Probabilistic Numerical Differential Equation solvers via Bayesian filtering and smoothing
ProbNumDiffEq.jl provides probabilistic numerical ODE solvers to the DifferentialEquations.jl ecosystem. The implemented ODE filters solve differential equations via Bayesian filtering and smoothing. The filters compute not just a single point estimate of the true solution, but a posterior distribution that contains an estimate of its numerical approximation error.
For a short intro video, check out the ProbNumDiffEq.jl poster presentation at JuliaCon2021.
Run Julia, enter ]
to bring up Julia's package manager, and add the ProbNumDiffEq.jl package:
julia> ]
(v1.8) pkg> add ProbNumDiffEq
using ProbNumDiffEq
# ODE definition as in DifferentialEquations.jl
function f(du, u, p, t)
a, b, c = p
du[1] = c * (u[1] - u[1]^3 / 3 + u[2])
du[2] = -(1 / c) * (u[1] - a - b * u[2])
end
u0 = [-1.0, 1.0]
tspan = (0.0, 20.0)
p = (0.2, 0.2, 3.0)
prob = ODEProblem(f, u0, tspan, p)
# Solve the ODE with a probabilistic numerical solver: EK1
sol = solve(prob, EK1())
# Plot the solution with Plots.jl
using Plots
plot(sol, color=["#CB3C33" "#389826" "#9558B2"])
In probabilistic numerics, the solution also contains error estimates - it just happens that they are too small to be visible in the plot above. But we can just plot them directly:
using Statistics
stds = std.(sol.pu)
plot(sol.t, hcat(stds...)', color=["#CB3C33" "#389826" "#9558B2"]
label=["std(u1(t))" "std(u2(t))"], xlabel="t", ylabel="standard-deviation")