Bouncing Ball Hybrid ODE Optimization

The bouncing ball is a classic hybrid ODE which can be represented in the DifferentialEquations.jl event handling system. This can be applied to ODEs, SDEs, DAEs, DDEs, and more. Let's now add the DiffEqFlux machinery to this problem in order to optimize the friction that's required to match data. Assume we have data for the ball's height after 15 seconds. Let's first start by implementing the ODE:

using DiffEqFlux, Optim, OrdinaryDiffEq, DiffEqSensitivity

function f(du,u,p,t)
  du[1] = u[2]
  du[2] = -p[1]
end

function condition(u,t,integrator) # Event when event_f(u,t) == 0
  u[1]
end

function affect!(integrator)
  integrator.u[2] = -integrator.p[2]*integrator.u[2]
end

cb = ContinuousCallback(condition,affect!)
u0 = [50.0,0.0]
tspan = (0.0,15.0)
p = [9.8, 0.8]
prob = ODEProblem(f,u0,tspan,p)
sol = solve(prob,Tsit5(),callback=cb)

Here we have a friction coefficient of 0.8. We want to refine this coefficient to find the value so that the predicted height of the ball at the endpoint is 20. We do this by minimizing a loss function against the value 20:

function loss(θ)
  sol = solve(prob,Tsit5(),p=[9.8,θ[1]],callback=cb,sensealg=ForwardDiffSensitivity())
  target = 20.0
  abs2(sol[end][1] - target)
end

loss([0.8])
res = DiffEqFlux.sciml_train(loss,[0.8],BFGS())

* Status: success

* Candidate solution
   Final objective value:     3.995469e-01

* Found with
   Algorithm:     BFGS

* Convergence measures
   |x - x'|               = 4.93e-07 ≰ 0.0e+00
   |x - x'|/|x'|          = 5.69e-07 ≰ 0.0e+00
   |f(x) - f(x')|         = 8.12e-10 ≰ 0.0e+00
   |f(x) - f(x')|/|f(x')| = 2.03e-09 ≰ 0.0e+00
   |g(x)|                 = 7.71e-12 ≤ 1.0e-08

* Work counters
   Seconds run:   0  (vs limit Inf)
   Iterations:    5
   f(x) calls:    16
   ∇f(x) calls:   16

Finding an optimal friction coefficient of approximately 0.866. In that version we showcased forward sensitivity analysis, but adjoints can be utilized as well:

function loss(θ)
  sol = solve(prob,Tsit5(),p=[9.8,θ[1]],callback=cb,sensealg=ReverseDiffAdjoint())
  target = 20.0
  abs2(sol[end][1] - target)
end

loss([0.8])
res = DiffEqFlux.sciml_train(loss,[0.8],BFGS())

* Status: success

* Candidate solution
   Final objective value:     3.995469e-01

* Found with
   Algorithm:     BFGS

* Convergence measures
   |x - x'|               = 4.93e-07 ≰ 0.0e+00
   |x - x'|/|x'|          = 5.69e-07 ≰ 0.0e+00
   |f(x) - f(x')|         = 8.12e-10 ≰ 0.0e+00
   |f(x) - f(x')|/|f(x')| = 2.03e-09 ≰ 0.0e+00
   |g(x)|                 = 7.71e-12 ≤ 1.0e-08

* Work counters
   Seconds run:   0  (vs limit Inf)
   Iterations:    5
   f(x) calls:    16
   ∇f(x) calls:   16

Note on Sensitivity Methods

Only some continuous adjoint sensitivities are compatible with callbacks, namely BacksolveAdjoint and InterpolatingAdjoint. All methods based on discrete sensitivity analysis via automatic differentiation, like ReverseDiffAdjoint, TrackerAdjoint, or ForwardDiffAdjoint are the methods to use (and ReverseDiffAdjoint is demonstrated above), are compatible with events. This applies to SDEs, DAEs, and DDEs as well.