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Reduced-order modeling (ROM) of fluid flows has been an active area of research for several decades. The huge computational cost of direct numerical simulations has motivated researchers to develop more efficient alternative methods, such as ROMs and other surrogate models. Similar to many application areas, such as computer vision and language modeling, machine learning and data-driven methods have played an important role in the development of novel models for fluid dynamics. The transformer is one of the state-of-the-art deep learning architectures that has made several breakthroughs in many application areas of artificial intelligence in recent years, including but not limited to natural language processing, image processing, and video processing. In this work, we investigate the capability of this architecture in learning the dynamics of fluid flows in a ROM framework. We use a convolutional autoencoder as a dimensionality reduction mechanism and train a transformer model to learn the system's dynamics in the encoded state space. The model shows competitive results even for turbulent datasets. 

Abstract Neurons exhibit complex geometry in their branched networks of neurites which is essential to the function of individual neuron but also brings challenges to transport a wide variety of essential materials throughout their neurite networks for their survival and function. While numerical methods like isogeometric analysis (IGA) have been used for modeling the material transport process via solving partial differential equations (PDEs), they require long computation time and huge computation resources to ensure accurate geometry representation and solution, thus limit their biomedical application. Here we present a graph neural network (GNN)-based deep learning model to learn the IGA-based material transport simulation and provide fast material concentration prediction within neurite networks of any topology. Given input boundary conditions and geometry configurations, the well-trained model can predict the dynamical concentration change during the transport process with an average error less than 10% and $$120 \sim 330$$ 120 ∼ 330 times faster compared to IGA simulations. The effectiveness of the proposed model is demonstrated within several complex neurite networks.