Composing Models and Building Reusable Components
The symbolic models of ModelingToolkit can be composed together to easily build large models. The composition is lazy and only instantiated at the time of conversion to numerical models, allowing a more performant way in terms of computation time and memory.
Simple Model Composition Example
The following is an example of building a model in a library with an optional forcing function, and allowing the user to specify the forcing later. Here, the library author defines a component named decay. The user then builds two decay components and connects them, saying the forcing term of decay1 is a constant while the forcing term of decay2 is the value of the state variable x.
using ModelingToolkit
function decay(;name)
@parameters t a
@variables x(t) f(t)
D = Differential(t)
ODESystem([
D(x) ~ -a*x + f
];
name=name)
end
@named decay1 = decay()
@named decay2 = decay()
@parameters t
D = Differential(t)
connected = compose(ODESystem([
decay2.f ~ decay1.x
D(decay1.f) ~ 0
], t; name=:connected), decay1, decay2)
equations(connected)
#4-element Vector{Equation}:
# Differential(t)(decay1₊f(t)) ~ 0
# decay2₊f(t) ~ decay1₊x(t)
# Differential(t)(decay1₊x(t)) ~ decay1₊f(t) - (decay1₊a*(decay1₊x(t)))
# Differential(t)(decay2₊x(t)) ~ decay2₊f(t) - (decay2₊a*(decay2₊x(t)))
simplified_sys = structural_simplify(connected)
equations(simplified_sys)\[ \begin{align} \frac{\mathrm{d} decay1_{+}f\left( t \right)}{\mathrm{d}t} =& -0.0 \\ \frac{\mathrm{d} decay1_{+}x\left( t \right)}{\mathrm{d}t} =& - decay1_{+}a decay1_{+}x\left( t \right) + decay1_{+}f\left( t \right) \\ \frac{\mathrm{d} decay2_{+}x\left( t \right)}{\mathrm{d}t} =& - decay2_{+}a decay2_{+}x\left( t \right) + decay1_{+}x\left( t \right) \end{align} \]
Now we can solve the system:
x0 = [
decay1.x => 1.0
decay1.f => 0.0
decay2.x => 1.0
]
p = [
decay1.a => 0.1
decay2.a => 0.2
]
using DifferentialEquations
prob = ODEProblem(simplified_sys, x0, (0.0, 100.0), p)
sol = solve(prob, Tsit5())
sol[decay2.f]23-element Vector{Float64}:
1.0
0.988412780354162
0.9276920413597063
0.83083442676386
0.7238511010482006
0.6064716273042755
0.4906266726783708
0.38123692999606124
0.2840663757976879
0.202505825379558
⋮
0.016682568907159138
0.008438451900843266
0.004154119364674387
0.001992945474243811
0.0009349151861883788
0.0004281899181999306
0.00019123722531327804
8.29837215148332e-5
4.5427656159678035e-5Basics of Model Composition
Every AbstractSystem has a system keyword argument for specifying subsystems. A model is the composition of itself and its subsystems. For example, if we have:
@named sys = compose(ODESystem(eqs,indepvar,states,ps),subsys)the equations of sys is the concatenation of get_eqs(sys) and equations(subsys), the states are the concatenation of their states, etc. When the ODEProblem or ODEFunction is generated from this system, it will build and compile the functions associated with this composition.
The new equations within the higher level system can access the variables in the lower level system by namespacing via the nameof(subsys). For example, let's say there is a variable x in states and a variable x in subsys. We can declare that these two variables are the same by specifying their equality: x ~ subsys.x in the eqs for sys. This algebraic relationship can then be simplified by transformations like structural_simplify which will be described later.
Numerics with Composed Models
These composed models can then be directly transformed into their associated SciMLProblem type using the standard constructors. When this is done, the initial conditions and parameters must be specified in their namespaced form. For example:
u0 = [
x => 2.0
subsys.x => 2.0
]Note that any default values within the given subcomponent will be used if no override is provided at construction time. If any values for initial conditions or parameters are unspecified an error will be thrown.
When the model is numerically solved, the solution can be accessed via its symbolic values. For example, if sol is the ODESolution, one can use sol[x] and sol[subsys.x] to access the respective timeseries in the solution. All other indexing rules stay the same, so sol[x,1:5] accesses the first through fifth values of x. Note that this can be done even if the variable x is eliminated from the system from transformations like alias_elimination or tearing: the variable will be lazily reconstructed on demand.
Variable scope and parameter expressions
In some scenarios, it could be useful for model parameters to be expressed in terms of other parameters, or shared between common subsystems. To facilitate this, ModelingToolkit supports symbolic expressions in default values, and scoped variables.
With symbolic parameters, it is possible to set the default value of a parameter or initial condition to an expression of other variables.
# ...
sys = ODESystem(
# ...
# directly in the defaults argument
defaults=Pair{Num, Any}[
x => u,
y => σ,
z => u-0.1,
])
# by assigning to the parameter
sys.y = u*1.1In a hierarchical system, variables of the subsystem get namespaced by the name of the system they are in. This prevents naming clashes, but also enforces that every state and parameter is local to the subsystem it is used in. In some cases it might be desirable to have variables and parameters that are shared between subsystems, or even global. This can be accomplished as follows.
@parameters t a b c d e f
p = [
a #a is a local variable
ParentScope(b) # b is a variable that belongs to one level up in the hierarchy
ParentScope(ParentScope(c))# ParentScope can be nested
DelayParentScope(d) # skips one level before applying ParentScope
DelayParentScope(e,2) # second argument allows skipping N levels
GlobalScope(f) # global variables will never be namespaced
]
level0 = ODESystem(Equation[],t,[],p; name = :level0)
level1 = ODESystem(Equation[],t,[],[];name = :level1) ∘ level0
parameters(level1)
#level0₊a
#b
#c
#level0₊d
#level0₊e
#f
level2 = ODESystem(Equation[],t,[],[];name = :level2) ∘ level1
parameters(level2)
#level1₊level0₊a
#level1₊b
#c
#level0₊d
#level1₊level0₊e
#f
level3 = ODESystem(Equation[],t,[],[];name = :level3) ∘ level2
parameters(level3)
#level2₊level1₊level0₊a
#level2₊level1₊b
#level2₊c
#level2₊level0₊d
#level1₊level0₊e
#fStructural Simplify
In many cases, the nicest way to build a model may leave a lot of unnecessary variables. Thus one may want to remove these equations before numerically solving. The structural_simplify function removes these trivial equality relationships and trivial singularity equations, i.e. equations which result in 0~0 expressions, in over-specified systems.
Inheritance and Combine
Model inheritance can be done in two ways: implicitly or explicitly. First, one can use the extend function to extend a base model with another set of equations, states, and parameters. An example can be found in the acausal components tutorial.
The explicit way is to shadow variables with equality expressions. For example, let's assume we have three separate systems which we want to compose to a single one. This is how one could explicitly forward all states and parameters to the higher level system:
using ModelingToolkit, OrdinaryDiffEq, Plots
## Library code
@parameters t
D = Differential(t)
@variables S(t), I(t), R(t)
N = S + I + R
@parameters β,γ
@named seqn = ODESystem([D(S) ~ -β*S*I/N])
@named ieqn = ODESystem([D(I) ~ β*S*I/N-γ*I])
@named reqn = ODESystem([D(R) ~ γ*I])
sir = compose(ODESystem([
S ~ ieqn.S,
I ~ seqn.I,
R ~ ieqn.R,
ieqn.S ~ seqn.S,
seqn.I ~ ieqn.I,
seqn.R ~ reqn.R,
ieqn.R ~ reqn.R,
reqn.I ~ ieqn.I], t, [S,I,R], [β,γ],
defaults = [
seqn.β => β
ieqn.β => β
ieqn.γ => γ
reqn.γ => γ
], name=:sir), seqn, ieqn, reqn)\[ \begin{align} S\left( t \right) =& ieqn_{+}S\left( t \right) \\ I\left( t \right) =& seqn_{+}I\left( t \right) \\ R\left( t \right) =& ieqn_{+}R\left( t \right) \\ ieqn_{+}S\left( t \right) =& seqn_{+}S\left( t \right) \\ seqn_{+}I\left( t \right) =& ieqn_{+}I\left( t \right) \\ seqn_{+}R\left( t \right) =& reqn_{+}R\left( t \right) \\ ieqn_{+}R\left( t \right) =& reqn_{+}R\left( t \right) \\ reqn_{+}I\left( t \right) =& ieqn_{+}I\left( t \right) \\ \frac{\mathrm{d} seqn_{+}S\left( t \right)}{\mathrm{d}t} =& \frac{ - seqn_+\beta seqn_{+}I\left( t \right) seqn_{+}S\left( t \right)}{seqn_{+}I\left( t \right) + seqn_{+}R\left( t \right) + seqn_{+}S\left( t \right)} \\ \frac{\mathrm{d} ieqn_{+}I\left( t \right)}{\mathrm{d}t} =& \frac{ieqn_+\beta ieqn_{+}I\left( t \right) ieqn_{+}S\left( t \right)}{ieqn_{+}I\left( t \right) + ieqn_{+}R\left( t \right) + ieqn_{+}S\left( t \right)} - ieqn_+\gamma ieqn_{+}I\left( t \right) \\ \frac{\mathrm{d} reqn_{+}R\left( t \right)}{\mathrm{d}t} =& reqn_+\gamma reqn_{+}I\left( t \right) \end{align} \]
Note that the states are forwarded by an equality relationship, while the parameters are forwarded through a relationship in their default values. The user of this model can then solve this model simply by specifying the values at the highest level:
sireqn_simple = structural_simplify(sir)
equations(sireqn_simple)\[ \begin{align} \frac{\mathrm{d} seqn_{+}S\left( t \right)}{\mathrm{d}t} =& \frac{ - seqn_+\beta ieqn_{+}I\left( t \right) seqn_{+}S\left( t \right)}{ieqn_{+}I\left( t \right) + reqn_{+}R\left( t \right) + seqn_{+}S\left( t \right)} \\ \frac{\mathrm{d} ieqn_{+}I\left( t \right)}{\mathrm{d}t} =& \frac{ieqn_+\beta ieqn_{+}I\left( t \right) seqn_{+}S\left( t \right)}{ieqn_{+}I\left( t \right) + reqn_{+}R\left( t \right) + seqn_{+}S\left( t \right)} - ieqn_+\gamma ieqn_{+}I\left( t \right) \\ \frac{\mathrm{d} reqn_{+}R\left( t \right)}{\mathrm{d}t} =& reqn_+\gamma ieqn_{+}I\left( t \right) \end{align} \]
## User Code
u0 = [seqn.S => 990.0,
ieqn.I => 10.0,
reqn.R => 0.0]
p = [
β => 0.5
γ => 0.25
]
tspan = (0.0,40.0)
prob = ODEProblem(sireqn_simple,u0,tspan,p,jac=true)
sol = solve(prob,Tsit5())
sol[reqn.R]18-element Vector{Float64}:
0.0
0.0014142401338982807
0.015567426114815186
0.1581830690301042
1.0627160216919251
3.3906763834079165
7.785038081421032
15.47626943415923
30.102184533124007
58.80475275825804
116.64582495682085
221.07866633033498
368.86570251966214
519.3989582784651
655.4762283913898
723.9496381572195
772.9458047004413
775.6835919121962Tearing Problem Construction
Some system types, specifically ODESystem and NonlinearSystem, can be further reduced if structural_simplify has already been applied to them. This is done by using the alternative problem constructors, ODAEProblem and BlockNonlinearProblem respectively. In these cases, the constructor uses the knowledge of the strongly connected components calculated during the process of simplification as the basis for building pre-simplified nonlinear systems in the implicit solving. In summary: these problems are structurally modified, but could be more efficient and more stable.
Components with discontinuous dynamics
When modeling, e.g., impacts, saturations or Coulomb friction, the dynamic equations are discontinuous in either the state or one of its derivatives. This causes the solver to take very small steps around the discontinuity, and sometimes leads to early stopping due to dt <= dt_min. The correct way to handle such dynamics is to tell the solver about the discontinuity by means of a root-finding equation, which can be modeling using ODESystem's event support. Please see the tutorial on Callbacks and Events for details and examples.