Operator learning regularization for macroscopic permeability prediction in dual-scale flow problem

TL;DR

This paper tackles predicting macroscopic permeability in dual-scale Stokes-Brinkman flow by learning the forward map from heterogeneous β to velocity u using Fourier Neural Operators (FNO/TFNO) and then deriving the permeability tensor K. Key contributions include a robust data-generation pipeline for β–u pairs, exploration of multiple regularized loss functions (notably H^1 and H^2 derivatives with βu terms), and demonstration that TFNO with an H^2-based loss yields the most accurate K, including successful zero-shot super-resolution from 64×64 to 128×128 grids. The findings show that second-derivative regularization is crucial for accurate K due to the Laplacian operation in the velocity field, enabling high-fidelity, computationally efficient surrogates for dual-scale textile flows. This method reduces reliance on resource-intensive FFT solvers and holds promise for 3D extensions with realistic microstructures.

Abstract

Liquid composites moulding is an important manufacturing technology for fibre reinforced composites, due to its cost-effectiveness. Challenges lie in the optimisation of the process due to the lack of understanding of key characteristic of textile fabrics - permeability. The problem of computing the permeability coefficient can be modelled as the well-known Stokes-Brinkman equation, which introduces a heterogeneous parameter $β$ distinguishing macropore regions and fibre-bundle regions. In the present work, we train a Fourier neural operator to learn the nonlinear map from the heterogeneous coefficient $β$ to the velocity field $u$, and recover the corresponding macroscopic permeability $K$. This is a challenging inverse problem since both the input and output fields span several order of magnitudes, we introduce different regularization techniques for the loss function and perform a quantitative comparison between them.

Figures (7)

• Figure 1: Distribution of heterogeneous coefficient $\beta$ in test dataset highlighting the scale of $\log_{10}\beta$. While many values of $\beta$ are zero and very close to zero, the coefficient can go up to values as large as $10^{14}$. The figure displays the distribution of magnitude values (in a logarithmic scale) on the training grid across the dataset considered.
• Figure 2: Examples of generated input: left – map of local permeability $k_s$; middle – map of $\mu/k_s$; right – map of heterogeneous coefficient $\beta$, the white region has value of $0$. All colour bars are shown in logarithm scale.
• Figure 3: Comparison of ground truth (GT) velocity field$u_1$ and $u_2$ vs. the output of the FNO model trained on the seven different loss functions (top on and bottom plots) depicted in section \ref{['sec:no_Stokes_Brinkman']} for an example input $\log_{10} \beta$ of the test dataset. The best results for this example can be achieved for training with a $H^1 \beta u \Delta u$ loss function or when optimizing with a $H^1 \beta u$ loss. The neural operators that have been trained on the $H^1$ loss function and the macroscopic pressure gradient loss perform worst.
• Figure 4: Comparison of ground truth (GT) velocity field$u_1$ and $u_2$ vs. the output of the TFNO model trained on the seven different loss functions depicted in Subsection \ref{['sec:no_Stokes_Brinkman']} for an example input $\log_{10} \beta$ of the test dataset. The best results for this example can be achieved for training with a $L^2$ loss function or when optimizing for the macroscopic pressure gradient. The neural operators that have been trained on the $H^1 \beta u$ and $H^2 \beta u$ loss function perform worst.
• Figure 5: Ground truth macroscopic permeability$K_{\text{GT}}$ (x-axis) vs. predicted $K_{\text{pred}}$ (y-axis) for the best performing model. Training the TFNO model with an $H^2$ loss function yields the best results out of all the models and regularizations tested. Except for a few outliers, the computed macroscopic permeability for the test samples is close to the diagonal, implying that $K_{\text{pred}}$ is close to $K_{\text{GT}}$.