Recurrent Localization Networks applied to the Lippmann-Schwinger Equation
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The process of discovering and designing new materials is very costly, both in terms of time and human effort. One of the most expensive parts is experimentation — before a new material can be trusted, it must first be tested extensively to understand all of its properties. These experiments usually take the form of either physical (real-world) tests or computer simulations. Unfortunately, classical physics-based simulations are still quite slow. In the process of designing a new material, a great number (e.g. thousands, millions) of simulations must be conducted. This constitutes a major bottleneck for discovering new materials. This work explores different methods to accelerate materials discovery by replacing slow, physics-based simulations with faster, reduced-order models based on machine learning. Specifically, the original physical governing equation is converted into an equivalent form more amenable for learning called the Lippmann-Schwinger (L-S) form. A recurrent series of convolutional neural networks (CNNs) is then used to approximately solve the L-S equation. This hybrid architecture, called a recurrent localization network (RLN), leverages the strengths of machine learning while still permitting a physics-based interpretation. As a demonstration, an RLN is trained to solve for interior strain fields of an elastic microstructure, producing high-accuracy results a thousand times faster than a classical Finite-Element simulation. Additionally, this methodology is faster, more accurate, and more interpretable than purely-data-driven models for the same problem. Since a wide range of physical systems can be converted into an equivalent L-S form, this architecture has potential applications across numerous problems in materials science.