The efficient design of novel materials with desired functionalities is a significant challenge in materials science. Traditional high-throughput computational and experimental methods struggle with the vastness of the chemical search space. Machine learning offers a promising solution by providing cheaper surrogates for computationally expensive calculations and generating new materials candidates with specified properties. Generative models, in particular, are highly appealing for their potential to realize inverse design—the ability to predict the materials parameters that will yield a desired property. This paper focuses on addressing the limitations of existing generative models such as variational autoencoders (VAEs) and generative adversarial networks (GANs), which often suffer from training instability and mode collapse. The authors propose leveraging invertible neural networks (INNs) and conditional INNs (cINNs) due to their intrinsic invertibility which enables both forward and reverse mappings between design space and target properties. This ensures stability and efficient generation of candidate materials. The framework, MatDesINNe (Materials Design with Invertible Neural Networks), is introduced, incorporating both INN/cINN for generation and additional steps like down-selection based on fitness criteria and optimization to refine candidates to achieve high-fidelity, chemically accurate results. The specific problem of band gap engineering in monolayer MoS2 is chosen as a case study to demonstrate the framework's capabilities. Band gap tuning in MoS2 is technologically relevant for various applications, including photocatalysis, electronics, sensors, and neuromorphic devices, and is achieved via the application of strain and electric fields.

The introduction extensively reviews existing data-driven approaches for accelerated materials design, highlighting the use of machine learning and generative models in overcoming the limitations of traditional high-throughput methods. It discusses the challenges and advantages of using generative models compared to discriminative models. The authors cite several papers exploring generative models for materials discovery, including variational autoencoders (VAEs) and generative adversarial networks (GANs). However, it highlights the limitations of existing methods, such as VAEs' difficulties in approximating inverse solutions and GANs' tendency towards mode collapse, which motivates the use of invertible neural networks.

The MatDesINNe framework comprises four main steps. First, training data is generated using Density Functional Theory (DFT) calculations, sampling a design space encompassing six strain parameters (variations in lattice constants a, b, c, angles α, β, γ) and an external electric field (E) applied to monolayer MoS2. Approximately 11,000 DFT calculations were performed. Second, an invertible neural network (INN or cINN) is trained on this data using a combination of forward and reverse training to learn both the forward (design parameters to band gap) and inverse (band gap to design parameters) mappings. The forward mapping serves as a surrogate model for subsequent steps. Third, down-selection is performed based on fitness criteria, selecting only samples that closely match the target band gap and lie within the training data distribution. Fourth, the selected samples are further refined using gradient descent-based optimization, guided by the gradients from the surrogate forward model, to arrive at highly accurate solutions. The authors compare their method (MatDesINNe-INN and MatDesINNe-cINN) against baseline methods: Mixture Density Networks (MDN), Conditional Variational Autoencoders (CVAE), basic INN, and basic cINN. The implementation details are provided, including the choice of architecture (affine coupling layers within the INNs), activation functions (ReLU), optimizers (Adam), and hyperparameters. The code and data are made publicly available. The baseline methods are also detailed including their loss functions and training parameters.

The key findings demonstrate the superior performance of MatDesINNe-cINN over the other tested models. Across multiple target band gaps (0, 0.5, and 1 eV), MatDesINNe-cINN achieved a significantly lower mean absolute error (MAE) compared to MDN, CVAE, INN, and cINN. For instance, for a target band gap of 1 eV, the MAE was reduced from approximately 0.8-0.9 eV for the baseline models to approximately 0.015 eV for MatDesINNe-cINN. DFT validation confirmed the high accuracy of the generated candidates, with MAEs of 0.1 eV or lower, approaching experimental error bars. The generated samples exhibited high diversity and were largely unique compared to the training data. Further analysis using MatDesINNe revealed interesting insights into the relationship between strain parameters (tensile/compressive and shear) and band gaps. The model demonstrated a capability of revealing regions of the design space where a metal-insulator transition (MIT) could occur with minimal perturbation, an important finding for neuromorphic applications. UMAP embeddings visualized these regions of the design space showing overlap between zero and non-zero band gap states indicating regions with easily induced MIT.

The superior performance of MatDesINNe is attributed to the combined use of cINNs and the iterative refinement process. The localization step utilizing the surrogate model and gradient descent plays a crucial role in pushing the initially generated samples towards chemically accurate solutions. The results highlight the ability of MatDesINNe to efficiently explore high-dimensional design spaces and generate diverse, high-fidelity materials candidates at a significantly lower computational cost compared to traditional DFT-based screening. The insights into strain-band gap relationships and MIT behavior in MoS2 demonstrate the potential of MatDesINNe for guiding experimental design and providing a deeper understanding of materials behavior. The ability to identify regions in design space where small changes can lead to significant property changes (like MIT) is particularly valuable for designing materials for applications such as neuromorphic computing.

This work successfully demonstrates MatDesINNe, a novel inverse design framework for generating high-fidelity materials candidates with target properties. The framework leverages the inherent invertibility of cINNs, coupled with a down-selection and localization process, to achieve remarkable accuracy and efficiency. The application to band gap engineering in MoS2 showcases its potential for various materials design problems. Future research directions could involve exploring more advanced invertible architectures, integrating additional design parameters (e.g., atomic composition), and incorporating experimental feedback loops to enhance the predictive power and applicability of MatDesINNe for real-world materials discovery.

The current implementation of MatDesINNe focuses on a relatively small design space involving strain and electric field parameters. The accuracy of the generated candidates is limited by the accuracy of the DFT surrogate model used for both training and the localization step. The model's ability to generalize to systems beyond the training data needs further investigation. While the model is able to generate high-fidelity structures it does not inherently model crystal structures and so further advancements are needed to enable the modeling of more complex materials.