E. Madenci

A. Mavi, A. C. Bekar, E. Haghighat et al.

This study presents a novel unsupervised convolutional Neural Network (NN) architecture with nonlocal interactions for solving Partial Differential Equations (PDEs). The nonlocal Peridynamic Differential Operator (PDDO) is employed as a convolutional filter for evaluating derivatives the field variable. The NN captures the time-dynamics in smaller latent space through encoder-decoder layers with a Convolutional Long-short Term Memory (ConvLSTM) layer between them. The ConvLSTM architecture is modified by employing a novel activation function to improve the predictive capability of the learning architecture for physics with periodic behavior. The physics is invoked in the form of governing equations at the output of the NN and in the latent (reduced) space. By considering a few benchmark PDEs, we demonstrate the training performance and extrapolation capability of this novel NN architecture by comparing against Physics Informed Neural Networks (PINN) type solvers. It is more capable of extrapolating the solution for future timesteps than the other existing architectures.

A. C. Bekar, E. Haghighat, E. Madenci

This study proposes a novel framework for learning the underlying physics of phenomena with moving boundaries. The proposed approach combines Ensemble SINDy and Peridynamic Differential Operator (PDDO) and imposes an inductive bias assuming the moving boundary physics evolve in its own corotational coordinate system. The robustness of the approach is demonstrated by considering various levels of noise in the measured data using the 2D Fisher-Stefan model. The confidence intervals of recovered coefficients are listed, and the uncertainties of the moving boundary positions are depicted by obtaining the solutions with the recovered coefficients. Although the main focus of this study is the Fisher-Stefan model, the proposed approach is applicable to any type of moving boundary problem with a smooth moving boundary front without a mushy region. The code and data for this framework is available at: https://github.com/alicanbekar/MB_PDDO-SINDy.