Parallel Ensemble Simulations Interface

Performing Monte Carlo simulations, solving with a predetermined set of initial conditions, and GPU-parallelizing a parameter search all fall under the ensemble simulation interface. This interface allows one to declare a template AbstractSciMLProblem to parallelize, tweak the template in trajectories for many trajectories, solve each in parallel batches, reduce the solutions down to specific answers, and compute summary statistics on the results.

Performing an Ensemble Simulation

Building a Problem

struct EnsembleProblem{T, T2, T3, T4, T5} <: SciMLBase.AbstractEnsembleProblem

- `prob`: The original base problem to replicate or modify.
- `prob_func`: A function that defines how to generate each subproblem.
- `output_func`: A function to post-process each individual simulation result.
- `reduction`: A function to combine results from all simulations.
- `u_init`: The initial container used to accumulate the results.
- `safetycopy`: Whether to copy the problem when creating subproblems (to avoid unintended modifications).

Solving the Problem

sim = solve(enprob, alg, ensemblealg = EnsembleThreads(), kwargs...)

EnsembleAlgorithms

The choice of ensemble algorithm allows for control over how the multiple trajectories are handled. Currently, the ensemble algorithm types are:

struct EnsembleSerial <: SciMLBase.BasicEnsembleAlgorithm

struct EnsembleThreads <: SciMLBase.BasicEnsembleAlgorithm

struct EnsembleDistributed <: SciMLBase.BasicEnsembleAlgorithm

struct EnsembleSplitThreads <: SciMLBase.BasicEnsembleAlgorithm

DiffEq Only (ODEProblem, SDEProblem)

• EnsembleGPUArray() - Requires installing and using DiffEqGPU. This uses a GPU for computing the ensemble with hyperparallelism. It will automatically recompile your Julia functions to the GPU. A standard GPU sees a 5x performance increase over a 16 core Xeon CPU. However, there are limitations on what functions can auto-compile in this fashion, please see DiffEqGPU for more details
• EnsembleGPUKernel() - Requires installing and using DiffEqGPU. This uses a GPU for computing the ensemble with hyperparallelism by building a custom GPU kernel. This can have drastically less overhead (for example, achieving 15x accelerating against Jax and PyTorch, see this paper for more details) but has limitations on what kinds of problems are compatible. See DiffEqGPU for more details

Choosing an Ensembler

For example, EnsembleThreads() is invoked by:

solve(ensembleprob, alg, EnsembleThreads(); trajectories = 1000)

Solution Type

The resulting type is a EnsembleSimulation, which includes the array of solutions.

Plot Recipe

There is a plot recipe for a AbstractEnsembleSimulation which composes all of the plot recipes for the component solutions. The keyword arguments are passed along. A useful argument to use is linealpha which will change the transparency of the plots. An additional argument is idxs which allows you to choose which components of the solution to plot. For example, if the differential equation is a vector of 9 values, idxs=1:2:9 will plot only the solutions of the odd components. Another additional argument is zcolors (an alias of marker_z) which allows you to pass a zcolor for each series. For details about zcolor see the Series documentation for Plots.jl.

Analyzing an Ensemble Experiment

Analysis tools are included for generating summary statistics and summary plots for a EnsembleSimulation.

To use this functionality, import the analysis module via:

using SciMLBase.EnsembleAnalysis

Time steps vs time points

For the summary statistics, there are two types. You can either summarize by time steps or by time points. Summarizing by time steps assumes that the time steps are all the same time point, i.e. the integrator used a fixed dt or the values were saved using saveat. Summarizing by time points requires interpolating the solution.

get_timestep(sim, i)

get_timepoint(sim, t)

componentwise_vectors_timestep(sim, i)

componentwise_vectors_timepoint(sim, t)

Summary Statistics Functions

Single Time Statistics

The available functions for time steps are:

SciMLBase.EnsembleAnalysis.timestep_mean
SciMLBase.EnsembleAnalysis.timestep_median
SciMLBase.EnsembleAnalysis.timestep_quantile
SciMLBase.EnsembleAnalysis.timestep_meanvar
SciMLBase.EnsembleAnalysis.timestep_meancov
SciMLBase.EnsembleAnalysis.timestep_meancor
SciMLBase.EnsembleAnalysis.timestep_weighted_meancov

The available functions for time points are:

timepoint_mean(sim, t)

timepoint_median(sim, t)

timepoint_quantile(sim, q, t)

timepoint_meanvar(sim, t)

timepoint_meancov(sim, t1, t2)

timepoint_meancor(sim, t1, t2)

timepoint_weighted_meancov(sim, W, t1, t2)

Full Timeseries Statistics

Additionally, the following functions are provided for analyzing the full timeseries. The mean and meanvar versions return a DiffEqArray which can be directly plotted. The meancov and meancor return a matrix of tuples, where the tuples are the (mean_t1,mean_t2,cov or cor).

The available functions for the time steps are:

The available functions for the time points are:

SciMLBase.EnsembleAnalysis.timeseries_point_mean
SciMLBase.EnsembleAnalysis.timeseries_point_median
SciMLBase.EnsembleAnalysis.timeseries_point_quantile
SciMLBase.EnsembleAnalysis.timeseries_point_meanvar
SciMLBase.EnsembleAnalysis.timeseries_point_meancov
SciMLBase.EnsembleAnalysis.timeseries_point_meancor
SciMLBase.EnsembleAnalysis.timeseries_point_weighted_meancov

### EnsembleSummary

@docs EnsembleSummary

## Example 1: Solving an ODE With Different Initial Conditions

### Random Initial Conditions

Let's test the sensitivity of the linear ODE to its initial condition. To do this,
we would like to solve the linear ODE 100 times and plot what the trajectories
look like. Let's start by opening up some extra processes so that way the computation
will be parallelized. Here we will choose to use distributed parallelism, which means
that the required functions must be made available to all processes. This can be
achieved with
[`@everywhere` macro](https://docs.julialang.org/en/v1.2/stdlib/Distributed/#Distributed.@everywhere):

julia using Distributed using OrdinaryDiffEq using Plots

addprocs() @everywhere using OrdinaryDiffEq

Now let's define the linear ODE, which is our base problem:

julia

Linear ODE which starts at 0.5 and solves from t=0.0 to t=1.0

prob = ODEProblem((u, p, t) -> 1.01u, 0.5, (0.0, 1.0))

For our ensemble simulation, we would like to change the initial condition around.
This is done through the `prob_func`. This function takes in the base problem
and modifies it to create the new problem that the trajectory actually solves.
Here, we will take the base problem, multiply the initial condition by a `rand()`,
and use that for calculating the trajectory:

julia @everywhere function prob_func(prob, i, repeat) remake(prob, u0 = rand() * prob.u0) end

Now we build and solve the `EnsembleProblem` with this base problem and `prob_func`:

julia ensembleprob = EnsembleProblem(prob, probfunc = probfunc) sim = solve(ensembleprob, Tsit5(), EnsembleDistributed(), trajectories = 10)

We can use the plot recipe to plot what the 10 ODEs look like:

julia plot(sim, linealpha = 0.4)

We note that if we wanted to find out what the initial condition was for a given
trajectory, we can retrieve it from the solution. `sim[i]` returns the `i`th
solution object. `sim[i].prob` is the problem that specific trajectory solved,
and `sim[i].prob.u0` would then be the initial condition used in the `i`th
trajectory.

Note: If the problem has callbacks, the functions for the `condition` and
`affect!` must be named functions (not anonymous functions).

### Using multithreading

The previous ensemble simulation can also be parallelized using a multithreading
approach, which will make use of the different cores within a single computer.
Because the memory is shared across the different threads, it is not necessary to
use the `@everywhere` macro. Instead, the same problem can be implemented simply as:

@example ensemble12 using OrdinaryDiffEq prob = ODEProblem((u, p, t) -> 1.01u, 0.5, (0.0, 1.0)) function probfunc(prob, i, repeat) remake(prob, u0 = rand() * prob.u0) end ensembleprob = EnsembleProblem(prob, probfunc = probfunc) sim = solve(ensembleprob, Tsit5(), EnsembleThreads(), trajectories = 10) using Plots; plot(sim);

The number of threads to be used has to be defined outside of Julia, in
the environmental variable `JULIA_NUM_THREADS` (see Julia's [documentation](https://docs.julialang.org/en/v1.1/manual/environment-variables/#JULIA_NUM_THREADS-1) for details).

### Pre-Determined Initial Conditions

Often, you may already know what initial conditions you want to use. This
can be specified by the `i` argument of the `prob_func`. This `i` is the unique
index of each trajectory. So, if we have `trajectories=100`, then we have `i` as
some index in `1:100`, and it's different for each trajectory.

So, if we wanted to use a grid of evenly spaced initial conditions from `0` to `1`,
we could simply index the `linspace` type:

@example ensemble13 initialconditions = range(0, stop = 1, length = 100) function probfunc(prob, i, repeat) remake(prob, u0 = initialconditions[i]) end

It's worth noting that if you run this code successfully, there will be no visible output.

## Example 2: Solving an SDE with Different Parameters

Let's solve the same SDE, but with varying parameters. Let's create a Lotka-Volterra
system with multiplicative noise. Our Lotka-Volterra system will have as its
drift component:

@example ensemble2 function f(du, u, p, t) du[1] = p[1] * u[1] - p[2] * u[1] * u[2] du[2] = -3 * u[2] + u[1] * u[2] end

For our noise function, we will use multiplicative noise:

@example ensemble2 function g(du, u, p, t) du[1] = p[3] * u[1] du[2] = p[4] * u[2] end

Now we build the SDE with these functions:

@example ensemble2 using StochasticDiffEq p = [1.5, 1.0, 0.1, 0.1] prob = SDEProblem(f, g, [1.0, 1.0], (0.0, 10.0), p)

This is the base problem for our study. What would like to do with this experiment
is keep the same parameters in the deterministic component each time, but vary
the parameters for the amount of noise using `0.3rand(2)` as our parameters.
Once again, we do this with a `prob_func`, and here we modify the parameters in
`prob.p`:

@example ensemble2

p is a global variable, referencing it would be type unstable.

Using a let block defines a small local scope in which we can

capture that local p which isn't redefined anywhere in that local scope.

This allows it to be type stable.

prob_func = let p = p (prob, i, repeat) -> begin x = 0.3rand(2) remake(prob, p = [p[1], p[2], x[1], x[2]]) end end

Now we solve the problem 10 times and plot all of the trajectories in phase space:

@example ensemble2 ensembleprob = EnsembleProblem(prob, probfunc = probfunc) sim = solve(ensembleprob, SRIW1(), trajectories = 10) using Plots; plot(sim, linealpha = 0.6, color = :blue, idxs = (0, 1), title = "Phase Space Plot"); plot!(sim, linealpha = 0.6, color = :red, idxs = (0, 2), title = "Phase Space Plot")

We can then summarize this information with the mean/variance bounds using a
`EnsembleSummary` plot. We will take the mean/quantile at every `0.1` time
units and directly plot the summary:

@example ensemble2 summ = EnsembleSummary(sim, 0:0.1:10) plot(summ, fillalpha = 0.5)

Note that here we used the quantile bounds, which default to `[0.05,0.95]` in
the `EnsembleSummary` constructor. We can change to standard error of the mean
bounds using `ci_type=:SEM` in the plot recipe.

## Example 3: Using the Reduction to Halt When Estimator is Within Tolerance

In this problem, we will solve the equation just as many times as needed to get
the standard error of the mean for the final time point below our tolerance
`0.5`. Since we only care about the endpoint, we can tell the `output_func`
to discard the rest of the data.

@example ensemble3 function output_func(sol, i) last(sol), false end

Our `prob_func` will simply randomize the initial condition:

@example ensemble3 using OrdinaryDiffEq

Linear ODE which starts at 0.5 and solves from t=0.0 to t=1.0

prob = ODEProblem((u, p, t) -> 1.01u, 0.5, (0.0, 1.0))

function prob_func(prob, i, repeat) remake(prob, u0 = rand() * prob.u0) end

Our reduction function will append the data from the current batch to the previous
batch, and declare convergence if the standard error of the mean is calculated
as sufficiently small:

@example ensemble3 using Statistics function reduction(u, batch, I) u = append!(u, batch) finished = (var(u) / sqrt(last(I))) / mean(u) < 0.5 u, finished end

Then we can define and solve the problem:

@example ensemble3 prob2 = EnsembleProblem(prob, probfunc = probfunc, outputfunc = outputfunc, reduction = reduction, uinit = Vector{Float64}()) sim = solve(prob2, Tsit5(), trajectories = 10000, batchsize = 20)

Since `batch_size=20`, this means that every 20 simulations, it will take this batch,
append the results to the previous batch, calculate `(var(u)/sqrt(last(I)))/mean(u)`,
and if that's small enough, exit the simulation. In this case, the simulation
exits only after 20 simulations (i.e. after calculating the first batch). This
can save a lot of time!

In addition to saving time by checking convergence, we can save memory by reducing
between batches. For example, say we only care about the mean at the end once
again. Instead of saving the solution at the end for each trajectory, we can instead
save the running summation of the endpoints:

@example ensemble3 function reduction(u, batch, I) u + sum(batch), false end prob2 = EnsembleProblem(prob, probfunc = probfunc, outputfunc = outputfunc, reduction = reduction, uinit = 0.0) sim2 = solve(prob2, Tsit5(), trajectories = 100, batchsize = 20)

this will sum up the endpoints after every 20 solutions, and save the running sum.
The final result will have `sim2.u` as simply a number, and thus `sim2.u/100` would
be the mean.

## Example 4: Using the Analysis Tools

In this example, we will show how to analyze a `EnsembleSolution`. First, let's
generate a 10 solution Monte Carlo experiment. For our problem, we will use a `4x2`
system of linear stochastic differential equations:

@example ensemble4 function f(du, u, p, t) for i in 1:length(u) du[i] = 1.01 * u[i] end end function σ(du, u, p, t) for i in 1:length(u) du[i] = 0.87 * u[i] end end using StochasticDiffEq prob = SDEProblem(f, σ, ones(4, 2) / 2, (0.0, 1.0)) #probsde2Dlinear

To solve this 10 times, we use the `EnsembleProblem` constructor and solve
with `trajectories=10`. Since we wish to compare values at the timesteps, we need
to make sure the steps all hit the same times. We thus set `adaptive=false` and
explicitly give a `dt`.

@example ensemble4 prob2 = EnsembleProblem(prob) sim = solve(prob2, SRIW1(), dt = 1 // 2^(3), trajectories = 10, adaptive = false)

**Note that if you don't do the `timeseries_steps` calculations, this code is
compatible with adaptive timestepping. Using adaptivity is usually more efficient!**

We can compute the mean and the variance at the 3rd timestep using:

@example ensemble4 using SciMLBase.EnsembleAnalysis m, v = timestep_meanvar(sim, 3)

or we can compute the mean and the variance at the `t=0.5` using:

@example ensemble4 m, v = timepoint_meanvar(sim, 0.5)

We can get a series for the mean and the variance at each time step using:

@example ensemble4 mseries, vseries = timeseriesstepsmeanvar(sim)

or at chosen values of `t`:

@example ensemble4 ts = 0:0.1:1 mseries = timeseriespoint_mean(sim, ts)

Note that these mean and variance series can be directly plotted. We can
compute covariance matrices similarly:

@example ensemble4 timeseriesstepsmeancov(sim) # Use the time steps, assume fixed dt timeseriespointmeancov(sim, 0:(1 // 2 ^ (3)):1, 0:(1 // 2 ^ (3)):1) # Use time points, interpolate

For general analysis, we can build a `EnsembleSummary` type.

@example ensemble4 summ = EnsembleSummary(sim)

will summarize at each time step, while

@example ensemble4 summ = EnsembleSummary(sim, 0.0:0.1:1.0)

will summarize at the `0.1` time points using the interpolations. To
visualize the results, we can plot it. Since there are 8 components to
the differential equation, this can get messy, so let's only plot the
3rd component:

@example ensemble4 using Plots; plot(summ; idxs = 3);

We can change to errorbars instead of ribbons and plot two different
indices:

@example ensemble4 plot(summ; idxs = (3, 5), error_style = :bars)

Or we can simply plot the mean of every component over time:

@example ensemble4 plot(summ; error_style = :none) ```