This tutorial gives and overview of how to leverage the efficient Koopman expectation method from DiffEqUncertainty to perform optimization under uncertainty. We demonstrate this by using a bouncing ball model with an uncertain model parameter. We also demonstrate its application to problems with probabilistic constraints, in particular a special class of constraints called chance constraints.
First lets consider a 2D bouncing ball, where the states are the horizontal position $x$, horizontal velocity $\dot{x}$, vertical position $y$, and vertical velocity $\dot{y}$. This model has two system parameters, acceleration due to gravity and coefficient of restitution (models energy loss when the ball impacts the ground). We can simulate such a system using ContinuousCallback
as
using OrdinaryDiffEq, Plots function ball!(du,u,p,t) du[1] = u[2] du[2] = 0.0 du[3] = u[4] du[4] = -p[1] end ground_condition(u,t,integrator) = u[3] ground_affect!(integrator) = integrator.u[4] = -integrator.p[2] * integrator.u[4] ground_cb = ContinuousCallback(ground_condition, ground_affect!) u0 = [0.0,2.0,50.0,0.0] tspan = (0.0,50.0) p = [9.807, 0.9] prob = ODEProblem(ball!,u0,tspan,p) sol = solve(prob,Tsit5(),callback=ground_cb) plot(sol, vars=(1,3), label = nothing, xlabel="x", ylabel="y")
For this particular problem, we wish to measure the impact distance from a point $y=25$ on a wall at $x=25$. So, we introduce an additional callback that terminates the simulation on wall impact.
stop_condition(u,t,integrator) = u[1] - 25.0 stop_cb = ContinuousCallback(stop_condition, terminate!) cbs = CallbackSet(ground_cb, stop_cb) tspan = (0.0, 1500.0) prob = ODEProblem(ball!,u0,tspan,p) sol = solve(prob,Tsit5(),callback=cbs)
To help visualize this problem, we plot as follows, where the star indicates a desired impace location
rectangle(xc, yc, w, h) = Shape(xc .+ [-w,w,w,-w]./2.0, yc .+ [-h,-h,h,h]./2.0) begin plot(sol, vars=(1,3), label=nothing, lw = 3, c=:black) xlabel!("x [m]") ylabel!("y [m]") plot!(rectangle(27.5, 25, 5, 50), c=:red, label = nothing) scatter!([25],[25],marker=:star, ms=10, label = nothing,c=:green) ylims!(0.0,50.0) end
We now wish to introduce uncertainty in p[2]
, the coefficient of restitution. This is defined via a continuous univiate distribution from Distributions.jl. We can then run a Monte Carlo simulation of 100,000 trajectories via the EnsembleProblem
interface.
using Distributions cor_dist = truncated(Normal(0.9, 0.02), 0.9-3*0.02, 1.0) trajectories = 100000 prob_func(prob,i,repeat) = remake(prob, p = [p[1], rand(cor_dist)]) ensemble_prob = EnsembleProblem(prob,prob_func=prob_func) ensemblesol = solve(ensemble_prob,Tsit5(),EnsembleThreads(),trajectories=trajectories, callback=cbs) begin # plot plot(ensemblesol, vars = (1,3), lw=1,alpha=0.2, label=nothing, idxs = 1:350) xlabel!("x [m]") ylabel!("y [m]") plot!(rectangle(27.5, 25, 5, 50), c=:red, label = nothing) scatter!([25],[25],marker=:star, ms=10, label = nothing, c=:green) plot!(sol, vars=(1,3), label=nothing, lw = 3, c=:black, ls=:dash) xlims!(0.0,27.5) end
Here, we plot the first 350 Monte Carlo simulations along with the trajectory corrresponding to the mean of the distribution (dashed line).
We now wish to compute the expected squared impact distance from the star. This is called an "observation" of our system or an "observable" of interest.
We define this observable as
obs(sol) = abs2(sol[3,end]-25)
obs (generic function with 1 method)
With the observable defined, we can compute the expected squared miss distance from our Monte Carlo simulation results as
mean_ensemble = mean([obs(sol) for sol in ensemblesol])
36.12309073549283
Alternatively, we can use the Koopman()
algorithm in DiffEqUncertainty.jl to compute this expectation much more efficiently as
using DiffEqUncertainty p_uncertain = [9.807, cor_dist] expectation(obs, prob, u0, p_uncertain, Koopman(), Tsit5(); ireltol = 1e-5, callback=cbs)
u: 36.008628214169704
We now wish to optimize the initial position ($x_0,y_0$) and horizontal velocity ($\dot{x}_0$) of the system to minimize the expected squared miss distance from the star, where $x_0\in\left[-100,0\right]$, $y_0\in\left[1,3\right]$, and $\dot{x}_0\in\left[10,50\right]$. We will demonstrate this using a gradient-based optimization approach from NLopt.jl using ForwardDiff.jl
AD through the expectation calculation.
First, we load the required packages and define our loss function
using NLopt, DiffEqSensitivity, ForwardDiff make_u0(θ) = [θ[1],θ[2],θ[3], 0.0] function 𝔼_loss(θ) # \bbE u0 = make_u0(θ) expectation(obs, prob, u0, p_uncertain, Koopman(), Tsit5(); ireltol = 1e-5, callback=cbs)[1] end
𝔼_loss (generic function with 1 method)
NLopt requires that this loss function return the loss as above, but also do an inplace update of the gradient. So, we wrap this function to put it in the form required by NLopt.
function 𝔼_loss_nlopt(x,∇) length(∇) > 0 ? ForwardDiff.gradient!(∇, 𝔼_loss,x) : nothing 𝔼_loss(x) end
𝔼_loss_nlopt (generic function with 1 method)
We then optimize using the Method of Moving Asymptotes algorithm (:LD_MMA
)
opt = Opt(:LD_MMA, 3) opt.lower_bounds = [-100.0,1.0, 10.0] opt.upper_bounds = [0.0,3.0, 50.0] opt.xtol_rel = 1e-3 opt.min_objective = 𝔼_loss_nlopt (minf,minx,ret) = NLopt.optimize(opt, [-1.0, 2.0, 50.0])
(0.0838151729557601, [-0.05713109086715892, 2.4366742783682063, 49.99835957 072762], :XTOL_REACHED)
Let's now visualize 350 Monte Carlo simulations
ensembleprob = EnsembleProblem(remake(prob,u0 = make_u0(minx)),prob_func=prob_func) ensemblesol = solve(ensembleprob,Tsit5(),EnsembleThreads(), trajectories=100_000, callback=cbs) begin plot(ensemblesol, vars = (1,3), lw=1,alpha=0.1, label=nothing, idxs = 1:350) plot!(solve(remake(prob, u0=make_u0(minx)),Tsit5(), callback=cbs), vars=(1,3),label = nothing, c=:black, lw=3,ls=:dash) xlabel!("x [m]") ylabel!("y [m]") plot!(rectangle(27.5, 25, 5, 50), c=:red, label = nothing) scatter!([25],[25],marker=:star, ms=10, label = nothing,c=:green) ylims!(0.0,50.0) xlims!(minx[1], 27.5) end
Looks pretty good! But, how long did it take? Let's benchmark.
using BenchmarkTools @btime NLopt.optimize($opt, $[-1.0, 2.0, 50.0])
127.903 ms (916193 allocations: 65.05 MiB) (0.0838151729557601, [-0.05713109086715892, 2.4366742783682063, 49.99835957 072762], :XTOL_REACHED)
Not bad for bound constrained optimization under uncertainty of a hybrid system!
With this approach we can also consider probabilistic constraints. Let us now consider a wall at $x=20$ with height 25.
constraint = [20.0, 25.0] begin plot(rectangle(27.5, 25, 5, 50), c=:red, label = nothing) xlabel!("x [m]") ylabel!("y [m]") plot!([constraint[1], constraint[1]],[0.0,constraint[2]], lw=5, c=:black, label=nothing) scatter!([25],[25],marker=:star, ms=10, label = nothing,c=:green) ylims!(0.0,50.0) xlims!(minx[1], 27.5) end
We now wish to minimize the same loss function as before, but introduce an inequality constraint such that the solution must have less than a 1% chance of colliding with the wall at $x=20$. This class of probabilistic constraints is called a chance constraint.
To do this, we first introduce a new callback and solve the system using the previous optimal solution
constraint_condition(u,t,integrator) = u[1] - constraint[1] constraint_affect!(integrator) = integrator.u[3] < constraint[2] ? terminate!(integrator) : nothing constraint_cb = ContinuousCallback(constraint_condition, constraint_affect!, save_positions=(true,false)); constraint_cbs = CallbackSet(ground_cb, stop_cb, constraint_cb) ensemblesol = solve(ensembleprob,Tsit5(),EnsembleThreads(), trajectories=350, callback=constraint_cbs, maxstep=0.1) begin plot(ensemblesol, vars = (1,3), lw=1,alpha=0.1, label=nothing) plot!(solve(remake(prob, u0=make_u0(minx)),Tsit5(), callback=constraint_cbs), vars=(1,3),label = nothing, c=:black, lw=3, ls=:dash) xlabel!("x [m]") ylabel!("y [m]") plot!(rectangle(27.5, 25, 5, 50), c=:red, label = nothing) plot!([constraint[1], constraint[1]],[0.0,constraint[2]], lw=5, c=:black) scatter!([25],[25],marker=:star, ms=10, label = nothing,c=:green) ylims!(0.0,50.0) xlims!(minx[1], 27.5) end
That doesn't look good!
We now need a second observable for the system. In order to compute a probability of impact, we use an indicator function for if a trajectory impacts the wall. In other words, this functions returns 1 if the trajectory hits the wall and 0 otherwise.
constraint_obs(sol) = sol[1,end] ≈ constraint[1] ? one(sol[1,end]) : zero(sol[1,end])
constraint_obs (generic function with 1 method)
Using the previously computed optimal initial conditions, lets compute the probability of hitting this wall
expectation(constraint_obs, prob, make_u0(minx), p_uncertain, Koopman(), Tsit5(); ireltol= 1e-9, iabstol = 1e-9, callback=constraint_cbs)[1]
0.9910305871387488
We then setup the constraint function for NLopt just as before.
function 𝔼_constraint(θ) u0 = [θ[1],θ[2],θ[3], 0.0] expectation(constraint_obs, prob, u0, p_uncertain, Koopman(), Tsit5(), ireltol= 1e-9, iabstol = 1e-9,callback=constraint_cbs)[1] end function 𝔼_constraint_nlopt(x,∇) length(∇) > 0 ? ForwardDiff.gradient!(∇, 𝔼_constraint,x) : nothing 𝔼_constraint(x) - 0.01 end
𝔼_constraint_nlopt (generic function with 1 method)
Note that NLopt requires the constraint function to be of the form $g(x) \leq 0$. Hence, why we return 𝔼_constraint(x) - 0.01
for the 1% chance constraint.
The rest of the NLopt setup looks the same as before with the exception of adding the inequality constraint
opt = Opt(:LD_MMA, 3) opt.lower_bounds = [-100.0, 1.0, 10.0] opt.upper_bounds = [0.0, 3.0, 50.0] opt.xtol_rel = 1e-3 opt.min_objective = 𝔼_loss_nlopt inequality_constraint!(opt,𝔼_constraint_nlopt, 1e-5) (minf2,minx2,ret2) = NLopt.optimize(opt, [-1.0, 2.0, 50.0])
(40.28996781875263, [-0.8176321801341233, 2.000952669391841, 50.0], :XTOL_R EACHED)
The probability of impacting the wall is now
λ = 𝔼_constraint(minx2)
0.010004498174330818
We can check if this is within tolerance by
λ - 0.01 <= 1e-5
true
Again, we plot some Monte Carlo simulations from this result as follows
ensembleprob = EnsembleProblem(remake(prob,u0 = make_u0(minx2)),prob_func=prob_func) ensemblesol = solve(ensembleprob,Tsit5(),EnsembleThreads(), trajectories=350, callback=constraint_cbs) begin plot(ensemblesol, vars = (1,3), lw=1,alpha=0.1, label=nothing) plot!(solve(remake(prob, u0=make_u0(minx2)),Tsit5(), callback=constraint_cbs), vars=(1,3),label = nothing, c=:black, lw=3, ls=:dash) plot!([constraint[1], constraint[1]],[0.0,constraint[2]], lw=5, c=:black) xlabel!("x [m]") ylabel!("y [m]") plot!(rectangle(27.5, 25, 5, 50), c=:red, label = nothing) scatter!([25],[25],marker=:star, ms=10, label = nothing,c=:green) ylims!(0.0,50.0) xlims!(minx[1], 27.5) end
This tutorial is part of the SciMLTutorials.jl repository, found at: https://github.com/SciML/SciMLTutorials.jl. For more information on doing scientific machine learning (SciML) with open source software, check out https://sciml.ai/.
To locally run this tutorial, do the following commands:
using SciMLTutorials
SciMLTutorials.weave_file("DiffEqUncertainty","02-AD_and_optimization.jmd")
Computer Information:
Julia Version 1.4.2
Commit 44fa15b150* (2020-05-23 18:35 UTC)
Platform Info:
OS: Linux (x86_64-pc-linux-gnu)
CPU: Intel(R) Core(TM) i7-9700K CPU @ 3.60GHz
WORD_SIZE: 64
LIBM: libopenlibm
LLVM: libLLVM-8.0.1 (ORCJIT, skylake)
Environment:
JULIA_LOAD_PATH = /builds/JuliaGPU/DiffEqTutorials.jl:
JULIA_DEPOT_PATH = /builds/JuliaGPU/DiffEqTutorials.jl/.julia
JULIA_CUDA_MEMORY_LIMIT = 2147483648
JULIA_NUM_THREADS = 8
Package Information:
Status `/builds/JuliaGPU/DiffEqTutorials.jl/tutorials/DiffEqUncertainty/Project.toml`
[6e4b80f9-dd63-53aa-95a3-0cdb28fa8baf] BenchmarkTools 0.5.0
[8a292aeb-7a57-582c-b821-06e4c11590b1] Cuba 2.1.0
[071ae1c0-96b5-11e9-1965-c90190d839ea] DiffEqGPU 1.5.0
[41bf760c-e81c-5289-8e54-58b1f1f8abe2] DiffEqSensitivity 6.28.0
[ef61062a-5684-51dc-bb67-a0fcdec5c97d] DiffEqUncertainty 1.5.0
[0c46a032-eb83-5123-abaf-570d42b7fbaa] DifferentialEquations 6.15.0
[31c24e10-a181-5473-b8eb-7969acd0382f] Distributions 0.23.8
[f6369f11-7733-5829-9624-2563aa707210] ForwardDiff 0.10.12
[76087f3c-5699-56af-9a33-bf431cd00edd] NLopt 0.6.0
[1dea7af3-3e70-54e6-95c3-0bf5283fa5ed] OrdinaryDiffEq 5.42.1
[91a5bcdd-55d7-5caf-9e0b-520d859cae80] Plots 1.5.8
[67601950-bd08-11e9-3c89-fd23fb4432d2] Quadrature 1.3.0