mutable struct Basis <: DataDrivenDiffEq.AbstractBasis

A basis over the states with parameters, independent variable, and possible exogenous controls. It extends an AbstractSystem as defined in ModelingToolkit.jl. f can either be a Julia function which is able to use ModelingToolkit variables or a vector of eqs. It can be called with the typical SciML signature, meaning out of place with f(u,p,t) or in place with f(du, u, p, t). If control inputs are present, it is assumed that no control corresponds to zero for all inputs. The corresponding function calls are f(u,p,t,inputs) and f(du,u,p,t,inputs) and need to be specified fully.

The optional implicits declare implicit variables in the Basis, meaning variables representing the (measured) target of the system. Right now only supported with the use of ImplicitOptimizers.

If linear_independent is set to true, a linear independent basis is created from all atom functions in f.

If simplify_eqs is set to true, simplify is called on f.

Additional keyworded arguments include name, which can be used to name the basis, and observed for defining observables.

Fields

• eqs

  The equations of the basis

• states

  Dependent (state) variables

• ctrls

  Control variables

• ps

  Parameters

• observed

  Observed

• iv

  Independent variable

• implicit

  Implicit variables of the basis

• f

  Internal function representation of the basis

• name

  Name of the basis

• systems

  Internal systems

Example

using ModelingToolkit
using DataDrivenDiffEq

@parameters w[1:2] t
@variables u[1:2](t)

Ψ = Basis([u; sin.(w.*u)], u, parameters = p, iv = t)

Note

The keyword argument eval_expression controls the function creation behavior. eval_expression=true means that eval is used, so normal world-age behavior applies (i.e. the functions cannot be called from the function that generates them). If eval_expression=false, then construction via GeneralizedGenerated.jl is utilized to allow for same world-age evaluation. However, this can cause Julia to segfault on sufficiently large basis functions. By default eval_expression=false.

monomial_basis(x)
monomial_basis(x, degree)

Constructs an array containing monomial basis in the variables x up to degree c of the form [x₁, x₁^2, ... , x₁^c, x₂, x₂^2, ...].

polynomial_basis(x)
polynomial_basis(x, degree)

Constructs an array containing a polynomial basis in the variables x up to degree c of the form [x₁, x₂, x₃, ..., x₁^1 * x₂^(c-1)]. Mixed terms are included.

sin_basis(x, coefficients)

Constructs an array containing a Sine basis in the variables x with coefficients c. If c is an Int returns all coefficients from 1 to c.

cos_basis(x, coefficients)

Constructs an array containing a Cosine basis in the variables x with coefficients c. If c is an Int returns all coefficients from 1 to c.

fourier_basis(x, coefficients)

Constructs an array containing a Fourier basis in the variables x with (integer) coefficients c. If c is an Int returns all coefficients from 1 to c.

chebyshev_basis(x, coefficients)

Constructs an array containing a Chebyshev basis in the variables x with coefficients c. If c is an Int returns all coefficients from 1 to c.

mutable struct Koopman{O, M, G, T} <: DataDrivenDiffEq.AbstractKoopman

A special basis over the states with parameters , independent variable and possible exogenous controls. It extends an AbstractBasis, which also stores information about the lifted dynamics, specified by a sufficient matrix factorization, an output mapping and internal variables to update the equations. It can be called with the typical SciML signature, meaning out of place with f(u,p,t) or in place with f(du, u, p, t). If control inputs are present, it is assumed that no control corresponds to zero for all inputs. The corresponding function calls are f(u,p,t,inputs) and f(du,u,p,t,inputs) and need to be specified fully.

If linear_independent is set to true, a linear independent basis is created from all atom functions in f.

If simplify_eqs is set to true, simplify is called on f.

Additional keyworded arguments include name, which can be used to name the basis, and observed for defining observeables.

Fields

• eqs

  The equations of the basis

• states

  Dependent (state) variables

• ctrls

  Control variables

• ps

  Parameters

• observed

  Observed

• iv

  Independent variable

• f

  Internal function representation of the basis

• lift

  Associated lifting of the operator

• name

  Name of the basis

• systems

  Internal systems

• is_discrete

  Discrete or time continuous

• K

  The operator/generator of the dynamics

• C

  Mapping back onto the observed states

• Q

  Internal matrix Q used for updating

• P

  Internal matrix P used for updating

Note

The keyword argument eval_expression controls the function creation behavior. eval_expression=true means that eval is used, so normal world-age behavior applies (i.e. the functions cannot be called from the function that generates them). If eval_expression=false, then construction via GeneralizedGenerated.jl is utilized to allow for same world-age evaluation. However, this can cause Julia to segfault on sufficiently large basis functions. By default eval_expression=false.

is_discrete(k)

Returns true if the AbstractKoopmanOperator k is discrete in time.

is_discrete(_)

Check if the problem is time discrete.

is_continuous(k)

Returns true if the AbstractKoopmanOperator k is continuous in time.

Check if the problem is time continuous.

eigen(k)

Return the eigendecomposition of the AbstractKoopmanOperator.

eigvals(k)

Return the eigenvalues of the AbstractKoopmanOperator.

eigvecs(k)

Return the eigenvectors of the AbstractKoopmanOperator.

modes(k)

Return the eigenvectors of a continuous AbstractKoopmanOperator.

operator(k)

Return the approximation of the discrete Koopman operator stored in k.

generator(k)

Return the approximation of the continuous Koopman generator stored in k.

updatable(k)

Returns true if the AbstractKoopmanOperator is updatable.

is_stable(k)

Returns true if either:

• the Koopman operator has just eigenvalues with magnitude less than one or
• the Koopman generator has just eigenvalues with a negative real part

update!(k, X, Y; p, t, U, threshold)

Update the Koopman k given new data X and Y. The operator is updated in place if the L2 error of the prediction exceeds the threshold.

p and t are the parameters of the basis and the vector of timepoints, if necessary.