add new topic

content/guide.rst

@@ -111,7 +111,7 @@ The next three days were scheduled as follows:
 +-------------+--------------------------------------------+
 | 10:45-11:00 | Break                                      |
 +-------------+--------------------------------------------+
-| 11:00-11:45 | Non-linear regression and Fourier methods  |
+| 11:00-11:45 | Scientific machine learning                |
 +-------------+--------------------------------------------+
 | 11:45-12:00 | Conclusions and outlook                    |
 +-------------+--------------------------------------------+

content/index.rst

@@ -53,6 +53,7 @@ please visit the lesson `Julia for high-performance scientific computing <https:
    linear-algebra
    data-science
    regression
+   sciml
 
 .. toctree::
    :maxdepth: 1

content/sciml.rst

@@ -0,0 +1,68 @@
+.. _sciml:
+
+Scientific Machine Learning
+===========================
+
+.. questions::
+
+   - What is the SciML Julia package and how to use it?
+   - How can I mix physical pre-knowledge and data to solve modelling problems in Julia?
+
+.. instructor-note::
+
+   - 35 min teaching
+   - 15 min exercises
+
+.. callout::
+
+   The code in this lession is written for Julia v1.11.2.
+
+A modelling problem
+-------------------
+
+In this session we will have a look at a physical modelling problem
+and investigate how Machine Learning can be used in combination with
+classical problems.
+
+Consider a spherical object moving in a viscous fluid under the influence
+of a drag force and gravity. We will consider this problem in 2 dimensions.
+
+.. code-block:: julia
+
+   using LinearAlgebra, Statistics, Random
+   using ComponentArrays, Lux, Zygote, Plots
+   using OrdinaryDiffEq, ModelingToolkit, DataDrivenDiffEq, SciMLSensitivity, DataDrivenSparse
+   using Optimization, OptimizationOptimisers, OptimizationOptimJL, LineSearches
+
+   rng = Random.default_rng()
+
+   function dynamics!(du, u, p, t)
+      m = 1.0
+      g = 10.0
+      
+      du[1] = -((u[1]^2 + u[3]^2)^0.5)*u[1] + 2*u[3]
+      du[3] = -((u[1]^2 + u[3]^2)^0.5)*u[3] - m*g - 2*u[1]
+      du[2] = u[1]
+      du[4] = u[3]
+   end
+
+   tspan = (0.0, 1.0)
+
+   sc = 1.0
+   deltat = 0.1
+
+   u0 = sc * rand(4)
+   inits_g = [rand(4) for ii in range(1,6)]
+
+   prob = ODEProblem(dynamics!, u0, tspan)
+
+   prbs = [ODEProblem(dynamics!, ui, tspan) for ui in inits_g]
+
+   times = Vector(0:deltat:1.0)
+
+   sols = [Array(solve(prb, Vern7(), abstol = 1e-12, reltol = 1e-12, saveat = 0.1)) for prb in prbs]
+
+   Xs = hcat(sols...)
+
+   # for axis equal; aspect_ratio = :equal
+   scatter(Xs[2,:], Xs[4,:], alpha = 0.75, color = :green, label = ["True Data" nothing])