Automating crystal-structure phase mapping by combining deep learning with constraint reasoning

1Department of Computer Science, Cornell University, Ithaca, NY, USA. 2Division of Engineering and Applied Science and Liquid Sunlight Alliance, California Institute of Technology, Pasadena, CA, USA. 3Department of Materials Science and Engineering, Cornell University, Ithaca, NY, USA. ✉e-mail: gregoire@caltech.edu; gomes@cs.cornell.edu Artificial intelligence (AI)1 aims to develop intelligent systems, inspired in part by human intelligence. AI systems are now performing at human and even superhuman levels on a range of tasks, such as image identification2 and face3 and speech recognition4. AI also has the potential to dramatically accelerate scientific discovery5–10. Recent AI achievements have been driven mainly by advances in supervised deep learning11, which requires large labelled datasets to supervise model training. However, in general, scientists do not have large amounts of labelled data for scientific discovery. They often solve complex tasks using only a few data samples by amplifying intuitive pattern recognition with detailed reasoning about prior knowledge to make sense of the data. Such a hybrid strategy has been difficult to automate. In this Article we consider crystal-structure phase mapping, a long-standing challenge in materials science that is emblematic of the class of scientific problems whose automation constitutes a substantial advancement with respect to the grand challenge of high-throughput unsupervised scientific data interpretation. Crystal-structure phase mapping involves separating noisy mixtures of X-ray diffraction (XRD) patterns into the source XRD signals of the corresponding crystal structures, a task for which labelled training data are typically not available. Furthermore, a valid phase diagram of the crystal structures of a given chemical system must satisfy thermodynamics rules (Fig. 1a–f). Here we provide a detailed description of how to formulate phase mapping as an unsupervised pattern demixing problem and how to solve it using deep reasoning networks (DRNets)12. DRNets are a general framework for combining deep learning with constraint reasoning for incorporating scientific prior knowledge. DRNets are designed with an interpretable latent space for encoding the prior-knowledge domain constraints, enabling seamless integration of constraint reasoning into neural network optimization. Constraint reasoning is a particular type of AI reasoning in which axioms and rules are expressed as constraints, and the inference procedure is a search method. The axioms and rules pertaining to a given task comprise the prior knowledge needed to identify valid solutions. In this Article, we show how DRNets require only a modest amount of (unlabelled) data and compensate for the limited data by exploiting and magnifying the rich scientific prior knowledge about the thermodynamic rules that govern the mixtures of crystals. We further provide insights concerning the interpretability and scalability of DRNets, as well as the role of data and the different DRNets’ modules, through a series of ablation studies. DRNets make this crystal-structure phase mapping advancement by combining learning with constraint reasoning, emulating the analysis of expert scientists and enabling interpretation of complex systems in high-dimensional composition spaces. Given the scientific complexity of crystal-structure phase mapping, we provide an initial intuitive explanation of the DRNets framework based on Multi-MNIST-Sudoku12, a variant of the Sudoku game that involves demixing two completed overlapping hand-written Sudokus (Fig. 1g–i). To demonstrate the scalability of DRNets, we also consider 9 × 9 Sudoku instances combining both digits and letters, beyond the 4 × 4 multi-MNIST-Sudoku instances Automating crystal-structure phase mapping by combining deep learning with constraint reasoning