Learning Homogenization for Elliptic Operators

TL;DR

This work studies the learnability of homogenized constitutive laws for divergence-form elliptic PDEs with discontinuous coefficients by framing the problem as learning maps between coefficient fields and the cell-problem solution $\chi$ and the homogenized tensor $\overline{A}$. It develops stability results establishing continuity and Lipschitz dependence of the cell problem on the coefficient in $L^2$ and $L^q$ norms, enabling rigorous universal approximation results for Fourier Neural Operators (FNOs) approximating the maps $A\mapsto\chi$ and $A\mapsto\overline{A}$. The authors introduce four microstructure classes (smooth, star-shaped, square, Voronoi) and demonstrate, through extensive numerical experiments, that FNOs can accurately learn the solution operator and effective properties, with larger errors concentrated near discontinuities and corners, while Voronoi geometries show robustness to discretization. The results provide a principled foundation for data-driven homogenization in continuum mechanics, offering guidance for learning constitutive models in materials with complex microscale features and suggesting extensions to locally periodic or random media. Overall, the paper combines rigorous stability analysis with practical operator-learning demonstrations to advance principled data-driven homogenization.

Abstract

Multiscale partial differential equations (PDEs) arise in various applications, and several schemes have been developed to solve them efficiently. Homogenization theory is a powerful methodology that eliminates the small-scale dependence, resulting in simplified equations that are computationally tractable while accurately predicting the macroscopic response. In the field of continuum mechanics, homogenization is crucial for deriving constitutive laws that incorporate microscale physics in order to formulate balance laws for the macroscopic quantities of interest. However, obtaining homogenized constitutive laws is often challenging as they do not in general have an analytic form and can exhibit phenomena not present on the microscale. In response, data-driven learning of the constitutive law has been proposed as appropriate for this task. However, a major challenge in data-driven learning approaches for this problem has remained unexplored: the impact of discontinuities and corner interfaces in the underlying material. These discontinuities in the coefficients affect the smoothness of the solutions of the underlying equations. Given the prevalence of discontinuous materials in continuum mechanics applications, it is important to address the challenge of learning in this context; in particular, to develop underpinning theory that establishes the reliability of data-driven methods in this scientific domain. The paper addresses this unexplored challenge by investigating the learnability of homogenized constitutive laws for elliptic operators in the presence of such complexities. Approximation theory is presented, and numerical experiments are performed which validate the theory in the context of learning the solution operator defined by the cell problem arising in homogenization for elliptic PDEs.

Figures (6)

• Figure 1: Microstructure Examples
• Figure 1: Visualization of the trained models evaluated on test samples that gave median relative $H^1$ error for each microstructure. The microstructure inputs of each row correspond to those of Figure \ref{['fig:microstructures']}. The first shows the true $\chi_1$, the second shows the $FNO$ predicted $\chi_1$, and the third shows the absolute value of the error between the true and predicted $\chi_1$. The fourth column shows the 2-norm of the gradient of the true $\chi_1$, and the fifth shows the 2-norm of the gradient of the predicted $\chi_1$. The last column shows the 2-norm of the difference between the two gradients.
• Figure 2: Errors for each each numerical experiment; five sample models are trained for each microstructure. The expressions for the RHE (Relative $H^1$ Error), RWE (Relative $W^{1,10}$ Error) and RAE (Relative $\overline{A}$ Error) may be found in equations \ref{['eqn:errors']} and \ref{['eqn:err_A']}. The errors are evaluated over a test set of size $500$. All examples have varying geometry except the second Voronoi example.
• Figure 3: Five sample models trained on Smooth and Voronoi data at $128 \times 128$ grid resolution evaluated at different resolutions.
• Figure 4: A comparison of test error for different amounts of training data for models trained on Voronoi and Smooth data. Five sample models are used for each data point.

Theorems & Definitions (43)

• Proposition 1.0
• Proposition 1.0
• Proposition 1.0
• Remark 1.1
• Definition 3.1: General Neural Operator
• Definition 3.2: Fourier Neural Operator
• Theorem 3.3
• Theorem 3.4
• Proposition A.0
• Proof 1