Basis

Many of the methods require the definition of a Basis on observables or functional forms. A Basis is generated via:

Basis(eqs::AbstractVector, states::AbstractVector; 
    parameters::AbstractArray = [], iv = nothing,
    simplify = false, linear_independent = false, name = gensym(:Basis), 
    pins = [], observed = [], eval_expression = false,
    kwargs...)

where eqs is either a vector containing symbolic functions using 'ModelingToolkit.jl' or a general function with the typical DiffEq signature h(u,p,t), which can be used with an Num or vector of Num. states are the dependent variables used to describe the Basis, and parameters are the optional parameters in the Basis. iv represents the independent variable of the system - in most cases the time. Additional arguments are simplify, which simplifies eqs before creating a Basis. linear_dependent breaks up eqs in linear independent elements which are unique. name is an optional name for the Basis, pins and observed can be using in accordance to ModelingToolkits documentation. eval_expression is used to generate a callable function from the eqs. If set to false, callable code will be returned. true will use eval on code returned from the function, which might cause worldage issues.

mutable struct Basis <: ModelingToolkit.AbstractSystem

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)

Example

We start by crearting some variables and parameters using ModelingToolkit.

using LinearAlgebra
using DataDrivenDiffEq
using Plots
using ModelingToolkit

@variables u[1:3]
@parameters w[1:2]

(ModelingToolkit.Num[w₁, w₂],)

To define a basis, simply write down the equations you want to be included as a Vector. Possible used parameters have to be given to the constructor.

h = [u[1]; u[2]; cos(w[1]*u[2]+w[2]*u[3])]
b = Basis(h, u, parameters = w)

\begin{align} \varphi{1} =& u{1} \ \varphi{2} =& u{2} \ \varphi{3} =& \cos\left( u{2} w{1} + u{3} w{_2} \right) \end{align}

Basis are callable with the signature of functions to be used in DifferentialEquations. So, the function value at a single point looks like:

x = b([1;2;3])

3-element Array{Any,1}:
 1
 2
  cos(2w₁ + 3w₂)

Or, in place

dx = similar(x)
b(dx, [1;2;3])

Notice that since we did not use any numerical values for the parameters, the basis uses the symbolic values in the result.

To use numerical values, simply pass this on in the function call. Here, we evaluate over a trajectory with two parameters and 40 timestamps.

X = randn(3, 40)
Y = b(X, [2;4], 0:39)

Suppose we want to add another equation, say sin(u[1]). A Basis behaves like an array, so we can simply

push!(b, sin(u[1]))
size(b)

(4,)

To ensure that a basis is well-behaved, functions already present are not included again.

push!(b, sin(u[1]))
size(b)

(4,)

We can also define functions of the independent variable and add them

t = independent_variable(b)
push!(b, cos(t*π))
println(b)

##Basis#253 : 5 dimensional basis in ["u₁", "u₂", "u₃"]
Parameters : SymbolicUtils.Sym{ModelingToolkit.Parameter{Real}}[w₁, w₂]

Independent variable: t
Equations
φ₁ = u₁
φ₂ = u₂
φ₃ = cos(u₂*w₁ + u₃*w₂)
φ₅ = sin(u₁)
φ₅ = cos(πt)

Additionally, we can iterate over a Basis using [eq for eq in basis] or index specific equations, like basis[2].

We can also chain Basis via just using it in the constructor

@variables x[1:2]
y = [sin(x[1]); cos(x[1]); x[2]]
t = independent_variable(b)
b2 = Basis(b(y, parameters(b), t), x, parameters = w, iv = t)
println(b2)

##Basis#282 : 5 dimensional basis in ["x₁", "x₂"]
Parameters : SymbolicUtils.Sym{ModelingToolkit.Parameter{Real}}[w₁, w₂]

Independent variable: t
Equations
φ₁ = sin(x₁)
φ₂ = cos(x₁)
φ₃ = cos(w₁*cos(x₁) + w₂*x₂)
φ₄ = sin(sin(x₁))
φ₅ = cos(3.141592653589793getindex(t, 1))

You can also use merge to create the union of two Basis:

b3 = merge(b, b2)
println(b3)

##Basis#290 : 10 dimensional basis in ["u₁", "u₂", "u₃", "x₁", "x₂"]
Parameters : SymbolicUtils.Sym{ModelingToolkit.Parameter{Real}}[w₁, w₂]

Independent variable: t
Equations
φ₁ = u₁
φ₂ = u₂
φ₃ = cos(u₂*w₁ + u₃*w₂)
φ₄ = sin(u₁)
...
φ₁₀ = cos(3.141592653589793getindex(t, 1))

which combines all the used variables and parameters ( and assumes the same independent_variable ):

variables(b)

3-element Array{SymbolicUtils.Sym{Real},1}:
 u₁
 u₂
 u₃

parameters(b)

2-element Array{SymbolicUtils.Sym{ModelingToolkit.Parameter{Real}},1}:
 w₁
 w₂

If you have a function already defined as pure code, you can use this also to create a Basis. Only the signature has to be consistent, so use f(u,p,t).

f(u, p, t) = [u[1]; u[2]; cos(p[1]*u[2]+p[2]*u[3])]
b_f = Basis(f, u, parameters = w)
println(b_f)

##Basis#298 : 3 dimensional basis in ["u₁", "u₂", "u₃"]
Parameters : SymbolicUtils.Sym{ModelingToolkit.Parameter{Real}}[w₁, w₂]

Independent variable: t
Equations
φ₁ = u₁
φ₂ = u₂
φ₃ = cos(u₂*w₁ + u₃*w₂)

This works for every function defined over Nums. So to create a Basis from a Flux model, simply extend the activations used:

using Flux
NNlib.σ(x::Num) = 1 / (1+exp(-x))

c = Chain(Dense(3,2,σ), Dense(2, 1, σ))
ps, re = Flux.destructure(c)

@parameters p[1:length(ps)]

g(u, p, t) = re(p)(u)
b = Basis(g, u, parameters = p)

Functions

jacobian(basis)

Returns a function representing the jacobian matrix / gradient of the `Basis` with respect to the
dependent variables as a function with the common signature `f(u,p,t)` for out of place and `f(du, u, p, t)` for in place computation.

dynamics(basis)

Returns the internal function representing the dynamics of the `Basis`.

push!(basis, eq, simplify_eqs = true; eval_expression = false)

Push the equations(s) in `eq` into the basis and update all internal fields accordingly.
`eq` can either be a single equation or an array. If `simplify_eq` is true, the equation will be simplified.

deleteat!(basis, inds, eval_expression = false)

Delete the entries specified by `inds` and update the `Basis` accordingly.

merge(x::Basis, y::Basis; eval_expression = false)

Return a new `Basis`, which is defined via the union of `x` and `y` .

merge!(x::Basis, y::Basis; eval_expression = false)

Updates `x` to include the union of both `x` and `y`.