Phase diagrams of polymer-containing liquid mixtures with a theory-embedded neural network


There are frequent inconsistent agreements among experimental, theoretical and simulation results, the study of phase separation and diagram often face numerous challenges: difficulty in solving the complex equations derived from thermodynamic theories, difficulty in constructing molecular theory to account for the experimental data due to unclear phase separation mechanism and limited experimental data for the phase diagram of the phase separation. To address these challenges, a more practical approach based on model-independent methods with less computational burden and simple modeling and analysis is highly desirable. With the growing application of neural networks in solving real-world problems across numerous research areas, its application in polymer-containing liquid mixtures has recently attracted significant research attention.

In a research paper recently published in the New Journal of Physics, Professor Issei Nakamura from Michigan Technological University developed a deep neural network to account for the phase behaviors of polymer-containing liquid mixtures. This model-independent approach was utilized in the phase separation of both salt-free and salt-doped polymer-containing liquids. The deep neural network comprised of a theory-embedded layer constructed via coarse-grained mean-field theory and scaling laws to enhance its accuracy. It was necessary to perform a random search of the weights without backpropagation of the algorithms to minimize the technical and programming burden. The main aim was to provide a tractable method for studying the phase behaviors that could not be accessed by the present molecular theory and simulations as well as providing a platform for approximating the existing statistical thermodynamics models while retaining the characteristic features.

Results showed that the theory-embedded layer substantially improved the accuracy of the deep neural network by simplifying the local minimum problems in the loss functions for accurate evaluations. Consequently, the layer successfully captured the important thermodynamic features of the target systems i.e. the entropy and enthalpy of the polymer mixtures which enable the description of the phase diagrams of polymer solutions and salt-doped and salt-free diblock copolymers melts. On the other hand, a Gaussian function with values assigned to the phase behavior types was used to express the output of the deep neural network because the phases could be easily identified from molecular properties.

Even though the layer design was not unique, it was noteworthy that the design and construction of the layer exhibited significant effects of the size and speed of the deep neural network. For instance, appropriate construction did not only reduce the size of the deep neural network but also improved its optimization speed. Therefore, the design approach may not be perfect in proving optimal solutions to other problems.

In a nutshell, Professor Issei Nakamura’s study used a theory-embedded neural network to study the phase diagrams of polymer-containing liquid mixtures. By tuning the deep neural network, it was possible to reproduce the phase diagrams without advanced theories and molecular simulations. The predictive power of the deep neural network-assisted in exploring new phase behaviors. Based on the presented findings, Professor Issei Nakamura in a statement to Advances in Engineering stated that the proposed deep neural network will open doors for more studies that can be extended to other disciplines like solving the inverse matter problems in soft-matter physics.

Phase diagrams of polymer-containing liquid mixtures with a theory-embedded neural network - Advances in Engineering

About the author

Issei Nakamura is assistant professor of physics at Michigan Technological University. He earned his Ph.D. in Physics at McMaster University in Canada. Before coming to Michigan Tech, Nakamura was a postdoctoral scholar at the California Institute of Technology and then worked as a professor at the Changchun Institute of Applied Chemistry, the Chinese Academy of Sciences. He also has experience as a system engineer in Japan.

His research focuses on soft-condensed matter theory and simulation with particular emphasis on electrostatic interactions. His research group employs a broad spectrum of theoretical and computational methods, such as Monte Carlo simulation, molecular dynamics simulation, statistical field theory, and machine learning. He also focuses on active teaching/learning methodology based on pre-class exercises for both undergraduate and graduate students.


Nakamura, I. (2020). Phase diagrams of polymer-containing liquid mixtures with a theory-embedded neural network. New Journal of Physics, 22(1), 015001.

Go To New Journal of Physics

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