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, DifferentialEquations

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)
  target = 20.0
  abs2(sol[end][1] - target)
end

loss([0.8])
@time res = DiffEqFlux.sciml_train(loss,[0.8])
@show res.u # [0.866554105436901]

This runs in about 0.091215 seconds (533.45 k allocations: 80.717 MiB) and finds an optimal drag coefficient.

Note on Sensitivity Methods

The continuous adjoint sensitivities BacksolveAdjoint, InterpolatingAdjoint, and QuadratureAdjoint are compatible with events for ODEs. BacksolveAdjoint and InterpolatingAdjoint can also handle events for SDEs. Use BacksolveAdjoint if the event terminates the time evolution and several states are saved. Currently, the continuous adjoint sensitivities do not support multiple events per time point.

All methods based on discrete sensitivity analysis via automatic differentiation, like ReverseDiffAdjoint,mTrackerAdjoint, 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.