Toward Training Physics-Informed Neural Networks without Automatic Differentiation

Author(s):

Bradley T. Burchett

Abstract:

Artificial neural networks have been used for the past four decades for pattern recognition, system modeling, and feedback control. A recent variant called physics-informed neural network (PINN) incorporates constitutive physical laws into network training to limit solutions to a subspace of physically meaningful models. This report seeks to unpack PINNs by presenting the basic building blocks of shallow networks—neurons, synapses, activation functions, gradients, and backpropagation—and showing some low-dimensional applications thereof. Throughout this work, I use small, shallow networks and analytic gradients to demonstrate fast, smooth solutions without overtraining. I conclude the report with three applications of PINNs: trajectory reconstruction, drag modeling, and optimal control.