ProbNumDiffEq.jl

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 our poster presentation at JuliaCon2021.

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For more probabilistic numerics check out the ProbNum Python package. It implements probabilistic ODE solvers, but also probabilistic linear solvers, Bayesian quadrature, and many filtering and smoothing implementations.

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Installation

Run Julia, enter ] to bring up Julia's package manager, and add the ProbNumDiffEq.jl package:

julia> ]
(v1.7) pkg> add ProbNumDiffEq.jl

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])
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: EK0
sol = solve(prob, EK0(order=1), abstol=1e-2, reltol=1e-1)

# Plot the solution with Plots.jl
using Plots
plot(sol, color=["#107D79" "#FF9933"])

Benchmarks

• Multi-Language Wrapper Benchmark: ProbNumDiffEq.jl vs. OrdinaryDiffEq.jl, Hairer's FORTRAN solvers, Sundials, LSODA, MATLAB, and SciPy.

References

The main references for this package include:

• M. Schober, S. Särkkä, and P. Hennig: A Probabilistic Model for the Numerical Solution of Initial Value Problems (2018) (link)
• F. Tronarp, H. Kersting, S. Särkkä, and P. Hennig: Probabilistic Solutions To Ordinary Differential Equations As Non-Linear Bayesian Filtering: A New Perspective (2019) (link)
• N. Krämer, P. Hennig: Stable Implementation of Probabilistic ODE Solvers (2020) (link)
• N. Bosch, P. Hennig, F. Tronarp: Calibrated Adaptive Probabilistic ODE Solvers (2021) (link)
• N. Bosch, F. Tronarp, P. Hennig: Pick-and-Mix Information Operators for Probabilistic ODE Solvers (2022) (link)

A more extensive list of references relevant to ODE filters is provided here.