A Study of Causal Physics-Informed Neural Network (PINN) methods applied to the 2-Dimensional Cahn-Hilliard (2D-CH) Equation — 64a — Caden Fischer, Yangxiao Bai, Dr. Kaiqun Fu, and Dr. Nathan McClanahan

Artificial Neural Networks (ANNs) have been at the forefront of development because of  advancements in computing resources. However, the applications of ANNs work well because  of the vast amount of data available for these problems. Physics-informed Neural Networks  (PINNs) allow the power of ANNs to be applied to physics problems which typically have a  much lower amount of data or none at all. Some PINNs try to find an approximate solution to  Partial Differential Equations (PDEs). The solutions to PDEs are typically approximated by  classical numerical methods, which can be computationally expensive. The PINN used in this  study is meshless, meaning the solution is not confined to a grid of distinct points. Thus, this  type of PINN promises less expensive solutions for high-dimensional problems, which is  extremely expensive for classical methods with a mesh. Causal PINN algorithms reintroduce  some causality since vanilla PINNs disregard causality in the training algorithm. This study  investigates Causal Loss and Time Slab techniques within causal PINN algorithms and applies  them to the 2-Dimensional Cahn-Hilliard Equation (2D-CH), a classical PDE used to model  phase separation, such as in biofilms. After the training, the model is compared to a Finite  Element Method code simulating the same equations. The results of this study show that the  causal methods perform better than the vanilla PINNs for the 2D-CH equation and can speed  up simulation computation time compared to the Finite Element Method.

South Dakota State University
Dr. Jung-Han Kimn