Multifidelity POD

Proper Orthogonal Decomposition Surrogate for Rarefied Gas Dynamics

Python PyTorch Reduced-Order Modeling LBM
PODMM prediction vs LBM ground truth across 5 cases

Coarse-to-Fine Resolution Mapping with 8× Speedup

Coarse In,
Fine Out

Running high-resolution LBM simulations of rarefied gas transport is expensive: a $1024 \times 1024$ grid can take 10+ minutes to converge. The Proper Orthogonal Decomposition Mapping Method (PODMM) learns a multifidelity mapping from cheap coarse ($256 \times 256$) snapshots to the expensive fine ($1024 \times 1024$) steady-state solution.

Trained on paired coarse/fine snapshots via SVD, PODMM is non-intrusive, requires no modification to the underlying solver, and reduces the total computational cost by up to 8×.

Key Results

  • Avg. Rel. Error (synthetic): 6.3%
  • Speedup (synthetic): 8×
  • Error (Mowry shale): 15%
  • Speedup (Mowry shale): 3.4×
  • $R^2$ (Mowry shale): ≥ 0.98
01. Methodology

The PODMM Framework

PODMM stacks paired coarse ($\mathbf{g}$) and fine ($\mathbf{f}$) snapshot matrices into a joint matrix $W^{\mathrm{PODMM}}$ and performs a Singular Value Decomposition. The resulting shared POD basis $U_M$ is partitioned into fine-resolution modes $\zeta_f$ and coarse-resolution modes $\zeta_g$:

$$ W^{\mathrm{PODMM}} = \begin{bmatrix} \mathbf{f}_1 - \bar{\mathbf{f}} & \cdots & \mathbf{f}_N - \bar{\mathbf{f}} \\ \mathbf{g}_1 - \bar{\mathbf{g}} & \cdots & \mathbf{g}_N - \bar{\mathbf{g}} \end{bmatrix} = U\Sigma V^T $$

At prediction time, a new coarse snapshot $\mathbf{g}_t$ is projected onto $\zeta_g$ to find reduced coordinates $\boldsymbol{\alpha}_t$, then the same coordinates reconstruct the fine-resolution prediction via $\zeta_f$:

$$ \boldsymbol{\alpha}_t = \arg\min_{\boldsymbol{\alpha}} \|\mathbf{g}_t - \bar{\mathbf{g}} - \zeta_g \boldsymbol{\alpha}\|_2^2 $$
$$ \mathbf{f}_t^{\mathrm{ROM}} = \bar{\mathbf{f}} + \zeta_f \boldsymbol{\alpha}_t $$

The operator is provably BIBO stable and Lipschitz continuous — bounded inputs always yield bounded outputs.

02. Synthetic Training Cases
Five synthetic pore geometries used for PODMM training

Five synthetic pore geometries (coarse 256² and fine 1024²)

Coarse → Fine: 4× Resolution

Five pore geometries were constructed on $256 \times 256$ (coarse) and $1024 \times 1024$ (fine) grids at 1 nm and 0.25 nm pixel resolution, respectively. The methane mean free path at $2\,\mathrm{MPa}$ and $300\,\mathrm{K}$ gives local Knudsen numbers spanning 0.02–0.2.

PODMM is trained on only 10% of available snapshots for most cases. Using coarse velocity fields as input, it predicts the fine-resolution converged velocity field — without ever running the expensive fine simulation to completion.

Coarse grid256 × 256
Fine grid1024 × 1024
Res. ratio4×
Training data10% of snaps
03. Results
PODMM relative error field for Case 1

Relative error field: PODMM vs. LBM ground truth (Case 1, 1.94% mean error)

Case Relative Error (%) Mean (%) Median (%)
11.946.292.64
22.13
37.62
417.11
52.64
04. Real-World Validation: Mowry Shale

From Synthetic to Real Rock

A scanning electron microscopy (SEM) image of Mowry shale — a formation of major interest for natural gas and CO₂ storage in Wyoming — was binarized and used to generate a $6142 \times 6142$ fine-resolution LBM simulation with 2.8 million active lattice nodes.

The full fine-resolution simulation requires ~41 hours of CPU time. PODMM, trained on 30% of coarse snapshots, predicts the final state in under one minute — achieving a 3.4× total speedup with only 15% relative error and $R^2 \approx 0.98$.

Fine grid6142 × 6142
Active nodes2.8 million
LBM wall-time~41 hours
PODMM total~12 hours
PODMM prediction vs LBM for Mowry shale PODMM vs LBM scatter plot for Mowry shale

Mowry shale: velocity field prediction and scatter validation ($R^2 \geq 0.98$)

Publication

Application of Multifidelity Proper Orthogonal Decomposition-Based Surrogate Models to Rarefied Gas Dynamics

Kitterman, A., Rustamov, N., Liu, Y., & Aryana, S. A.

Published in Physics of Fluids (2026).

Read Paper → doi.org/10.1063/5.0333761

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