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 = u du = -p end function condition(u,t,integrator) # Event when event_f(u,t) == 0 u end function affect!(integrator) integrator.u = -integrator.p*integrator.u 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],callback=cb) target = 20.0 abs2(sol[end] - 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.
The continuous adjoint sensitivities
QuadratureAdjoint are compatible with events for ODEs.
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
ForwardDiffAdjoint are the methods to use (and
ReverseDiffAdjoint is demonstrated above), are compatible with events. This applies to SDEs, DAEs, and DDEs as well.