A Pretraining-Finetuning Computational Framework for Material Homogenization

TL;DR

This work tackles the computational bottleneck of numerical homogenization for multiscale elasticity by introducing PreFine-Homo, a two-phase framework that first pretrains a Fourier Neural Operator (FNO) to map geometry and material properties to displacement fields, then optionally fine-tunes the prediction with a physics-based iterative solver to obtain the homogenized tensor $E_{ijkl}^H$. The pretraining phase yields dramatic speedups (up to around 1,000×) while maintaining high accuracy, and the fine-tuning phase provides substantial accuracy improvements and enables unlimited extrapolation by leveraging FEM-like iterations with a strong initial guess. The model demonstrates robust performance across TPMS geometries, multiple materials, and varying resolutions, including cross-resolution and extrapolation to GRF-generated structures, with systematic analyses of iteration savings and convergence behavior. The framework also discusses theoretical and practical aspects of combining data-driven operators with physics-based solvers, and outlines potential extensions to nonlinear problems and physics-informed operator learning, along with a planned self-learning loop to continuously improve the pretrained operator as more data become available.

Abstract

Homogenization is a fundamental tool for studying multiscale physical phenomena. Traditional numerical homogenization methods, heavily reliant on finite element analysis, demand significant computational resources, especially for complex geometries, materials, and high-resolution problems. To address these challenges, we propose PreFine-Homo, a novel numerical homogenization framework comprising two phases: pretraining and fine-tuning. In the pretraining phase, a Fourier Neural Operator (FNO) is trained on large datasets to learn the mapping from input geometries and material properties to displacement fields. In the fine-tuning phase, the pretrained predictions serve as initial solutions for iterative algorithms, drastically reducing the number of iterations needed for convergence. The pretraining phase of PreFine-Homo delivers homogenization results up to 1000 times faster than conventional methods, while the fine-tuning phase further enhances accuracy. Moreover, the fine-tuning phase grants PreFine-Homo unlimited generalization capabilities, enabling continuous learning and improvement as data availability increases. We validate PreFine-Homo by predicting the effective elastic tensor for 3D periodic materials, specifically Triply Periodic Minimal Surfaces (TPMS). The results demonstrate that PreFine-Homo achieves high precision, exceptional efficiency, robust learning capabilities, and strong extrapolation ability, establishing it as a powerful tool for multiscale homogenization tasks.

Figures (17)

• Figure 1: Schematic diagram of the Fourier neural operator (FNO) in the neural operator including GNO, LNO, and FNO: $\boldsymbol{P}$ is used to upsample the input data for better processing by the Fourier layer. $\boldsymbol{Q}$ downsamples the data processed by the Fourier layer to the output space. $\boldsymbol{v}^{(t-1)}$ and $\boldsymbol{v}^{(t)}$ are the input and output of the $t$-th Fourier layer, respectively. $\mathcal{F}$ is the fast Fourier transform. $\Re$ performs a linear transformation on the output after the fast Fourier transform, and then applies the inverse Fourier transform $\mathcal{F}^{-1}$. $\boldsymbol{W}$ performs a linear transformation on $\boldsymbol{v}^{(t-1)}$. $\sigma$ is a nonlinear transformation.
• Figure 2: PreFine-Homo consists of two key steps: pretraining and fine-tuning. In the pretraining phase, a large amount of data with different materials and geometries is obtained using traditional numerical solvers. Subsequently, the Fourier Neural Operator (FNO) is employed to fit the large dataset. The fine-tuning phase involves using the FNO to provide an initial solution, which is then used as the initial vector in a traditional iterative algorithm for further refinement. The pretraining phase of PreFine-Homo enables rapid prediction of displacement fields for new geometries and materials in the test set. Depending on the requirements, the fine-tuning phase may be optionally applied, and the effective elastic tensor is subsequently obtained based on the homogenization formula.
• Figure 3: Prediction results of the proposed PreFine-Homo model in the pretraining phase across different geometries (Resolution 128) for Schoen Gyroid of "sheet-networks" in the test set under uniaxial tension in the x-direction. (a) The left column shows the geometric input for PreFine-Homo, and the right two columns show the FEM reference solution and the prediction by PreFine-Homo for the $x$-direction displacement field, $y$-direction displacement field(b), $z$-direction displacement field(c). (d) The relative error of the effective elastic tensor in the corresponding positions by PreFine-Homo, using the traditional finite element method as the reference solution. The second, third, and fourth rows of (a, b, c) show the results for different 2D cross-sections.
• Figure 4: Prediction results of the proposed PreFine-Homo model in the pretraining phase across different geometries (Resolution 128) for Schwarz Diamond (a,b,c,d) and Fischer-Koch S (e,f,g,h) of both "sheet-networks" in the test set under uniaxial tension in the x-direction.
• Figure 5: Training and testing errors of the pretraining phase of the PreFine-Homo for the three displacement fields under six loading conditions in different geometries (Resolution 128). The numbers in the figures represent the final converged values of the testing errors: (a) $x$-direction tension, (b) $y$-direction tension, (c) $z$-direction tension, (d) $xy$-direction shear, (e) $xz$-direction shear, (f) $yz$-direction shear.