Functions with parameters for differential equations



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ParameterizedFunctions.jl is a component of the JuliaDiffEq ecosystem which allows for parameters to be explicitly present within functions. The interface which ParameterizedFunctions describes allows for functionality which requires parameters, such as parameter sensitivity analysis and parameter estimation, to be added to the differential equation solvers of DifferentialEquations.jl. While the interface itself is of importance to ecosystem developers, ParameterizedFunctions.jl provides user-facing macros which make a ParameterizedFunction easy to define, and automatically include optimizations like explicit Jacobian functions and explicit inverse Jacobian functions for the differential equation solvers to take advantage of. The result is an easy to use API which allows for more functionality and more performance optimizations than could traditionally be offered.


The Basic Idea

ParameterizedFunction is a type which can be used in various JuliaDiffEq solvers where the parameters must be accessible by the solver function. These use call overloading generate a type which acts like a function f(t,u,du) but has access to many more features. For example, a ParameterizedFunction can contain a function for the Jacobian or Inverse Jacobian. If such functions exist, the solvers can use them to increase the speed of computations. If they don't exist, the solvers will ignore them. Since ParameterizedFunction is a subtype of Function, these can be used anywhere that a function can be used, just with the extra functionality ignored.

Basic Usage

ParameterizedFunction Constructor

The easiest way to make a ParameterizedFunction is to use the constructor:

pf = ParameterizedFunction(f,params)

The form for f is f(t,u,params,du) where params is any type which defines the parameters. The resulting ParameterizedFunction has the function call pf(t,u,params,du) which matches the original function, and a call pf(t,u,du) which uses internal parmaeters which can be used with a differential equation solver. Note that the internal parameters can be modified at any time via the field: pf.p = ....

An additional version exists for f(t,u,params) which will then act as the not inplace version f(t,u) in the differential equation solvers.


pf_func = function (t,u,p,du)
  du[1] = p[1] * u[1] - p[2] * u[1]*u[2]
  du[2] = -3 * u[2] + u[1]*u[2]

pf = ParameterizedFunction(pf_func,[1.5,1.0])

And now pf can be used in the differential equation solvers and the ecosystem functionality which requires explicit parameters (parameter estimation, etc.).

Note that the not inplace version works the same:

pf_func2 = function (t,u,p)
  [p[1] * u[1] - p[2] * u[1]*u[2];-3 * u[2] + u[1]*u[2]]

pf2 = ParameterizedFunction(pf_func2,[1.5,1.0])

ODE Macros

A helper macro is provided to make it easier to define a ParameterizedFunction, and it will symbolically compute a bunch of extra functions to make the differential equation solvers run faster. For example, to define the previous LotkaVolterra, you can use the following command:

f = @ode_def LotkaVolterra begin
  dx = a*x - b*x*y
  dy = -c*y + d*x*y
end a=>1.5 b=>1 c=3 d=1

Note that the syntax for parameters here is that => will put these inside the parameter type, while = will inline the number (i.e. replace each instance of c with 3). Inlining slightly decreases the function cost and so is preferred in any case where you know that the parameter will always be constant. This will silently create the LotkaVolterra type and thus g=LotkaVolterra(a=1.0,b=2.0) will create a different function where a=1.0 and b=2.0. However, at any time the parameters of f can be changed by using f.a = or f.b = .

The macro also defines the Jacobian f'. This is defined as an in-place Jacobian f(Val{:jac},t,u,J). This is calculated using SymEngine.jl automatically, so it's no effort on your part. The symbolic inverse of the Jacobian is also computed, and an in-place function for this is available as well as f(Val{:invjac},t,u,iJ). If the Jacobians cannot be computed, a warning is thrown and only the function itself is usable. The functions jac_exists(f) and invjac_exists(f) can be used to see whether the Jacobian and the function for its inverse exist.

Extra Options

In most cases the @ode_def macro should be sufficient. This is because by default the macro will simply calculate each function symbolically, and if it can't it will simply throw a warning and move on. However, in extreme cases the symbolic calculations may take a long time, in which case it is necessary to turn them off. To do this, use the ode_def_opts function. The @ode_def macro simply defines the specifiable options:

opts = Dict{Symbol,Bool}(
      :build_tgrad => true,
      :build_jac => true,
      :build_expjac => false,
      :build_invjac => true,
      :build_invW => true,
      :build_invW_t => true,
      :build_hes => false,
      :build_invhes => false,
      :build_dpfuncs => true)

and calls the function ode_def_opts(name::Symbol,opts,ex::Expr,params). Note that params is an iterator holding expressions for the parameters.

Extra Little Tricks

There are some extra little tricks you can do. Since @ode_def is a macro, you cannot directly make the parameters something that requires a runtime value. Thus the following will error:

vec = rand(1,4)
f = @ode_def LotkaVolterraExample begin
dx = ax - bxy
dy = -cy + dxy
end a=>vec[1] b=>vec[2] c=>vec[3] d=vec[4]

To do the same thing, instead initialize it with values of the same type, and simply replace them:

vec = rand(1,4)
f = @ode_def LotkaVolterraExample begin
dx = ax - bxy
dy = -cy + dxy
end a=>1.0 b=>1.0 c=>1.0 d=vec[4]
f.a,f.b,f.c = vec[1:3]

Notice that when using =, it can inline expressions. It can even inline expressions of time, like d=3*t or d=2π. However, do not use something like d=3*x as that will fail to transform the x.

In addition, one can also use their own function inside of the macro. For example:

f(x,y,d) = erf(x*y/d)
NJ = @ode_def FuncTest begin
  dx = a*x - b*x*y
  dy = -c*y + f(x,y,d)
end a=>1.5 b=>1 c=3 d=4

will do fine. The symbolic derivatives will not work unless you define a derivative for f.

Extra Macros

Instead of using ode_def_opts directly, one can use one of the following macros to be more specific about what to not calculate. In increasing order of calculations:


Finite Element PDEs

Similar macros for finite element method definitions also exist. For the finite element solvers, the definitions use x[:,1] instead of x and x[:,2] instead of y. To more easily define systems of equations for finite element solvers, we can use the @fem_def macro. The first argument is the function signature. This is required in order to tell the solver linearity. Other than that, the macro usage is similar to before. For example,

l = @fem_def (t,x,u) BirthDeath begin
  du = 1-x*α*u
  dv = 1-y*v
end α=0.5

defines a system of equations

l = (t,x,u)  -> [1-.5*x[:,1]*u[:,1]   1-x[:,2]*u[:,2]]

which is in the form for the FEM solver.

The ParameterizedFunction Interface

The ParameterizedFunction interface is as follows:

  • ParameterizedFunction is a type which is a subtype of Function
  • The type must hold the parameters.
  • Hessians, Inverse Jacobians, Inverse Hessians, explicit parameter functions, parameter derivatives, and parameter Jacobians.
  • The standard call (p::TypeName)(t,u,du) must be overloaded for the function calculation. All other functions are optional.

Solvers can interface with ParameterizedFunctions as follows:

f.a # accesses the parameter a
f(t,u,du) # Call the function
f(t,u,params,du) # Call the function to calculate with parameters params (vector)
f(Val{:tgrad},t,u,J) # Call the explicit t-gradient function
f(Val{:a},t,u,2.0,du) # Call the explicit parameter function with a=2.0
f(Val{:deriv},Val{:a},t,u,2.0,df) # Call the explicit parameter derivative function with a=2.0
f(Val{:paramjac},t,u,params,J) # Call the explicit parameter Jacobian function
f(Val{:jac},t,u,J) # Call the explicit Jacobian function
f(Val{:expjac},t,u,γ,J) # Call the explicit exponential Jacobian function exp(γJ)
f(Val{:invjac},t,u,iJ) # Call the explicit Inverse Jacobian function
f(Val{:invW},t,u,γ,iW) # Call the explicit inverse Rosenbrock-W function (M - γJ)^(-1)
f(Val{:invW_t},t,u,γ,iW) # Call the explicit transformed inverse Rosenbrock-W function (M/γ - J)^(-1)
f(Val{:hes},t,u,H) # Call the explicit Hessian function
f(Val{:invhes},t,u,iH) # Call the explicit Inverse Hessian function

To test for whether certain overloads exist, the following functions are provided by traits in DiffEqBase.jl:


These are compile-time checks and thus the inappropriate branches will compile way when a function (usually an ODE/SDE solver) is dispatched on f. It is requested that solvers should only use the explicit functions when they exist to help with performance.

In addition, the following functions are provided:

  • param_values(f) : Returns an array of the values for each of the parameters
  • num_params(f) : Returns the number of parameters for f

Internals: How it Works

This shows how to manually build a ParameterizedFunction to give to a solver.


An example of explicitly defining a parameterized function is as follows. This serves as a general template for doing so:

type  LotkaVolterra <: AbstractParameterizedFunction{true}
f = LotkaVolterra(0.0,0.0)
(p::LotkaVolterra)(t,u,du) = begin
         du[1] = p.a * u[1] - p.b * u[1]*u[2]
         du[2] = -3 * u[2] + u[1]*u[2]


Let's go step by step to see what this template does. The first part defines a type:

type  LotkaVolterra <: AbstractParameterizedFunction{true}

The fields are the parameters for our function. The abstract type is parameterized by whether the function is written in-place or not. Then we built the type:

f = LotkaVolterra(0.0,0.0)

We put in values for the parameters and told it that we will be defining each of those functions. First we define the main overload. This is required even if none of the other functions are provided. The function for the main overload is the differential equation, so for the Lotka-Volterra equation:

(p::LotkaVolterra)(t,u,du) = begin
         du[1] = p.a * u[1] - p.b * u[1]*u[2]
         du[2] = -3 * u[2] + u[1]*u[2]

Note how we represented the parameters in the equation. If you did this and set the booleans to false, the result is f is a ParameterizedFunction, but f(t,u,du) acts like the function:

function f(t,u,du)
         du[1] = 0.0 * u[1] - 0.0 * u[1]*u[2]
         du[2] = -3 * u[2] + u[1]*u[2]

At anytime the function parameters can be accessed by the fields (f.a, f.b).

Extra Functions

Jacobian Function

The Jacobian overload is provided by overloading in the following manner:

function (p::LotkaVolterra)(::Type{Val{:jac}},t,u,J)
  J[1,1] = p.a - p.b * u[2]
  J[1,2] = -(p.b) * u[1]
  J[2,1] = 1 * u[2]
  J[2,2] = -3 + u[1]

Inverse Jacobian

The Inverse Jacobian overload is provided by overloading in the following manner:

function (p::LotkaVolterra)(::Type{Val{:invjac}},t,u,J)
  J[1,1] = (1 - (p.b * u[1] * u[2]) / ((p.a - p.b * u[2]) * (-3 + u[1] + (p.b * u[1] * u[2]) / (p.a - p.b * u[2])))) / (p.a - p.b * u[2])
  J[1,2] = (p.b * u[1]) / ((p.a - p.b * u[2]) * (-3 + u[1] + (p.b * u[1] * u[2]) / (p.a - p.b * u[2])))
  J[2,1] = -(u[2]) / ((p.a - p.b * u[2]) * (-3 + u[1] + (p.b * u[1] * u[2]) / (p.a - p.b * u[2])))
  J[2,2] = (-3 + u[1] + (p.b * u[1] * u[2]) / (p.a - p.b * u[2])) ^ -1

Hessian and Inverse Hessian

These are the same as the Jacobians, except with value types :hes and :invhes.

Explicit Parameter Functions

For solvers which need to auto-differentiate parameters (local sensitivity analysis), explicit parameter functions are required. For our example, we do the following:

function (p::LotkaVolterra)(::Type{Val{:a}},t,u,a,du)
  du[1] = a * u[1] - p.b * u[1] * u[2]
  du[2] = -3 * u[2] + 1 * u[1] * u[2]
function (p::LotkaVolterra)(::Type{Val{:b}},t,u,b,du)
  du[1] = p.a * u[1] - b * u[1] * u[2]
  du[2] = -3 * u[2] + 1 * u[1] * u[2]

Explicit Parameter Derivatives

For solvers which need parameters derivatives, specifying the functions can increase performance. For our example, we allow the solvers to use the explicit derivatives in the parameters a and b by:

function (p::LotkaVolterra)(::Type{Val{:deriv}},::Type{Val{:a}},t,u,a,du)
  du[1] = 1 * u[1]
  du[2] = 1 * 0
function (p::LotkaVolterra)(::Type{Val{:deriv}},::Type{Val{:b}},t,u,b,du)
  du[1] = -(u[1]) * u[2]
  du[2] = 1 * 0

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