In the conference HSCC'17-19, Goubault and Putot presented a series of papers listed in the related works section and an accompanying demonstration software called RINO. This project uses RINO to create a numerical solver that computes inner and outer approximations of flowpipes for ODEs. Currently it only implements the algorithm present in HSCC'17.

A modified version of JuliaDiff/ForwardDiff.jl
AffineArithmetic.jl
ModalIntervalArithmetic.jl
IntervalArithmetic.jl
TaylorSeries.jl used in some example code

1. Assuming Julia is already installed, install IntervalArithmetic and TaylorSeries from official repositories.
2. Install ForwardDiff dependencies DiffResults, DiffRules, StaticArrays, SpecialFunctions, NaNMath , CommonSubexpressions from official repositories.
3. Clone AffineArithmetic.jl, ModalIntervalArithmetic.jl and ForwardDiff.jl from the Github repositories.
4. So the Julia REPL knows where to find the modules you've cloned, add the following to ~/.julia/config/startup.jl (Julia startup file):

verificationModulePaths = [
    "/path/to/AffineArithmetic.jl/src"
    "/path/to/ModalIntervalArithmetic.jl/src"
    "/path/to/ForwardDiff.jl/src"
]

for path in verificationModulePaths
    if(!(path in LOAD_PATH))
        push!(LOAD_PATH, path)
    end
end

5. Clone Flowpipes.jl from this repository. To test that it works do include("test/runtests.jl") in the REPL.

Flowpipes comes with a single function approximate(f::Function, tspan::NTuple{2,<:Real}, τ::Real, z₀::Vector{<:Interval}; order::Int=4)

Arguments:

• f is the function representing the system of autonomous first order ODEs \dot{x} = f(x)
• tspan=(tstart, tend) contains the start and end time the solution of the system is defined on. We build the inner and outer approximation of flowpipe over this time period.
• τ is the time step
• z₀ is an interval representing the bounds of the initial values that forms the flowpipe.
• order (optional) is the order of taylor approximation we use to build the outer approximation at each step.

Returns:

• st vector of evenly spaced points of time from t₀ to tn
• sz vector of intervals representing outer approximations at each time point. sz[i] is the outer approximation at st[i]
• sia vector of intervals representing of inner approximationss, or NaN if does not exist at each time point. sia[i] is the inner approimxation/NaN at st[i]

To use simply add include("~/path/to/Flowpipes.jl") in your file. For a runnable example take a look at ./examples/approx_2.jl.

• ForwardDiff takes up much CPU cycles and has a long runtime. See ./benchmarks/profiler.txt for Profiler results. Most of the cycles are spent in jacobian.jl and partials.jl. This is most likely because ForwardDiff does not use source code transformation, rather it computes derivatives (and higher-order derivatives) with operator overloading using dual numbers. This means for each higher-order derivative, hyper-dual numbers must be instantiated and calculated instead of one pass with pre-constructed functions f', f'', f''', etc.

• ForwardDiff does not natively support obtaining derivatives f'(x) of functions evaluated with types other than real numbers (i.e. Interval and Affine). Instead ForwardDiff must be altered by replacing instances of Real with ResolvableType = Union{Affine, Taylor1, Real}. This may mean that

• Flowpipes gives woefully poor inner/outer approximations compared to RINO. I'm not sure exactly why and it's something to investigate.

Eric Goubault and Sylvie Putot. 2017. Forward inner-approximated reachability of non-linear continuous systems. In Proceedings of the 20th. ACM International Conference on Hybrid Systems: Computation and Control (HSCC '17). ACM Press, New York, NY, 1-10. DOI:https://doi.org/10.1145/3049797.3049811 PDF:http://www.lix.polytechnique.fr/Labo/Sylvie.Putot/Publications/hscc17.pdf

Alexandre Goldsztejn1, David Daney1, Michel Rueher1, and Patrick Taillibert. 2005. Modal Intervals Revisited: a mean-value extension to generalized intervals. In Proceedings of the First International Workshop on Quantification in Constraint Programming (QCP '05). ??? PDF:http://goldsztejn.com/publications/QCP2005.Goldsztejn-Daney-Rueher-Taillibert.pdf

Miguel Sainz et al. 2014. Modal Interval Analysis: New Tools for Numerical Information. Springer Lecture Notes in Mathematics, New York, NY. PDF:https://www.springer.com/gp/book/9783319017204

Jorge Stolfi and Luiz Henrique de Figueiredo. 1997. Self-Validated Numberical Methods and Applications. In Proceedings of the 21st Brazilian Mathematics Colloquium. ??? PDF:http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.36.8089&rep=rep1&type=pdf

Robust INner and Outer Approximated Reachability (RINO) https://github.com/cosynus-lix/RINO

Affine Arithmetic C++ Library (aaflib) http://aaflib.sourceforge.net

TaylorModels: Rigorous function approximation using Taylor models in Julia. This package produces tight outer approximations to flowpipes similar to Flowpipes. <https://github.com/JuliaIntervals/TaylorModels.jl >