Understanding the relationships between processing, structure, properties, and performance (PSPP) [

1,

2] is critical in material science. In general, there are two broad modeling paradigms: forward and inverse. Forward modeling is to predict the effects or results given a set of known causes, e.g., exploring the relationships from processing to performance in materials. As different sets of inputs might cause the same result, forward modeling usually learns a many-to-one mapping. Forward modeling has been widely studied in various fields of machine learning, such as object detection [

3,

4], image segmentation [

5,

6], machine translation [

7,

8] and some prediction tasks in scientific computing [

9‐

16]. Inverse modeling is the process to infer the causes based on results or observations, e.g., exploring the relationships from performance to processing in materials. Inverse problems are one of the most important problems in science as they can help us understand the unknown causes leading to the observations. Thus, it is extensively used in various scientific fields, such as geophysics, health care and materials science [

17‐

24]. Discovering microstructures that exhibit given properties is one of the inverse problems focused on structure–property linkage in materials, which is explored in this work. Variation in microstructure leads to a wide range of material properties, which in turn impacts the performance. Thus, inferring possible microstructures for a given property can help domain scientists improve the materials’ performance and accelerate materials discovery, and design. Traditional approaches [

25,

26] for inverse modeling mainly rely on human analysis and experiments, which are extremely expensive in terms of cost and time. With the availability of large amounts of reliable data, data-driven methods have been tried to solve inverse problems. However, there are still many challenges for inverse modeling. The challenges for inverse modeling are threefold: (1) Inverse modeling usually requires learning a one-to-many nonlinear mapping. Because it is possible that different input combinations from many causal factors might cause the same output, there may be more than one microstructure that has a given property. This lack of uniqueness makes it difficult to train inverse models. (2) Inverse models usually need to learn a mapping from low-dimension inputs to high-dimension outputs, which means important missing information needs to be recovered from less informational inputs to produce high informational outputs. Thus, if the inverse model directly learns the mapping from inputs to outputs, the outputs might have limited diversity and only cover a small portion of real data distribution, especially when the difference of dimensionality between inputs and outputs is significant. In this work, the microstructures are represented by images, which are much more high-dimensional as compared to properties. (3) Traditional approaches for inverse modeling usually involve an iterative learning process, such as optimization, so that optimal or near-optimal solution can gradually be achieved by minimizing the error between candidate solution and target. However, due to the fact that the space of all possible causal factors can be extremely large, inverse modeling requires significant computing time. To overcome the above challenges, we propose a framework that combines generative adversarial networks (GAN) [

27] and mixture density networks (MDN) [

28] for inverse modeling. More specifically, a GAN is first trained so that the high-dimensional (i.e., high-resolution) microstructure image

x can be represented by low-dimensional latent variable vector

z. Then, we can utilize MDN, a neural network attempting to learn one-to-many nonlinear mapping (i.e., address challenge 1), to model the mapping from image property

y to latent variable vector

z instead of directly mapping from image property

y to image

x. Because latent variable vector

z has similar dimensionality as the image property

y, it is easier and more stable to train the MDN by using latent variable vector

z as an immediate representation of image

x (i.e., address challenge 2). Also, it is expected to increase the diversity of the outputs of the inverse model to cover a wider range of real data distribution. After the proposed framework is well trained, given a desired image property

y, the MDN can produce various sets of latent variable vector

z, which can be further used by GAN to generate corresponding images

x to solve the inverse problem. Because the proposed framework is based on deep learning, it only requires one forward pass to produce various predictions, which means it can quickly produce possible solutions using modern computation resources (i.e., address challenge 3). We apply the proposed framework on a materials science inverse problem where microstructure images

x need to be designed given a desired material’s optical absorption property value

y. Three baseline methods are used to evaluate the performance of the proposed framework: (1) Optimization-based inverse modeling method; (2) Deep learning-based inverse modeling method that directly maps from material’s optical absorption property value

y to microstructure images

x; (3) The third baseline combines traditional dimensionality reduction, such as principal component analysis (PCA), and MDN to illustrate the advantage of using GAN. Compared with baseline methods, the results show that the proposed framework can not only generate solutions with properties closer to the target properties, but also produce more candidate solutions in an efficient manner. A conference version of this work appeared in [

29], and the current article significantly expands on the conference paper with more background and details on the framework, subsequent analysis of results as well as significant insights and discussion.