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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

Example: Solving the FitzHugh-Nagumo ODE

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])
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"])

Fitzhugh-Nagumo Solution

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")

Fitzhugh-Nagumo Standard-Deviations

Related packages

  • probdiffeq: Fast and feature-rich filtering-based probabilistic ODE solvers in JAX.
  • ProbNum: Probabilistic numerics in Python. It has not only probabilistic ODE solvers, but also probabilistic linear solvers, Bayesian quadrature, and many filtering and smoothing implementations.
  • Fenrir.jl: Parameter-inference in ODEs with probabilistic ODE solvers. This package builds on ProbNumDiffEq.jl to provide a negative marginal log-likelihood function, which can then be used with an optimizer or with MCMC for parameter inference.