This page collects simple test problems that illustrate the use of SciML tools. Each of these “hello world” problems has an accompanying Jupyter notebook hosted by Google Collaboratory, which contains technical details about the simulation and allows the user to experiment with the code.

## Flow Around a Cylinder

Jupyter Notebook on Google Collaboratory

Predicting the flow of an incompressible fluid around a a bluff cylindrical object is a classical problem in fluid dynamics. Under appropriate conditions, the fluid downstream of the cylinder displays a complex pattern, referred to as a “Karman Vortex Street.”

The results displayed on the left were obtained by a neural network informed by the Navier-Stokes equations, using only the inflow velocity along a line upstream of the cylinder as boundary condition, along with relatively small number of scattered measurements of the concentration of a passive tracer.

This demo is an independent Tensorflow 2.x implementation of an experiment described in Raissi, Yazdani, and Karniadakis, “Hidden Fluid Mechanics: Learning velocity and pressure fields from flow visualizations,” Science 367:1026-1030 (2020).

Contributed by: Levi McClenny and Ulisses Braga-Neto, Aug 2021.

## Viscous Burgers PDE

Jupyter Notebook on Google Collaboratory

The one-dimensional Viscous Burgers PDE is a classical nonlinear time-dependent PDE that models the evolution of waves in incompressible fluids. It has been extensively used as a test problem in numerical methods for PDEs.

This demo compares the solution obtained by a baseline physically-informed neural network (PINN) to a very accurate Gaussian-Hermite quadrature of the analytical solution. The PINN approximation matches the quadrature solution quite well, while taking less time to compute.

Contributed by: Ulisses Braga-Neto, Jun 2021.