Theory-training deep neural networks for an alloy solidification benchmark problem

Deep neural networks are machine learning tools that are transforming fields ranging from speech recognition to computational medicine. In this study, we extend their application to the field of alloy solidification modeling. To that end, and for the first time in the field, theory-trained deep neural networks (TTNs) for solidification are introduced. These networks are trained using the framework founded by Raissi et al.[1-3] and a theory that consists of a mathematical macroscale solidification model and the boundary and initial conditions of a well-known solidification benchmark problem. One of the main advantages of TTNs is that they do not need any prior knowledge of the solution of the governing equations or any external data for training. Using the built-in capabilities in TensorFlow, networks with different widths and depths are trained, and their predictions are examined in detail to verify that they satisfy both the model equations and the initial and boundary conditions of the benchmark problem. Issues that are critical in theory-training are identified, and guidelines that can be used in the future for successful and efficient training of similar networks are proposed. Through this study, theory-trained deep neural networks are shown to be a viable tool to simulate alloy solidification problems.Comment: 43 pages, 9 figure