Trajectory Uncertainty
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When the vector field $\dot{x} = f(x)$ is a Gaussian process, we have uncertainty in the vector field, which we can lift to the trajectory. In this example we show how to use the approximate PULL solvers, introduced in this paper, to propagate this uncertainty.
The implementation of the PULL solvers is still in progress and the interface may change.
1D Example
Setup
using GPDiffEq
using Plots
using LinearAlgebrausing StatsPlotsDefining a simple GP
In this example we sample the vector field values directly instead of learning them from a trajectory.
ts = range(-4.25, 4.252; length=100)
f(x) = x * cos(x)
X = range(-3.0, 3.0; length=10)
σ_n = 0.1
y = f.(X) .+ σ_n * randn(length(X))
p = plot(ts, f.(ts); label="truth", xlabel="x", ylabel="f(x)")
scatter!(p, X, y; label="data")
With this data, we define the GP.
ker = SqExponentialKernel()
gp = GP(ker)
fx = gp(X, σ_n^2)
fp = posterior(fx, y)
plot!(p, ts, mean(fp, ts); ribbons=sqrt.(var(fp, ts)), label="GP posterior")
A GPODE problem
We define a GPODEProblem with the GP as the vector field.
u0 = 1.0
tspan = (0.0, 8.0)
ff = GPODEFunction(fp)
prob = GPODEProblem(ff, u0, tspan)GPODEProblem with uType Float64 and tType Float64. In-place: false
timespan: (0.0, 8.0)
u0: 1.0and integrate with the PULL Euler solver.
sol = solve(prob, PULLEuler(); dt=0.1)This results in a trajectory where each state is a normal distribution, and we can plot the trajectories of the means and their standard deviations.
plot(sol.t, mean.(sol.u); ribbons=std.(sol.u), label="μ ± σ", xlabel="t", ylabel="u(t)")
2D Example
Setup
function fun(x)
return [-0.1 2.0; -2.0 -0.1] * (x .^ 3)
end
xrange = range(-2.2, 2.2; length=6)
x = collect.(Iterators.product(xrange, xrange))[:]
y = fun.(x)
# This is annoying UX, needs fix
y = ColVecs(reduce(hcat, y))
xMO, yMO = prepare_isotopic_multi_output_data(x, y)Defining a Multi-Output GP
σ_n = 3e-2
ker = SqExponentialKernel()
mker = IndependentMOKernel(ker)
gp = GP(mker)
fx = gp(xMO, σ_n)
fp = posterior(fx, yMO)AbstractGPs.PosteriorGP{AbstractGPs.GP{AbstractGPs.ZeroMean{Float64}, KernelFunctions.IndependentMOKernel{KernelFunctions.SqExponentialKernel{Distances.Euclidean}}}, NamedTuple{(:α, :C, :x, :δ), Tuple{Vector{Float64}, LinearAlgebra.Cholesky{Float64, Matrix{Float64}}, KernelFunctions.MOInputIsotopicByFeatures{Vector{Float64}, Vector{Vector{Float64}}, Int64}, Vector{Float64}}}}(AbstractGPs.GP{AbstractGPs.ZeroMean{Float64}, KernelFunctions.IndependentMOKernel{KernelFunctions.SqExponentialKernel{Distances.Euclidean}}}(AbstractGPs.ZeroMean{Float64}(), Independent Multi-Output Kernel
Squared Exponential Kernel (metric = Distances.Euclidean(0.0))), (α = [-29.2124703130299, 32.2874671880857, -7.965684324886952, -23.297037809412693, -16.60262823825942, 17.04785570559112, -18.22035634735454, -15.306706476310413, -5.602061180182959, 23.975425084666156, -32.28746718808571, -29.21247031302988, 23.975425084666156, 5.6020611801829645, 12.631131521828273, -13.960724313599602, 15.498661385982752, 14.41021704844255, 13.983933194686868, -15.884346777476033, 13.960724313599629, 12.63113152182824, 23.297037809412682, -7.965684324886963, -15.306706476310406, 18.220356347354503, -15.884346777476066, -13.983933194686836, -10.937943645620454, 12.08930613463313, -12.089306134633127, -10.937943645620477, -14.410217048442616, 15.4986613859828, -17.04785570559108, -16.602628238259427, 17.04785570559113, 16.602628238259474, 14.410217048442554, -15.498661385982798, 12.089306134633159, 10.937943645620454, 10.937943645620438, -12.089306134633114, 15.88434677747611, 13.983933194686822, 15.306706476310364, -18.22035634735452, -23.297037809412696, 7.9656843248868965, -13.960724313599627, -12.631131521828209, -13.983933194686838, 15.884346777476038, -15.49866138598281, -14.410217048442561, -12.631131521828255, 13.960724313599634, -23.975425084666128, -5.602061180182979, 32.28746718808571, 29.212470313029925, 5.602061180182997, -23.975425084666178, 18.22035634735448, 15.306706476310405, 16.602628238259502, -17.04785570559111, 7.965684324886902, 23.29703780941267, 29.212470313029897, -32.287467188085685], C = LinearAlgebra.Cholesky{Float64, Matrix{Float64}}([1.014889156509222 0.0 0.6689945260858627 0.0 0.20938525445078104 0.0 0.030210087079326205 0.0 0.0020092788525361955 0.0 6.160426818418967e-5 0.0 0.6689945260858627 0.0 0.4542173726600899 0.0 0.1421632262177757 0.0 0.02051129844258684 0.0 0.0013642105065943936 0.0 4.1826543788011286e-5 0.0 0.20938525445078104 0.0 0.1421632262177757 0.0 0.044494957932335086 0.0 0.006419728825951168 0.0 0.00042697743092001683 0.0 1.309108097653619e-5 0.0 0.030210087079326205 0.0 0.02051129844258684 0.0 0.006419728825951168 0.0 0.0009262379405195118 0.0 6.160426818418967e-5 0.0 1.8887800733677394e-6 0.0 0.0020092788525361955 0.0 0.0013642105065943936 0.0 0.00042697743092001683 0.0 6.160426818418967e-5 0.0 4.097312032349871e-6 0.0 1.2562313536350808e-7 0.0 6.160426818418967e-5 0.0 4.1826543788011286e-5 0.0 1.309108097653619e-5 0.0 1.8887800733677394e-6 0.0 1.2562313536350808e-7 0.0 3.851591485822847e-9 0.0; 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0.0 1.9169024154916113e-6 0.0 1.3285996130070735e-5 0.0 4.244930574471081e-5 0.0 6.252150377482015e-5 0.0 4.244930574471081e-5 0.0 1.3285996130070735e-5 0.0 6.252150377482015e-5 0.0 0.00043333476471489087 0.0 0.0013845224503386026 0.0 0.002039195319842277 0.0 0.0013845224503386026 0.0 0.0004333347647148905 0.0 0.0009400288421806862 0.0 0.006515313173187524 0.0 0.020816694375305873 0.0 0.030659889794007515 0.0 0.020816694375305873 0.0 0.006515313173187519 0.0 0.006515313173187519 0.0 0.04515745032486089 0.0 0.1442799167427881 0.0 0.21250282427502198 0.0 0.14427991674278806 0.0 0.04515745032486087 0.0 0.020816694375305873 0.0 0.14427991674278812 0.0 0.46098028621083353 0.0 0.6789552902885679 0.0 0.46098028621083353 0.0 0.14427991674278806 0.0 0.030659889794007515 0.0 0.21250282427502207 0.0 0.6789552902885679 0.0 0.5039665508436914 0.0 0.41246887335006166 0.0 0.1276169543823575; 1.2749355788711452e-7 0.0 1.9169024154916113e-6 0.0 1.3285996130070735e-5 0.0 4.244930574471081e-5 0.0 6.252150377482015e-5 0.0 4.244930574471081e-5 0.0 4.158317552466647e-6 0.0 6.252150377482015e-5 0.0 0.00043333476471489087 0.0 0.0013845224503386026 0.0 0.002039195319842277 0.0 0.0013845224503386026 0.0 6.252150377482015e-5 0.0 0.0009400288421806862 0.0 0.006515313173187524 0.0 0.020816694375305873 0.0 0.030659889794007515 0.0 0.020816694375305873 0.0 0.0004333347647148905 0.0 0.006515313173187519 0.0 0.04515745032486089 0.0 0.14427991674278806 0.0 0.21250282427502198 0.0 0.14427991674278806 0.0 0.0013845224503386026 0.0 0.020816694375305873 0.0 0.14427991674278812 0.0 0.46098028621083353 0.0 0.6789552902885679 0.0 0.46098028621083353 0.0 0.002039195319842277 0.0 0.030659889794007515 0.0 0.21250282427502207 0.0 0.6789552902885679 0.0 0.5039515815308104 0.0 0.4068063536781695 0.0; 0.0 1.2749355788711452e-7 0.0 1.9169024154916113e-6 0.0 1.3285996130070735e-5 0.0 4.244930574471081e-5 0.0 6.252150377482015e-5 0.0 4.244930574471081e-5 0.0 4.158317552466647e-6 0.0 6.252150377482015e-5 0.0 0.00043333476471489087 0.0 0.0013845224503386026 0.0 0.002039195319842277 0.0 0.0013845224503386026 0.0 6.252150377482015e-5 0.0 0.0009400288421806862 0.0 0.006515313173187524 0.0 0.020816694375305873 0.0 0.030659889794007515 0.0 0.020816694375305873 0.0 0.0004333347647148905 0.0 0.006515313173187519 0.0 0.04515745032486089 0.0 0.14427991674278806 0.0 0.21250282427502198 0.0 0.14427991674278806 0.0 0.0013845224503386026 0.0 0.020816694375305873 0.0 0.14427991674278812 0.0 0.46098028621083353 0.0 0.6789552902885679 0.0 0.46098028621083353 0.0 0.002039195319842277 0.0 0.030659889794007515 0.0 0.21250282427502207 0.0 0.6789552902885679 0.0 0.5039515815308104 0.0 0.4068063536781695; 3.90893843426485e-9 0.0 1.2749355788711452e-7 0.0 1.9169024154916113e-6 0.0 1.3285996130070735e-5 0.0 4.244930574471081e-5 0.0 6.252150377482015e-5 0.0 1.2749355788711452e-7 0.0 4.158317552466647e-6 0.0 6.252150377482015e-5 0.0 0.0004333347647148905 0.0 0.0013845224503386026 0.0 0.002039195319842277 0.0 1.9169024154916113e-6 0.0 6.252150377482015e-5 0.0 0.0009400288421806862 0.0 0.006515313173187519 0.0 0.020816694375305873 0.0 0.030659889794007515 0.0 1.3285996130070735e-5 0.0 0.0004333347647148905 0.0 0.006515313173187519 0.0 0.04515745032486087 0.0 0.14427991674278806 0.0 0.21250282427502198 0.0 4.244930574471081e-5 0.0 0.0013845224503386026 0.0 0.020816694375305873 0.0 0.14427991674278806 0.0 0.46098028621083353 0.0 0.6789552902885679 0.0 6.252150377482015e-5 0.0 0.002039195319842277 0.0 0.030659889794007515 0.0 0.21250282427502198 0.0 0.6789552902885679 0.0 0.5172216476909152 0.0; 0.0 3.90893843426485e-9 0.0 1.2749355788711452e-7 0.0 1.9169024154916113e-6 0.0 1.3285996130070735e-5 0.0 4.244930574471081e-5 0.0 6.252150377482015e-5 0.0 1.2749355788711452e-7 0.0 4.158317552466647e-6 0.0 6.252150377482015e-5 0.0 0.0004333347647148905 0.0 0.0013845224503386026 0.0 0.002039195319842277 0.0 1.9169024154916113e-6 0.0 6.252150377482015e-5 0.0 0.0009400288421806862 0.0 0.006515313173187519 0.0 0.020816694375305873 0.0 0.030659889794007515 0.0 1.3285996130070735e-5 0.0 0.0004333347647148905 0.0 0.006515313173187519 0.0 0.04515745032486087 0.0 0.14427991674278806 0.0 0.21250282427502198 0.0 4.244930574471081e-5 0.0 0.0013845224503386026 0.0 0.020816694375305873 0.0 0.14427991674278806 0.0 0.46098028621083353 0.0 0.6789552902885679 0.0 6.252150377482015e-5 0.0 0.002039195319842277 0.0 0.030659889794007515 0.0 0.21250282427502198 0.0 0.6789552902885679 0.0 0.5172216476909152], 'U', 0), x = [([-2.2, -2.2], 1), ([-2.2, -2.2], 2), ([-1.32, -2.2], 1), ([-1.32, -2.2], 2), ([-0.44, -2.2], 1), ([-0.44, -2.2], 2), ([0.44, -2.2], 1), ([0.44, -2.2], 2), ([1.32, -2.2], 1), ([1.32, -2.2], 2), ([2.2, -2.2], 1), ([2.2, -2.2], 2), ([-2.2, -1.32], 1), ([-2.2, -1.32], 2), ([-1.32, -1.32], 1), ([-1.32, -1.32], 2), ([-0.44, -1.32], 1), ([-0.44, -1.32], 2), ([0.44, -1.32], 1), ([0.44, -1.32], 2), ([1.32, -1.32], 1), ([1.32, -1.32], 2), ([2.2, -1.32], 1), ([2.2, -1.32], 2), ([-2.2, -0.44], 1), ([-2.2, -0.44], 2), ([-1.32, -0.44], 1), ([-1.32, -0.44], 2), ([-0.44, -0.44], 1), ([-0.44, -0.44], 2), ([0.44, -0.44], 1), ([0.44, -0.44], 2), ([1.32, -0.44], 1), ([1.32, -0.44], 2), ([2.2, -0.44], 1), ([2.2, -0.44], 2), ([-2.2, 0.44], 1), ([-2.2, 0.44], 2), ([-1.32, 0.44], 1), ([-1.32, 0.44], 2), ([-0.44, 0.44], 1), ([-0.44, 0.44], 2), ([0.44, 0.44], 1), ([0.44, 0.44], 2), ([1.32, 0.44], 1), ([1.32, 0.44], 2), ([2.2, 0.44], 1), ([2.2, 0.44], 2), ([-2.2, 1.32], 1), ([-2.2, 1.32], 2), ([-1.32, 1.32], 1), ([-1.32, 1.32], 2), ([-0.44, 1.32], 1), ([-0.44, 1.32], 2), ([0.44, 1.32], 1), ([0.44, 1.32], 2), ([1.32, 1.32], 1), ([1.32, 1.32], 2), ([2.2, 1.32], 1), ([2.2, 1.32], 2), ([-2.2, 2.2], 1), ([-2.2, 2.2], 2), ([-1.32, 2.2], 1), ([-1.32, 2.2], 2), ([-0.44, 2.2], 1), ([-0.44, 2.2], 2), ([0.44, 2.2], 1), ([0.44, 2.2], 2), ([1.32, 2.2], 1), ([1.32, 2.2], 2), ([2.2, 2.2], 1), ([2.2, 2.2], 2)], δ = [-20.231200000000005, 22.36080000000001, -21.066003200000008, 5.664736, -21.287481600000007, 1.2351680000000003, -21.304518400000006, 0.8944320000000003, -21.525996800000005, -3.535136, -22.36080000000001, -20.231200000000005, -3.535136, 21.525996800000005, -4.3699392, 4.829932800000001, -4.591417600000001, 0.4003648, -4.6084544, 0.059628800000000044, -4.829932800000001, -4.3699392, -5.664736000000001, -21.066003200000008, 0.8944320000000004, 21.304518400000006, 0.05962880000000004, 4.6084544, -0.16184959999999998, 0.1788864, -0.1788864, -0.16184959999999998, -0.4003648, -4.591417600000001, -1.2351680000000005, -21.287481600000007, 1.2351680000000005, 21.287481600000007, 0.4003648, 4.591417600000001, 0.1788864, 0.16184959999999998, 0.16184959999999998, -0.1788864, -0.05962880000000004, -4.6084544, -0.8944320000000004, -21.304518400000006, 5.664736000000001, 21.066003200000008, 4.829932800000001, 4.3699392, 4.6084544, -0.059628800000000044, 4.591417600000001, -0.4003648, 4.3699392, -4.829932800000001, 3.535136, -21.525996800000005, 22.36080000000001, 20.231200000000005, 21.525996800000005, 3.535136, 21.304518400000006, -0.8944320000000003, 21.287481600000007, -1.2351680000000003, 21.066003200000008, -5.664736, 20.231200000000005, -22.36080000000001]))A GPODE problem
We define a GPODEProblem with the GP as the vector field.
h = 0.002
u0 = [2.0; 0.0]
tspan = (0.0, 2.0)
gpff = GPODEFunction(fp)(::GPDiffEq.PullSolversModule.GPODEFunction{false, AbstractGPs.PosteriorGP{AbstractGPs.GP{AbstractGPs.ZeroMean{Float64}, KernelFunctions.IndependentMOKernel{KernelFunctions.SqExponentialKernel{Distances.Euclidean}}}, NamedTuple{(:α, :C, :x, :δ), Tuple{Vector{Float64}, LinearAlgebra.Cholesky{Float64, Matrix{Float64}}, KernelFunctions.MOInputIsotopicByFeatures{Vector{Float64}, Vector{Vector{Float64}}, Int64}, Vector{Float64}}}}, AbstractGPs.PosteriorGP{GPDiffEq.DerivativeGPModule.DerivativeGP{AbstractGPs.GP{AbstractGPs.ZeroMean{Float64}, KernelFunctions.IndependentMOKernel{KernelFunctions.SqExponentialKernel{Distances.Euclidean}}}, AbstractGPs.ZeroMean{Float64}, GPDiffEq.DerivativeGPModule.DerivativeKernelCollection{KernelFunctions.IndependentMOKernel{KernelFunctions.SqExponentialKernel{Distances.Euclidean}}}}, NamedTuple{(:α, :C, :x, :δ), Tuple{Vector{Float64}, LinearAlgebra.Cholesky{Float64, Matrix{Float64}}, KernelFunctions.MOInputIsotopicByFeatures{Vector{Float64}, Vector{Vector{Float64}}, Int64}, Vector{Float64}}}}}) (generic function with 1 method)Plot the GP
ug = range(-2.0, 2.0; length=6)
ug = vcat.(ug, ug')[:]
gp_pred_mean = gpff.(ug)
sf = maximum(norm.(gp_pred_mean))
dug = fun.(ug)
p = quiver(
getindex.(ug, 1),
getindex.(ug, 2);
quiver=(getindex.(dug, 1) / sf, getindex.(dug, 2) / sf),
)
quiver!(
p,
getindex.(ug, 1),
getindex.(ug, 2);
quiver=(getindex.(gp_pred_mean, 1) / sf, getindex.(gp_pred_mean, 2) / sf),
)
Define a GPODEProblem with the GP as the vector field.
prob = GPODEProblem(gpff, u0, tspan)GPODEProblem with uType Vector{Float64} and tType Float64. In-place: false
timespan: (0.0, 2.0)
u0: 2-element Vector{Float64}:
2.0
0.0Solving it with a non-PULL method uses the vector field defined by the mean of the GP.
det_gpsol = solve(prob, Euler(); dt=h)However, integrating with the PULL Euler solver returns a trajectory where each state is a multi-variate normal distribution.
gpsol = solve(prob, PULLEuler(); dt=h)Time series
We visualize the trajectories as time series, showing the standard deviation of each component.
m = mean.(gpsol.u)
s = var.(gpsol.u)
plot(
gpsol.t,
getindex.(m, 1);
ribbons=sqrt.(getindex.(s, 1)),
label="u1",
xlabel="t",
ylabel="mean",
)
plot!(gpsol.t, getindex.(m, 2); ribbons=sqrt.(getindex.(s, 2)), label="u2")
Phase plot
The components of each state are not independent, which we show in the phase plot by showing the ellipses of the covariance matrix for a subset of states.
μ_res = mean.(gpsol.u)
ell_intervall = 60
begin
p2 = deepcopy(p)
plot!(
p2,
getindex.(μ_res, 1),
getindex.(μ_res, 2);
linewidth=2.8,
label="PULL Euler",
color=:goldenrod4,
)
plot!(p2, det_gpsol; idxs=(1, 2), linewidth=2.8, label="GP Mean", linestyle=:dash)
for i in 1:ell_intervall:length(gpsol.u)
mv = gpsol[i]
covellipse!(p2, mv.μ, 2 * mv.Σ; label="", color=:goldenrod4)
end
p2
end
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