Neural Contrast Expansion for Explainable Structure-Property Prediction and Random Microstructure Design

Effective properties of composite materials are defined as the ensemble average of property-specific PDE solutions over the underlying microstructure distributions. Traditionally, predicting such properties can be done by solving PDEs derived from microstructure samples or building data-driven models that directly map microstructure samples to properties. The former has a higher running cost, but provides explainable sensitivity information that may guide material design; the latter could be more cost-effective if the data overhead is amortized, but its learned sensitivities are often less explainable. With a focus on properties governed by linear self-adjoint PDEs (e.g., Laplace, Helmholtz, and Maxwell curl-curl) defined on bi-phase microstructures, we propose a structure-property model that is both cost-effective and explainable. Our method is built on top of the strong contrast expansion (SCE) formalism, which analytically maps $N$-point correlations of an unbounded random field to its effective properties. Since real-world material samples have finite sizes and analytical PDE kernels are not always available, we propose Neural Contrast Expansion (NCE), an SCE-inspired architecture to learn surrogate PDE kernels from structure-property data. For static conduction and electromagnetic wave propagation cases, we show that NCE models reveal accurate and insightful sensitivity information useful for material design. Compared with other PDE kernel learning methods, our method does not require measurements about the PDE solution fields, but rather only requires macroscopic property measurements that are more accessible in material development contexts.