This page collects simple test problems that illustrate the use of SciML tools. Each of these 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.

## Wave Equation

Jupyter Notebook on Google Collaboratory

This demo shows how the classic wave equation can be solved accurately by Gaussian process regression. This is known to be a difficult problem to solve with most numerical methods, due to the complexity and speed of the solution time evolution.

The results displayed on the left were obtained by an Iterated Extended Kalman Smoother (IEKS) applied in conjunction with a semi-discretization approach to obtain an accurate solution with well-calibrated uncertainty quantification. The latter is an important feature offered by this probabilistic approach, which allows effective error control (the solution is more accurate when the confidence band is small, and vice-versa).

Contributed by: Durward Cator, Simo Särkkä, and Ulisses Braga-Neto, Nov 2023.

Acknowledgement: NSF-AoF:A Bayesian Paradigm for Physics-Informed Machine Learning. Award Number: 2225507.

## 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.