Distributional Inverse Homogenization

2026-04-15

AI Summary

This work addresses the longstanding challenge of non-invasively inferring statistical characteristics of material microstructures from macroscopic mechanical responses. The authors propose a distributional inverse homogenization framework that, for the first time, integrates probabilistic modeling with homogenization theory to formulate a novel class of inverse problems. By leveraging large datasets of macroscopic measurements, the method learns global statistical properties of underlying microstructures. Built upon Voronoi tessellations and accelerated via surrogate models, the approach is theoretically grounded in the one-dimensional setting. Numerical experiments on two-dimensional Voronoi-based material systems demonstrate that the statistical distribution of microstructural features can be accurately recovered solely from macroscopic responses, enabling non-invasive, data-driven inference of microstructural variability.

Technology Category

• Machine Learning: Learning with Manifolds
• Reasoning under Uncertainty: Probabilistic Inference
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

For many materials, macroscopic mechanical behavior is determined by an intricate microstructure. Understanding the relation between these two scales helps scientists and engineers design better materials. The relation which maps microstructure to bulk mechanical properties can be understood via the well-established theory of homogenization. However inverting the homogenization process, to recover microstructural information from measured macroscopic properties, is fraught with difficulties because of the averaging processes that underlie homogenization. Therefore, scientists and engineers usually need recourse to more invasive, often highly localized, investigations to learn about a microstructure. In this work, we develop a noninvasive methodology by which one can leverage large collections of measured bulk mechanical properties to learn information about the statistics of microstructure at a global level. We call this, distributional inverse homogenization. We study this problem in one and two dimensions, considering both periodic and stochastic homogenization. We demonstrate the methodology in the context of 2D Voronoi constructions and underpin the observed empirical success with theory in 1D. We also show how the natural spatial variability of microstructure can be exploited to gather data that enables distributional inversion. And we concurrently learn a surrogate model, approximating the homogenization map, that accelerates the resulting computations in this setting. The work formulates a new class of inverse problems, bridging ideas from probability and homogenization to facilitate the learning of microstructural material variability from macroscopic measurements.

Problem

• inverse homogenization
• microstructure
• macroscopic properties
• distributional inference
• noninvasive characterization

Innovation

• distributional inverse homogenization
• microstructure statistics
• surrogate modeling
• stochastic homogenization
• noninvasive inference