SciMLWorkshop: Workshop Materials for Training in Scientific Computing and Scientific Machine Learning (SciML)
These exercises teach common workflows which involve SciML's tools like
DifferentialEquations.jl, ModelingToolkit.jl, DiffEqFlux.jl, and the connections to parts
like stochastic differential equations and Bayesian estimation.
Difficulty Levels
The designation (B) is for "Beginner", meaning that a user new to the package
should feel comfortable trying this exercise. An exercise designated (I) is
for "Intermediate", meaning the user may want to have some previous background
in DifferentialEquations.jl or try some (B) exercises first. The additional
(E) designation is for "Experienced", which are portions of exercises which may
take some work.
Overview
The exercises are described as follows:
- Exercise 1 is solving a simple stiff ordinary differential equation. This is an
introductory exercise to get users acquainted with the syntax.
- Exercise 2 takes the user through solving a stiff ordinary differential equation
and using the ModelingToolkit.jl to automatically convert the function to a
symbolic form to derive the analytical Jacobian to speed up the solver. The
same biological system is then solved with stochasticity, utilizing
EnsembleProblems to understand 95% bounds on the solution. Finally,
probabilistic programming is employed to perform Bayesian parameter estimation
of the parameters against data.
- Exercise 3 takes the user through defining hybrid delay differential equation,
that is a differential equation with events, and using differentiable programming
techniques (automatic differentiation) to to perform gradient-based parameter
estimation.
- Exercise 4 takes the user through differential-algebraic equation (DAE)
modeling, the concept of index, and using both mass-matrix and implicit
ODE representations. This will require doing a bit of math, but the student
will understand how to change their equations to make their DAE numerically
easier for the integrators.
- Exercise 5 has one build an acausal model, a DAE system through a component-based
modeling approach. Using a tutorial model of an RC circuit (resistor and capacitor)
plus some information about inductors, the user then builds new ModelingToolkit
components for an inductor and generates an RLC circuit to be simulated.
- Exercise 6 takes the user through optimizing a PDE solver, utilizing
automatic sparsity pattern recognition, automatic conversion of numerical
codes to symbolic codes for analytical construction of the Jacobian,
preconditioned GMRES, and setting up a solver for IMEX and GPUs, and compute
adjoints of PDEs.
- Exercise 7 focuses on a chaotic orbit, utilizing parallel ensembles across
supercomputers and GPUs to quickly describe phase space.
- Exercise 8 takes the user through training a neural stochastic differential
equation, using GPU-acceleration and adjoints through Flux.jl's neural
network framework to build efficient training codes.
- Exercise 9 is a deep dive into controls analysis on nonlinear models. The
user starts by building an acausal model of a DC motor, which is augmented
with a PI-controller that is tuned to control the actions of the motor.
This exercise worksheet is meant to be a living document leading new users through
a deep dive of the SciML feature set. If you further suggestions
or want to contribute new problems, please
open an issue or PR at the SciMLWorkshop.jl repository.