ceyron / pinns-in-julia

Simple implementation of Physics-Informed Neural Networks for the solution of Partial Differential Equations in Julia

Physics-Informed Neural Networks (PINNs) in Julia

PINNs are coordinate networks trained to be the solution to an initial-boundary value problem. The optimization problem is based on minimizing the residuum of the PDE in the domain, as well as residuums on the boundary and initial conditions. Additional supervised data (i.e., labeled point-wise solution values) are optional. The classical approach to PINNs is to use the automatic differentiation capabilities of deep learning frameworks to compute the input-output derivatives of the network. For a second-order PDE, that requires three hierarchical autodiff passes (two to obtain the residuum loss and a final one to backpropagate into the parameter space). As of the creation of this repo, Julia does not yet properly support higher-order autodiff, so we implement the input-output derivatives of the network manually. For a simple MLP, this is still manageable.

This repository contains a simple PINN example for the 1d Poisson equation with homogeneous Dirichlet boundary conditions. Check out the intro part of the Jupyter notebook for more details. You might also find the accompanying YouTube video helpful to code along.

Generally interested in Scientific Machine Learning? Check out my YouTube channel and the corresponding GitHub repository.