Dynamical Systems Co-Integration Framework

About | Context | A Single Sketch | Documentation


A lightweight framework to enable hierarchical, heterogeneous dynamical systems co-integration. Batteries included!


Entangle dynamical systems

  • Simulate joint evolution of dynamical systems from across the Julia ecosystem
  • Each of the systems can evolve at its own pace - the framework will keep them in sync
  • The systems may access each other's (interpolated) observables

Organize systems into hierarchies

  • A dynamical system may contain an embedded hierarchy of agents
  • The systems are indexed as folders in a file system - even wildcards are supported
  • You may take sums of dynamical systems - and override default ⊕ to embrace custom system classes

The package solves a typical scaling problem: while modeling a complex system, one tears the joint dynamics into several parts, but inevitably runs into troubles when trying to put these parts together.

We provide a minimalistic interface featuring compositions of dynamical systems, to approximate the joint dynamics from separate systems' evolutionary laws. In practice, one only has to define a wrap type for a custom dynamical system class and implement a handful of common interface methods. So far, integrations of DifferentialEquations.jl, Agents.jl, and AlgebraicDynamics.jl are provided, see the docs.

The dynamical systems are organized into hierarchies. That is, a dynamical system may contain an embedded hierarchy of dynamical systems. Each system is then assigned a path in the overarching hierarchy, like a folder in a file system. A dynamical system typically indexes and retrieves other systems in the hierarchy via relative paths.

The underlying concept of a sum (a join) of dynamical systems is manifested at two distinct levels: first, the dynamical systems are implicitly coupled as they generally observe (and interact with) each other's state. Second, a structural sum operator is provided; this outputs another dynamical system which usually contains the summands within a hierarchy of embedded systems. Importantly, using Julia's dynamic dispatch, it is possible to fully customize this behavior for user-defined dynamical system classes.

Context: Dynamics of Value Evolution (DyVE)

The package is an integral part of the Dynamics of Value Evolution (DyVE) computational framework for learning, designing, integrating, simulating, and optimizing R&D process models, to better inform strategic decisions in science and business.

As the framework evolves, multiple functionalities have matured enough to become standalone packages.

This includes ReactiveDynamics.jl, a package which implements a category of reaction (transportation) network-type dynamical systems. The central concept of the package is of a stateful, parametric transition; simultaneous action of the transitions then evolves the dynamical system. Moreover, a network's dynamics can be specified using a compact modeling metalanguage.

CEEDesigns.jl is a decision-making framework for the cost-efficient design of experiments, with direct applications in drug research, healthcare, and various other fields. Typically, a design consists of multiple experiments. Each experiment is regarded as an option to acquire additional experimental evidence and is associated with a monetary cost and an execution time. The framework, then, aims to select experiments that balance the value of acquired information and the incurred costs.

Another package is GeneratedExpressions.jl, which implements a metalanguage to support code-less expression comprehensions. In the context of ReactiveDynamics.jl, expression comprehensions are used to generate complex dynamics from user-specified template transitions.

A Single Sketch

Integrating a Custom Dynamical System

# drug entity, lives in a therapeutic area
@aagent FreeAgent Molecule struct SmallMolecule

    profile::NTuple{N, Float64}


Note the use of a conveniency macro @aagent which appends additional fields expected (not required, though) by default interface methods.

Next we provide an evolutionary law for SmallMolecule type. This is done by extending the interface function AlgebraicAgents._step!(agent).

Here, t is the maximal time to which all the systems in a hierarchy have been projected. Once t reaches the time point to which mol has been projected, then it's time for another step:

# implement evolution
function AlgebraicAgents._step!(mol::SmallMolecule)
   t = projected_to(mol) # get current time; this equals the time point up to which the mol agent has been projected (enforced via `AlgebraicAgents.step!`)

   # log sales volume at time t
   push!(mol.df_sales, (t, mol.sales))
   # increment mol's age - by default, mols will evolve by unit step
   mol.age += 1
   # apply sales decay in time
   mol.sales *= sales_decay_small

   # remove mol 1) once sales volume drops below a given level
   # 2) also account for some random effect - prob of removal increases in time
   if (mol.sales <= 10) || (rand() >= exp(-0.2*mol.age))
      mol.kill_time = t
      push!(getagent(mol, "../dx").removed_mols, (mol.mol, t))
      # remove mol from the system

We provide additional methods required by the common interface:

# to reinit the system's state - since we have cold starts here, let's simply remove the mol
AlgebraicAgents._reinit!(mol::Molecule) = disentangle!(mol)
# return time to which the system has been projected
AlgebraicAgents._projected_to(mol::Molecule) = mol.age + mol.birth_time

Adding SDE Models

Let's define toy market demand model and represent this as a stochastic differential equation defined in DifferentialEquations.jl

# add SDE models for drug demand in respective areas
using DifferentialEquations

dt = 1//2^(4); tspan = (0.0,100.)
f(u,p,t) = p[]*u; g(u,p,t) = p[]*u

prob_1 = SDEProblem(f,g,.9,tspan,Dict{Symbol, Any}(=>.01, =>.01))
prob_2 = SDEProblem(f,g,1.2,tspan,Dict{Symbol, Any}(=>.005, =>.01))
SDEProblem with uType Float64 and tType Float64. In-place: false
timespan: (0.0, 100.0)
u0: 1.2

Internally, a discovery unit will adjust its productivity according to the observed market demand:

# sync with market demand
dx.productivity, = getobservable(getagent(dx,  "../demand"), "demand")

Defining & Entangling the Systems

Next step is to initiate the actual dynamical systems.

# define therapeutic areas
therapeutic_area1 = FreeAgent("therapeutic_area1")
therapeutic_area2 = FreeAgent("therapeutic_area2")

# join therapeutic models into a pharma model
pharma_model = (therapeutic_area1, therapeutic_area2; name="pharma_model")

# initialize and push discovery units to therapeutic areas
# discovery units evolve at different pace
entangle!(therapeutic_area1, Discovery("dx", 5.2, 10.; dt=3.))
entangle!(therapeutic_area2, Discovery("dx", 6., 8.; dt=5.))

# add SDE models for drug demand in respective areas
demand_model_1 = DiffEqAgent("demand", prob_1, EM(); observables=Dict("demand" => 1), dt)
demand_model_2 = DiffEqAgent("demand", prob_2, EM(); observables=Dict("demand" => 1), dt)

# push market demand units to therapeutic areas
entangle!(therapeutic_area1, demand_model_1)
entangle!(therapeutic_area2, demand_model_2)

# show the model
agent pharma_model with uuid 05e79978 of type FreeAgent 
   inner agents: 
    agent therapeutic_area1 with uuid 8b7ac56f of type FreeAgent 
       inner agents: demand, dx
    agent therapeutic_area2 with uuid 6c749c0f of type FreeAgent 
       inner agents: demand, dx
getagent(pharma_model, glob"therapeutic_area?/")
2-element Vector{FreeAgent}:
 FreeAgent{name=therapeutic_area2, uuid=6c749c0f, parent=FreeAgent{name=pharma_model, uuid=05e79978, parent=nothing}}
 FreeAgent{name=therapeutic_area1, uuid=8b7ac56f, parent=FreeAgent{name=pharma_model, uuid=05e79978, parent=nothing}}

Simulating the System

Let's next evolve the compound model over a hundred time units. The last argument is optional here; see ?simulate for the details.

# let the problem evolve
simulate(pharma_model, 100)
agent pharma_model with uuid 05e79978 of type FreeAgent 
   inner agents: 
    agent therapeutic_area1 with uuid 8b7ac56f of type FreeAgent 
       inner agents: yBh8n, NL9OQ, IfnJm, jB2XS, YKXlR, hEjuU, TPT6A, G3KGF, dx, 0CBpW, Vj3wm, LgNCo, jBF5x, ankr2, nN8iW, 9XjQX, y3YVk, PRejO, KR4Vl, 5HktC, 4dHax, arRnv, AhPe7, 5tnsD, qqkxY, kOjsf, q2zm5, Np78L, Jlzf7, DYVji, iafx4, b1WsV, ph5PO, NmmaJ, ZQrZK, kyEmh, Szewc, Y6i1A, EEFIu, g8Kuc, p8lgB, RBFqP, 3ndHM, oqWwt, HHKxT, ck6Kh, kQ9lL, qUr9S, GWYI8, PApQn, yitPS, iD5oS, 4EQqz, pCUle, rpXFX, 9295B, aLG3w, z2dTa, T2Cg7, nF00k, demand, g3vDH, KZ423, HcGru, yh9qU, m9u9r, HNZPr, 2p5VX, WKh1Y, DXarI, TFmHm, uGhAj, Medas, MgoKV, KGgP4, llnyd, lTTiZ, doSkK, ZfhWv, XRbpr, FKCI3, ys9UH, l5gVu, N4xmD, PjsRa, ouR7H, ym4Tb, KVYTB, 6YEHf, eyprJ, WBV4y, J92xS, bfsxh, Z57dn, smhsk, ZjoM4, tu9vN, lnUVB, PqUab, oiJOb, rZHlz
    agent therapeutic_area2 with uuid 6c749c0f of type FreeAgent 
       inner agents: dx, aIcFE, qDfar, demand
getagent(pharma_model, "therapeutic_area1/dx")
agent dx with uuid af488692 of type Main.Discovery 
   custom properties:
   rate_small: 5.2
   rate_large: 10.0
   productivity: 2.3297332635836594
   t: 102.0
   dt: 3.0
   t0: 0.0
   removed_mols: [("DGSLp", 99.0), ("Grjue", 99.0), ("s8c3x", 99.0), ("NLbHs", 99.0), ("DsTSW", 99.0), ("Qzokp", 99.0), ("8BFqZ", 99.0), ("YVKb0", 99.0), ("pRKig", 99.0), ("x3Sbn", 99.0)    ("Qq6Rs", 99.0), ("C4bqV", 99.0), ("Ic9YH", 99.0), ("D5emm", 99.0), ("VW0lS", 99.0), ("Dez7v", 99.0), ("z0VjL", 99.0), ("w9LBP", 99.0), ("0Aq1f", 99.0), ("i27ZG", 99.0)]
   df_output: 34×4 DataFrame
 Row  time     small  large  removed
      Float64  Int64  Int64  Int64
   1      0.0     16     32        0
   2      3.0     21     26       38
   3      6.0      8     22       46
   4      9.0     22     22       33
   5     12.0     16     34       45
   6     15.0     13     39       44
   7     18.0     14     27       56
   8     21.0     18     31       42
  28     81.0     14     58       98
  29     84.0     24     62       80
  30     87.0     39     70       74
  31     90.0     42     69      121
  32     93.0     38     74      107
  33     96.0     38     87       95
  34     99.0     42     74      133
                       19 rows omitted
   parent: therapeutic_area1
getagent(pharma_model, "therapeutic_area1/demand")
agent demand with uuid 18d7fbd7 of type DiffEqAgent 
   custom properties:
t: 100.0
u: 2.3580108744816726
   observables: demand (index: 1)
   parent: therapeutic_area1


It's possible to provide custom plotting recipes by specializing the interface method AlgebraicAgents._draw(agent). Whenever a dynamical system's state is logged into a single DataFrame - as is the case with Discovery type - you may utilize a convenience macro @draw_df. To that end, we need to load both DataFrames and Plots.

# implement plots
using DataFrames, Plots
AlgebraicAgents.@draw_df Discovery df_output

To see this in action, call

draw(getagent(pharma_model, "therapeutic_area1/dx"))