Can neural networks learn to solve partial differential equations (PDEs)? We investigate this question for two (systems of) PDEs, namely, the Poisson equation and the steady Navier–Stokes equations.

• Tim Dockhorn. "A Discussion on Solving Partial Differential Equations using Neural Networks." https://arxiv.org/abs/1904.07200

You can hear me talking about this work and machine learning in general on Ashwin's podcast at https://www.youtube.com/watch?v=3c6YXfgi46Q&t=9s.

A neural network (with two fully connected layers of size 16) for the manufactured Poisson problem (using dataset of 2000 interior and boundary points) can be trained using the following command:

foo@bar:~$ python3 poisson.py -b 2000 -n 2

A velocity and a pressure neural network (with two fully connected layers size 16 each) for the Kovasznay problem (using dataset of 4000 interior and boundary points) can be trained using the following command:

foo@bar:~$ python3 kovasznay_flow.py -b 4000 -u 2 -p 2

We recommend the following package versions to reproduce the results of the paper

• Tensorflow: 1.12.0
• Numpy: 1.16.1
• Scipy: 1.2.0
• Matplotlib: 3.0.2