consistency of the kernel ridge regression used for latent learning. The framework is then numerically demonstrated on two classic flow configurations: flow past a cylinder and backward-facing step flow. Results confirm that PPMD achieves superior accuracy and generalization beyond the training parameter range compared to the conventional proper orthogonal decomposition with Gaussian process regression (POD+GPR) method.

Parametric Probabilistic Manifold Decomposition for Nonlinear Model Reduction

TL;DR

The proposed Parametric Probabilistic Manifold Decomposition (PPMD) achieves superior accuracy and generalization beyond the training parameter range compared to the conventional proper orthogonal decomposition with Gaussian process regression (POD+GPR) method.

Abstract

Probabilistic Manifold Decomposition (PMD)\cite{doi:10.1137/25M1738863}, developed in our earlier work, provides a nonlinear model reduction by embedding high-dimensional dynamics onto low-dimensional probabilistic manifolds. The PMD has demonstrated strong performance for time-dependent systems. However, its formulation is for temporal dynamics and does not directly accommodate parametric variability, which limits its applicability to tasks such as design optimization, control, and uncertainty quantification. In order to address the limitations, a \emph{Parametric Probabilistic Manifold Decomposition} (PPMD) is presented to deal with parametric problems. The central advantage of PPMD is its ability to construct continuous, high-fidelity parametric surrogates while retaining the transparency and non-intrusive workflow of PMD. By integrating probabilistic-manifold embeddings with parameter-aware latent learning, PPMD enables smooth predictions across unseen parameter values (such as different boundary or initial conditions). To validate the proposed method, a comprehensive convergence analysis is established for PPMD, covering the approximation of the linear principal subspace, the geometric recovery of the nonlinear solution manifold, and the statistical consistency of the kernel ridge regression used for latent learning. The framework is then numerically demonstrated on two classic flow configurations: flow past a cylinder and backward-facing step flow. Results confirm that PPMD achieves superior accuracy and generalization beyond the training parameter range compared to the conventional proper orthogonal decomposition with Gaussian process regression (POD+GPR) method.
Paper Structure (35 sections, 5 theorems, 97 equations, 8 figures, 1 table)

This paper contains 35 sections, 5 theorems, 97 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Governing equations
  3. Probabilistic Manifold Decomposition (PMD)
  4. Linear feature extraction and residual separation
  5. Probabilistic manifold construction
  6. Dynamic prediction in latent spaces
  7. Linear latent dynamics
  8. Nonlinear dynamics on the probabilistic manifold
  9. Reconstruction via lifting
  10. Parametric probabilistic manifold decomposition(PPMD)
  11. Snapshot Matrix Construction
  12. Dimensionality Reduction and Manifold Embedding
  13. Parametric Prediction in Latent Space
  14. From discrete latent samples to a dense continuous parameter trajectory
  15. Smoothing splines and model selection
  16. ...and 20 more sections

Key Result

Lemma 5.1

Assuming the residual vectors $\{r_i\}_{i=1}^{n_s}$ are i.i.d. samples from a compact $d$-dimensional Riemannian manifold $\mathcal{M} \subset \mathbb{R}^N$ with a smooth density $p: \mathcal{M} \to (0,\infty)$ satisfying $0 < p_{\min} \leq p(x) \leq p_{\max} < \infty$ for all $x \in \mathcal{M}$. L

Figures (8)

  • Figure 1: Flow past a cylinder at Re=250: Comparison of velocity reconstruction solutions at $t=15s$ from the high-fidelity full model, POD+GPR, and PPMD using 6 and 12 basis functions.
  • Figure 2: Flow past a cylinder: reconstruction errors at Re=250 and $t=15s$ for POD+GPR and PPMD using 6 and 12 basis functions.
  • Figure 3: Flow past a cylinder: velocity solutions at $Re=392$ and $Re=104$ at $t=15s$ from the high-fidelity full model, POD+GPR, and PPMD using 6 and 12 basis functions.
  • Figure 4: Flow past a cylinder test case: prediction errors at $Re=392$ and $Re=104$ at $t=15s$ from the high-fidelity full model for POD+GPR and PPMD using 6 and 12 basis functions.
  • Figure 5: Backward-facing step flow at Re=2300: Comparison of reconstruction velocity solutions at $t=25s$ from the high-fidelity full model, POD+GPR, and PPMD using 6 and 12 basis functions.
  • ...and 3 more figures

Theorems & Definitions (9)

  • Lemma 5.1: Graph-geodesic approximation
  • Proof 1
  • Theorem 5.2: Convergence of the probability transition operator
  • Proof 2
  • Corollary 5.3: Eigenvector convergence
  • Theorem 5.4: Convergence of kernel ridge regression
  • Proof 3
  • Corollary 5.5: Vector-valued KRR convergence
  • Proof 4