WENDy.solve — Function

Solve the inverse problem for the unknown parameters solve( wendyProb::WENDyProblem, p₀::AbstractVector{<:Real}, params::WENDyParameters=WENDyParameters(); alg::Symbol=:trustRegion, kwargs... )

Arguments

  • wendyProblem::WENDyProblem : An instance of a WENDyProblem for the ODE that you wish to estimate parameters for
  • p₀::AbstractVector{<:Real} : Inital guess for the parameters
  • params::WENDyParameters : hyperparameters for the WENDy Algorithm
  • alg::AbstractWENDySolver=TrustRegion() : (optional) Choice of solver
    • OELS : output error least squares
    • WLS : weak form least squares
    • IRLS : WENDy generalized least squares solver via iterative reweighted least squares
    • TrustRegion : (default) optimize over the weak form negative log-likelihood with a trust region solver. This approximates the maximum likelhood estimator. Note: this is the only solver that will respect the constraints
    • ARCqK : optimize over the weak form negative log-likelihood with adaptive regularized cubics algorithm
    • HybridTrustRegionOELS : hybrid solver that first optimizes with the trust region solver then passes the result as an intialization to the output error least squares problem
    • HybridWLSTrustRegion : hybrid solver that first optimizes with the weak form least squares solver then passes the result as an intialization to the trust region weak form negative log-likelihood solver
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WENDy.WENDyParameters — Type

Hyper-parameters for the WENDy Algorithm

Constructor

WENDyParameters(;   
    diagReg::Real=1.0e-10,
    radiusMinTime::Real=0.01,
    radiusMaxTime::Real=5.0,
    numRadii::Int=100,
    radiiParams::AbstractVector{<:Real}=2 .^(0:3),
    testFunSubRate::Real=2.0,
    maxTestFunCondNum::Real=1e4,
    minTestFunInfoNum::Real=0.95,
    Kmax::Int=200,
    Kᵣ::Union{Nothing,Int}=100,
    fsAbstol::Real=1e-8,
    fsReltol::Real=1e-8,
    nlsAbstol::Real=1e-8,
    nlsReltol::Real=1e-8,
    nlsMaxiters::Int=1000,
    optimAbstol::Real=1e-8,
    optimReltol::Real=1e-8,
    optimMaxiters::Int=500,
    optimTimelimit::Real=200.0,
    fsAlg::OrdinaryDiffEqAlgorithm=Rodas4P(),
    fsU0Free::Bool=true
)

Fields

  • diagReg::Real = 1.0e-10 : Regularization constant for the covariance computations
  • radiusMinTime::Real = 0.01 : Minimum test function radius (in time units matching _tt)
  • radiusMaxTime::Real = 5.0 : Minimum test function radius (in time units matching _tt)
  • numRadii::Int = 100 : Maximum number of radii to be checked in the min radii detection sub-algorithm
  • radiiParams::AbstractVector{<:Real} = 2 .^(0:3) : Multiplied by the minRadius to give a list of radii to use when building the test function matrix
  • testFunSubRate::Real = 2.0 : Corresponds to how much we should sup-sample in the min radii detection sub-algorithm
  • maxTestFunCondNum::Real = 1e4 : Maximum Condition number of the test function matrix after svd reduction
  • minTestFunInfoNum::Real = 0.95 : Minimum information (σₖ/σ₁)in the test function matrix after svd reduction
  • Kmax::Int = 200 : Hard maximum size on the test function matrix
  • Kᵣ::Union{Nothing,Int} = 100 : how many test function to spread through the time domain in the min radii detection sub-algorithm
  • fsAbstol::Real = 1e-8 : forward solve absolute tolerance for solving ordinary differential equation
  • fsReltol::Real = 1e-8 : forward solve relative tolerance for solving ordinary differntial equation
  • nlsAbstol::Real = 1e-8 : nonlinear least squares absolute tolerance (only used in the IRLS WENDy algorithm)
  • nlsReltol::Real = 1e-8 : nonlinear least squares relative tolerance (only used in the IRLS WENDy algorithm)
  • nlsMaxiters::Int = 1000 : nonlinear least squares maximum iterations (only used in the IRLS WENDy algorithm)
  • optimAbstol::Real = 1e-8 : absolute tolerance (used by all other optimization algorithms)
  • optimReltol::Real = 1e-8 : relative tolerance (used by all other optimization algorithms)
  • optimMaxiters::Int = 200 : maximum iterations (used by all other optimization algorithms)
  • optimTimelimit::Real = 500.0 : maximum time in seconds (used by all other optimization algorithms)
  • fsAlg::OrdinaryDiffEqAlgorithm = Rodas4P() : forward solve algorithm used by the forward solve nonlinear least squares algorithm
  • fsU0Free::Bool = true : Specifies if the forward solve algorithm should also optimize over the initial condition
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WENDy.WENDyProblem — Type
WENDyProblem{lip, DistType}(...)

A WENDyProblem struct pre-computes and allocates data structures for efficient solving of the parameter inverse problem

Constructor

WENDyProblem(
    _tt::AbstractVector{<:Real}, 
    U::AbstractVecOrMat{<:Real}, 
    _f!::Function, 
    J::Int; 
    linearInParameters::Val{lip}=Val(false), 
    noiseDist::Val{DistType}=Val(Normal), 
    params::WENDyParameters=WENDyParameters(), 
    constraints::Union{Nothing,AbstractVector{Tuple{<:Real,<:Real}}}=nothing, 
    ll::LogLevel=Warn
)

Arguments

  • _tt::AbstractVector{<:Real} : vector of times (equispaced)
  • U::AbstractVecOrMat{<:Real} : Corrupted state variable data
  • _f!::Function : Right hand-side of the differential equation Must be of the form f!(du, u, p, t)
  • J::Int : number of parameters (to be estimated)
  • linearInParameters::Val{lip}=Val(false) : (optional) specify whether the right hand side is `linear in parameters' for improved computational efficiency
  • noiseDist::Val{DistType}=Val(Normal) : (optional) specify the distribution of the measurement noise. Choose either Val(Normal) for additive Gaussian noise of Val(LogNormal) for multiplicative LogNormal noise.
  • params::WENDyParameters : (optional) struct of hyper-parameters for the WENDy Algorithm (see the doc for WENDyParameters)
  • constraints=nothing : (optional) Linear box constraints for each parameter, ∀j ∈ [1, ⋯,J], ℓⱼ ≤ pⱼ ≤ uⱼ. Accepts constraints as a list of tuples, [(ℓ₁,u₁), ⋯]. Note: this only is compatible with the TrustRegion solver.
  • ll::LogLevel=Warn : (optional) see additional algorithm information by setting ll=Info

Fields

  • D::Int : number of state variables
  • J::Int : number of parameters (to be estimated)
  • Mp1::Int : (Mp1+1) number of data points in time
  • K::Int : number of test functions
  • u₀::AbstractVector{<:Real} : Initial Condition of the ODE (Necessary for the forward solver)
  • constraints : vector of tuples containing linear constraints for each parameter
  • data : Internal data structure
  • oels::LeastSquaresCostFunction : Cost function for the comparison method
  • wlsq::LeastSquaresCostFunction : Cost function for the weak form least squares problem
  • wnll::SecondOrderCostFunction : Cost function for the weak form negative log-likelihood
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