Adam P. Generale

Forecasting stochastic nonlinear dynamical systems under the influence of conditioning variables is a fundamental challenge repeatedly encountered across the biological and physical sciences. While flow-based models can impressively predict the temporal evolution of probability distributions representing possible outcomes of a specific process, existing frameworks cannot satisfactorily account for the impact of conditioning variables on these dynamics. Amongst several limitations, existing methods require training data with paired conditions and are developed for discrete conditioning variables. We propose Conditional Variable Flow Matching (CVFM), a framework for learning flows transforming conditional distributions with amortization across continuous conditioning variables - permitting predictions across the conditional density manifold. This is accomplished through several novel advances. In particular, simultaneous sample conditioned flows over the main and conditioning variables. In addition, motivated by theoretical analysis, a conditional Wasserstein distance combined with a loss reweighting kernel facilitating conditional optimal transport. Collectively, these advances allow for learning system dynamics provided measurement data whose states and conditioning variables are not in correspondence. We demonstrate CVFM on a suite of increasingly challenging problems, including discrete and continuous conditional mapping benchmarks, image-to-image domain transfer, and modeling the temporal evolution of materials internal structure during manufacturing processes. We observe that CVFM results in improved performance and convergence characteristics over alternative conditional variants.

Inverse problems are central to material design. While numerous studies have focused on designing microstructures by inverting structure-property linkages for various material systems, such efforts stop short of providing realizable paths to manufacture such structures. Accomplishing the dual task of designing a microstructure and a feasible manufacturing pathway to achieve a target property requires inverting the complete process-structure-property linkage. However, this inversion is complicated by a variety of challenges such as inherent microstructure stochasticity, high-dimensionality, and ill-conditioning of the inversion. In this work, we propose a Bayesian framework leveraging a lightweight flow-based generative approach for the stochastic inversion of the complete process-structure-property linkage. This inversion identifies a solution distribution in the processing parameter space; utilizing these processing conditions realizes materials with the target property sets. Our modular framework readily incorporates the output of stochastic forward models as conditioning variables for a flow-based generative model, thereby learning the complete joint distribution over processing parameters and properties. We demonstrate its application to the multi-objective task of designing processing routes of heterogeneous materials given target sets of bulk elastic moduli and thermal conductivities.

Inverse Microstructure Design problems are ubiquitous in materials science; for example, property-driven microstructure design requires the inversion of a structure–property linkage. However, prior frameworks have struggled to address this problem’s unique combination of challenges: the high dimensionality and stochasticity of microstructures, under sampled initial datasets, and ill-conditioning of the inversion. In this work, we propose a computational framework for Inverse Microstructure Design problems using a Bayesian methodology. We construct this framework from three modular components, enabling flexible extension and re-use. First, we define a low-dimensional, informative microstructure prior by integrating domain knowledge (i.e., statistical continuum mechanics) into a distributional learning scheme. This scheme includes multiple latent representations which address the challenges inherent to representing microstructures. Second, we define a property-specific likelihood using a multi-output Gaussian process regression surrogate model. Finally, we efficiently learn the conditional posterior density for a given target property, and generate samples using deep variational inference. We demonstrate our proposed method for solving stochastic microstructure design problems by identifying woven ceramic matrix composites matching target anisotropic thermal conductivities. Through this example, we analyze the integral role of each component in the inversion framework.

Hierarchical multiscale modeling of heterogeneous materials has traditionally relied upon a deterministic estimation of constitutive properties when making microstructure-sensitive predictions of effective response at each subsequent length-scale. Such an approach is wholly unsuitable for a variety of material classes, such as ceramic matrix composites, which exhibit large variability at multiple length-scales. This work demonstrates a framework for approaching two open problems towards improved microstructuresensitive predictions, namely, (i) probabilistically calibrating complex constitutive models at the mesoscale to sparsely observed macroscale experimental data, and (ii) propagating this stochastic constituent behavior at the mesoscale towards low-cost homogenized predictions for unseen microstructures. The proposed stochastic scale-bridging framework displays a continuity of information flow where no portion of the experimental data is neglected out of convenience, facilitating the greatest information gain from oftentimes costly experiments. In this paper, suitable protocols were developed to address the challenges described above. The protocols were subsequently demonstrated on ceramic matrix composite’s uniaxial tensile stress–strain response, where constituent behavior at the mesoscale was described using continuum damage mechanics, and predictions encapsulating constitutive model parameter uncertainty were made for novel microstructures. The methodology presented in this work is broadly applicable to various material classes and constitutive models with high-dimensional parameter sets.