Probabilistic Learning Framework for Inverse Material Design

Inverse material design, which seeks microstructures yielding desired macroscopic properties, is inherently ill-posed due to non-uniqueness and the existence of unattainable targets. While data-driven generative models offer powerful empirical solutions, they often lack rigorous theoretical guarantees. This paper establishes a formal probabilistic learning framework to address these challenges. We first prove that the forward homogenization map, which predicts effective properties from a microstructure descriptor field, is Frechet differentiable and locally Lipschitz continuous under physically reasonable assumptions. This foundational result justifies the use of gradient-based methods and sensitivity analysis. Building upon this, we demonstrate the probabilistic well-posedness of the Bayesian inverse problem: the posterior distribution over microstructures is well-defined and depends continuously on the target property data. Furthermore, we prove Bayesian consistency, showing that as observational uncertainty vanishes, the posterior measure concentrates on the true set of solutions. This theoretical foundation validates advanced probabilistic inference techniques (e.g., MCMC, variational inference) for robustly exploring the solution manifold and quantifying uncertainty, thereby enabling a more complete and practical exploration of the material design space.