Deep Learning and PDE: a quick and practical introduction
Talk by Dr. Dominique Zosso (Dept. of Mathematical Sciences, MSU)
9/17/2020 3:10 pm Webex Meeting
In this talk I will try to provide a very quick introduction from generic artificial neural networks (ANN) to their application to numerically solving PDE. This less of a research talk, and more like a tutorial and practical in nature.
In the first part, I will describe a simple model of feed-forward artificial neural network, starting from the biologically motivated individual perceptron as its core unit. If technology permits, I will include a MATLAB demonstration for some simple regression and classification problems, and mention the universal representation theorem. In the intermission, I will provide a superficial overview of more specific/complicated network architectures that are currently being employed, such as convolutional NN, recurrent NN, auto-encoder networks, generative adversarial networks (GAN). I will also hint at the difference between supervised and unsupervised (or self-supervised) training, as well as talk about a menu of activation functions and network regularizers.
In the second part, we will focus on a relatively recent approach to use "Deep learning" to numerically tackle (certain) PDE. We will specifically look at the core ideas in the 2018 paper on "Deep Galerkin Methods"
. There, we construct an ANN to represent the unknown function u: for arbitrary input values x, the network will evaluate to an approximation of u(x). This DGM network will be trained using a loss function defined based on the PDE terms, and converge to the solution of the PDE. Again, if technology permits, I will provide a live demo of a (few) simple examples.  Sirignano and Spiliopoulos, "DGM: A deep learning algorithm for solving partial differential equations", Journal of Computational Physics 375 (2018):1339-1364.