Basis

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.

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

• 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.