(455d) Inverse Design of Materials with Globally Optimal Topology and Geometry through Mixed Integer Linear Programming (MILP)

The rapid discovery and optimization of materials is key to the development of sustainable, safe, and economic advancements in technologies that can improve human health, create and store clean energy, and mitigate the impact of pollutants and contaminants. There have been great advances in the automated, high-throughput exploration of material design spaces that enable these developments but the massive scale of these design spaces means guidance is needed in their exploration. Inverse design allows us to quantify relationships between a material's intrinsic characteristics (e.g., geometry, topology, molecular structure) and its target properties (e.g., catalytic efficiency, energy storage capability). These identified relationships can then be exploited to design new materials that optimize a target property. These methods are particularly applicable to materials where shape is a key indicator of performance, such as metal organic frameworks (MOFs), zeolites, and self-assembled systems such as soft gels and polymer networks. In this talk we demonstrate how a materials topology and geometry, measured via Minkowski functionals, can be used for the inverse design and generation of new material structures through a mixed integer linear programming (MILP) approach.

Minkowski functionals, also known as intrinsic volumes, concisely quantify the topology and geometry of complex materials. For 2-Dimensional systems these characteristics represent the area, perimeter, and Euler characteristic of a material. These measures have been used to discover connections between a material’s multi-scale structure and emergent physical properties such as porosity, viscosity, and permeability. The Minkowski functionals are concise, scalable in computation, and are directly interpretable. They also act as a powerful dimensionality reduction technique, allowing the use of simpler data centric models for prediction of material properties with equal or higher levels of accuracy (e.g., linear regression model as opposed to complex convolutional neural networks).

However, one drawback of the dimensionality reduction of the Minkowski functionals is the inability to reconstruct a material directly from these measures. Generative models, such as variational autoencoders and generative adversarial networks (GANs) can be used for this purpose, but they often require a tremendous amount of training data, do not directly account for physical constraints, and are black box models that are difficult to interpret. To address these challenges, we propose a framework leveraging mixed integer linear programming (MILP) for the generation of materials from a set of Minkowski measures.

Utilizing training data, we regress how a material's physical and chemical characteristics depend on the Minkowski measures. With material dependence as an objective function, our MILP framework is then able to create a material with globally optimal geometry and topology. This makes MILP an effective strategy for the rapid generation of material designs with specific Minkowski measures. It is also amenable to the direct incorporation of physically meaningful constraints to the final solution. This allows for the exploitation of the relationship between the Minkowski measures and physical properties of a material, allowing us to extrapolate and generate physically meaningful material structures with optimal properties. Furthermore, there is a potential for multiplicity of optimal solutions which can be useful in data augmentation tasks to aid in the training of other deep learning models. We demonstrate our MILP framework’s utility on a set of real images and show effective reproduction of materials with complex geometry and topology. With this demonstration, we show the MILP is an effective tool in problems that can use Minkowski measures or functions of the Minkowski measures to design topologically or geometrically optimal configurations.

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