Julia SciML Ecosystem Documentation

https://docs.sciml.ai/
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This project provides documentation for the Julia SciML ecosystem, covering various packages related...
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### Setup Environment and Install Packages

Source: https://docs.sciml.ai/Catalyst/stable/inverse_problems/examples/ode_fitting_oscillation

Sets up a temporary Julia environment and installs necessary packages for the example. This environment is deleted when the Julia session ends.

```julia
using Pkg
Pkg.activate(; temp = true) # Creates a temporary environment, which is deleted when the Julia session ends.
Pkg.add("Catalyst")
Pkg.add("OptimizationBase")
Pkg.add("OptimizationOptimisers")
Pkg.add("OrdinaryDiffEqRosenbrock")
Pkg.add("Plots")
Pkg.add("SciMLSensitivity")
```

--------------------------------

### Setup Environment and Install Packages

Source: https://docs.sciml.ai/Catalyst/stable/inverse_problems/turing_ode_param_fitting

Installs necessary Julia packages for running the ODE parameter fitting example. Ensure you are in the correct project directory before running.

```julia
using Pkg
Pkg.activate(".")
Pkg.add("Catalyst")
Pkg.add("Distributions")
Pkg.add("OrdinaryDiffEqDefault")
Pkg.add("Plots")
Pkg.add("StatsPlots")
Pkg.add("SymbolicIndexingInterface")
Pkg.add("Turing")
```

--------------------------------

### Quick-Start Simulation Examples

Source: https://docs.sciml.ai/Catalyst/stable/model_simulation/simulation_introduction

Demonstrates how to set up and run ODE, SDE, and jump simulations using Catalyst. Requires prior environment setup and package installation.

```julia
using Catalyst, JumpProcesses, OrdinaryDiffEqDefault, StochasticDiffEq

# First we designate our model, initial condition, time span, and parameter values.
rn = @reaction_network begin
    (p,d), 0 <--> X
end
u0 = [:X => 2]
tspan = (0.0, 10.0)
ps = [:p => 5.0, :d => 0.25]
# Now, for each simulation type, we first bundle these into a `XProblem`. 
# This can be simulated using `solve` and the solution plotted using `plot`.

# ODE simulation.
oprob = ODEProblem(rn, u0, tspan, ps)
osol = solve(oprob)
plot(osol)

# SDE simulation.
sprob = SDEProblem(rn, u0, tspan, ps)
ssol = solve(sprob)
plot(ssol)

# Jump simulation.
jprob = JumpProblem(rn, u0, tspan, ps)
jsol = solve(jprob)
plot(jsol)
```

--------------------------------

### Setup Environment and Install Packages

Source: https://docs.sciml.ai/Catalyst/stable/model_creation/compositional_modeling

Installs the necessary Catalyst, GraphRecipes, and Plots packages in a temporary environment.

```julia
using Pkg
Pkg.activate(; temp = true) # Creates a temporary environment, which is deleted when the Julia session ends.
Pkg.add("Catalyst")
Pkg.add("GraphRecipes")
Pkg.add("Plots")
```

--------------------------------

### Setup Environment and Install Packages

Source: https://docs.sciml.ai/Catalyst/stable/inverse_problems/optimization_ode_param_fitting

Installs necessary packages for ODE parameter fitting in a temporary Julia environment.

```julia
using Pkg
Pkg.activate(; temp = true) # Creates a temporary environment, which is deleted when the Julia session ends.
Pkg.add("Catalyst")
Pkg.add("OptimizationBase")
Pkg.add("OptimizationNLopt")
Pkg.add("OptimizationEvolutionary")
Pkg.add("OptimizationOptimJL")
Pkg.add("OrdinaryDiffEqDefault")
Pkg.add("Plots")
```

--------------------------------

### Environment Setup and Package Installation

Source: https://docs.sciml.ai/Catalyst/stable/model_simulation/examples/periodic_events_simulation

Sets up a temporary Julia environment and installs necessary packages: Catalyst, JumpProcesses, OrdinaryDiffEqDefault, and Plots. This is a prerequisite for running the examples in the tutorial.

```julia
using Pkg
Pkg.activate(; temp = true) # Creates a temporary environment, which is deleted when the Julia session ends.
Pkg.add("Catalyst")
Pkg.add("JumpProcesses")
Pkg.add("OrdinaryDiffEqDefault")
Pkg.add("Plots")
```

--------------------------------

### Environment Setup and Package Installation

Source: https://docs.sciml.ai/Catalyst/stable/model_creation/events

Installs Catalyst, OrdinaryDiffEqTsit5, and Plots in a temporary Julia environment.

```julia
using Pkg
Pkg.activate(; temp = true) # Creates a temporary environment, which is deleted when the Julia session ends.
Pkg.add("Catalyst")
Pkg.add("OrdinaryDiffEqTsit5")
Pkg.add("Plots")
```

--------------------------------

### Setup Environment and Install Packages

Source: https://docs.sciml.ai/Catalyst/stable/model_creation/examples/hodgkin_huxley_equation

Installs necessary packages for running the Hodgkin-Huxley model examples. Creates a temporary environment that is deleted upon session end.

```julia
using Pkg
Pkg.activate(; temp = true) # Creates a temporary environment, which is deleted when the Julia session ends.
Pkg.add("Catalyst")
Pkg.add("ModelingToolkitBase")
Pkg.add("NonlinearSolveFirstOrder")
Pkg.add("OrdinaryDiffEqRosenbrock")
Pkg.add("Plots")
```

--------------------------------

### Environment Setup and Package Installation

Source: https://docs.sciml.ai/Catalyst/stable/model_simulation/ode_simulation_performance

Sets up a temporary Julia environment and installs necessary packages for performing ODE simulations and benchmarking.

```julia
using Pkg
Pkg.activate(; temp = true) # Creates a temporary environment, which is deleted when the Julia session ends.
Pkg.add("BenchmarkTools")
Pkg.add("Catalyst")
Pkg.add("CUDA")
Pkg.add("DiffEqGPU")
Pkg.add("IncompleteLU")
Pkg.add("OrdinaryDiffEqBDF")
Pkg.add("OrdinaryDiffEqRosenbrock")
Pkg.add("OrdinaryDiffEqTsit5")
Pkg.add("OrdinaryDiffEqVerner")
Pkg.add("Plots")
Pkg.add("StaticArrays")
```

--------------------------------

### Environment Setup and Package Installation

Source: https://docs.sciml.ai/Catalyst/stable/steady_state_functionality/steady_state_stability_computation

Sets up a temporary Julia environment and installs the necessary Catalyst and HomotopyContinuation packages.

```julia
using Pkg
Pkg.activate(; temp = true) # Creates a temporary environment, which is deleted when the Julia session ends.
Pkg.add("Catalyst")
Pkg.add("HomotopyContinuation")
```

--------------------------------

### Setup Catalyst Environment

Source: https://docs.sciml.ai/Catalyst/stable/model_creation/programmatic_CRN_construction

Installs the Catalyst package in a temporary environment for running the code examples.

```julia
using Pkg
Pkg.activate(; temp = true)
Pkg.add("Catalyst")
```

--------------------------------

### Environment Setup and Package Installation

Source: https://docs.sciml.ai/Catalyst/stable/model_creation/parametric_stoichiometry

Sets up a temporary Julia environment and installs necessary packages for running Catalyst examples. This includes Catalyst, Distributions, JumpProcesses, Latexify, ModelingToolkitBase, OrdinaryDiffEqTsit5, and Plots.

```julia
using Pkg
Pkg.activate(; temp = true) # Creates a temporary environment, which is deleted when the Julia session ends.
Pkg.add("Catalyst")
Pkg.add("Distributions")
Pkg.add("JumpProcesses")
Pkg.add("Latexify")
Pkg.add("ModelingToolkitBase")
Pkg.add("OrdinaryDiffEqTsit5")
Pkg.add("Plots")
```

--------------------------------

### Environment Setup and Package Installation

Source: https://docs.sciml.ai/Catalyst/stable/inverse_problems/global_sensitivity_analysis

Sets up a temporary Julia environment and installs necessary packages for Global Sensitivity Analysis.

```julia
using Pkg
Pkg.activate(; temp = true) 
Mkdir.add("Catalyst")
Mkdir.add("GlobalSensitivity")
Mkdir.add("Plots")
Mkdir.add("OrdinaryDiffEqDefault")
```

--------------------------------

### Setup Environment and Install Packages

Source: https://docs.sciml.ai/Catalyst/stable/model_creation/reactionsystem_content_accessing

Installs necessary packages for running Catalyst code in a temporary environment.

```julia
using Pkg
Pkg.activate(; temp = true) # Creates a temporary environment, which is deleted when the Julia session ends.
Pkg.add("Catalyst")
Pkg.add("OrdinaryDiffEqDefault")
Pkg.add("Plots")
Pkg.add("SymbolicIndexingInterface")
```

--------------------------------

### Environment Setup and Package Installation

Source: https://docs.sciml.ai/Catalyst/stable/model_simulation/simulation_structure_interfacing

Sets up a temporary Julia environment and installs necessary packages: Catalyst, OrdinaryDiffEqDefault, and SymbolicIndexingInterface.

```julia
using Pkg
Pkg.activate(; temp = true) # Creates a temporary environment, which is deleted when the Julia session ends.
Pkg.add("Catalyst")
Pkg.add("OrdinaryDiffEqDefault")
Pkg.add("SymbolicIndexingInterface")
```

--------------------------------

### Environment Setup and Package Installation

Source: https://docs.sciml.ai/Catalyst/stable/steady_state_functionality/examples/nullcline_plotting

Sets up a temporary Julia environment and installs necessary packages: Catalyst, HomotopyContinuation, and Plots.

```julia
using Pkg
Pkg.activate(; temp = true) # Creates a temporary environment, which is deleted when the Julia session ends.
Pkg.add("Catalyst")
Pkg.add("HomotopyContinuation")
Pkg.add("Plots")
```

--------------------------------

### Setup Environment and Install Packages

Source: https://docs.sciml.ai/Catalyst/stable/model_simulation/examples/interactive_brusselator_simulation

Installs necessary packages for running the simulation and visualization. Creates a temporary environment that is deleted when the Julia session ends.

```julia
using Pkg
Pkg.activate(; temp = true) # Creates a temporary environment, which is deleted when the Julia session ends.
Pkg.add("CairoMakie")
Pkg.add("Catalyst")
Pkg.add("GLMakie")
Pkg.add("OrdinaryDiffEqTsit5")
```

--------------------------------

### Environment Setup and Package Installation

Source: https://docs.sciml.ai/Catalyst/stable/steady_state_functionality/dynamical_systems

This code block sets up a temporary Julia environment and installs necessary packages for running the examples on this page. The temporary environment is deleted upon session end.

```julia
using Pkg
Pkg.activate(; temp = true) # Creates a temporary environment, which is deleted when the Julia session ends.
Pkg.add("CairoMakie")
Pkg.add("Catalyst")
Pkg.add("DynamicalSystems")
Pkg.add("OrdinaryDiffEqRosenbrock")
Pkg.add("OrdinaryDiffEqTsit5")
Pkg.add("Plots")
```

--------------------------------

### Setup Environment and Install Catalyst

Source: https://docs.sciml.ai/Catalyst/stable/model_creation/chemistry_related_functionality

Sets up a temporary Julia environment and installs the Catalyst package. This is useful for ensuring a clean and isolated environment for running the code examples.

```julia
using Pkg
Pkg.activate(; temp = true) # Creates a temporary environment, which is deleted when the Julia session ends.
Pkg.add("Catalyst")
```

--------------------------------

### Julia Environment Setup for SDE Simulations

Source: https://docs.sciml.ai/Catalyst/stable/model_simulation/sde_simulation_performance

Installs necessary packages for SDE simulations in a temporary Julia environment. This setup is recommended before running the provided code examples.

```julia
using Pkg
Pkg.activate(; temp = true) # Creates a temporary environment, which is deleted when the Julia session ends.
Pkg.add("Catalyst")
Pkg.add("CUDA")
Pkg.add("DiffEqGPU")
Pkg.add("StaticArrays")
Pkg.add("StochasticDiffEq")
```

--------------------------------

### Environment Setup and Package Installation

Source: https://docs.sciml.ai/Catalyst/stable/steady_state_functionality/examples/bifurcationkit_codim2

Sets up a temporary Julia environment and installs necessary packages: BifurcationKit, Catalyst, OrdinaryDiffEqDefault, and Plots.

```julia
using Pkg
Pkg.activate(; temp = true) # Creates a temporary environment, which is deleted when the Julia session ends.
Pkg.add("BifurcationKit")
Pkg.add("Catalyst")
Pkg.add("OrdinaryDiffEqDefault")
Pkg.add("Plots")
```

--------------------------------

### Setup Environment and Install Packages

Source: https://docs.sciml.ai/Catalyst/stable/model_creation/conservation_laws

Installs necessary packages for running Catalyst models, including Catalyst itself, Latexify for equation formatting, OrdinaryDiffEqDefault for ODE solving, and Plots for visualization. Creates a temporary environment.

```julia
using Pkg
Pkg.activate(; temp = true) # Creates a temporary environment, which is deleted when the Julia session ends.
Pkg.add("Catalyst")
Pkg.add("Latexify")
Pkg.add("OrdinaryDiffEqDefault")
Pkg.add("Plots")
```

--------------------------------

### Setup Julia Environment and Install Packages

Source: https://docs.sciml.ai/Catalyst/stable/steady_state_functionality/examples/bifurcationkit_periodic_orbits

Installs necessary packages for BifurcationKit.jl and Catalyst.jl in a temporary environment.

```julia
using Pkg
Pkg.activate(; temp = true)
Pkg.add("BifurcationKit")
Pkg.add("Catalyst")
Pkg.add("OrdinaryDiffEqDefault")
Pkg.add("Plots")
```

--------------------------------

### Setup Environment and Install Packages

Source: https://docs.sciml.ai/Catalyst/stable/model_creation/examples/basic_CRN_library

Installs necessary packages for running CRN models and simulations. Ensure you are in the correct project directory.

```julia
using Pkg
Pkg.activate(".")
Pkg.add("Catalyst")
Pkg.add("OrdinaryDiffEqDefault")
Pkg.add("StochasticDiffEq")
Pkg.add("JumpProcesses")
Pkg.add("Plots")
```

--------------------------------

### Quick-start Deep Equilibrium Network Example

Source: https://docs.sciml.ai/DeepEquilibriumNetworks/stable

This example demonstrates setting up and running a Deep Equilibrium Network using Lux.jl and DifferentialEquations.jl. It includes model definition, GPU setup, forward pass, and gradient calculation.

```julia
using DeepEquilibriumNetworks, Lux, Random, NonlinearSolve, Zygote, SciMLSensitivity
using LuxCUDA  # For NVIDIA GPU support

seed = 0
rng = Random.default_rng()
Random.seed!(rng, seed)

model = Chain(Dense(2 => 2),
    DeepEquilibriumNetwork(
        Parallel(+, Dense(2 => 2; use_bias=false), Dense(2 => 2; use_bias=false)),
        NewtonRaphson()))

gdev = gpu_device()
cdev = cpu_device()

ps, st = Lux.setup(rng, model) |> gdev
x = rand(rng, Float32, 2, 3) |> gdev
y = rand(rng, Float32, 2, 3) |> gdev

res, st_ = model(x, ps, st)
st_.layer_2.solution
```

```text
DeepEquilibriumSolution
 * Initial Guess: Float32[0.0 0.0 0.0; 0.0 0.0 0.0]
 * Steady State: Float32[-0.219407 0.637479 -0.744242; 1.50412 -0.962679 3.12426]
 * Residual: Float32[-4.47035f-8 -5.96046f-8 -5.96046f-8; -1.19209f-7 5.96046f-8 -2.38419f-7]
 * Jacobian Loss: 0.0
 * NFE: 3
```

```julia
gs = only(Zygote.gradient(p -> sum(abs2, first(model(x, p, st)) .- y), ps))
```

```text
(layer_1 = (weight = Float32[-4.6991134 -8.227526; -13.163104 -20.53097], bias = Float32[-11.837078, -31.839073]), layer_2 = (model = (layer_1 = (weight = Float32[15.304397 -58.909546; -21.828897 81.71437],), layer_2 = (weight = Float32[17.71767 3.5711353; -23.398441 -5.474779],)), init = nothing))
```

--------------------------------

### Environment Setup and Package Installation

Source: https://docs.sciml.ai/Catalyst/stable/model_creation/functional_parameters

Sets up a temporary Julia environment and installs necessary packages: Catalyst, DataInterpolations, OrdinaryDiffEqDefault, and Plots.

```julia
using Pkg
Pkg.activate(; temp = true) # Creates a temporary environment, which is deleted when the Julia session ends.
Pkg.add("Catalyst")
Pkg.add("DataInterpolations")
Pkg.add("OrdinaryDiffEqDefault")
Pkg.add("Plots")
```

--------------------------------

### Environment Setup and Package Installation

Source: https://docs.sciml.ai/Catalyst/stable/model_simulation/examples/activation_time_distribution_measurement

Sets up a temporary Julia environment and installs necessary packages like Catalyst, Plots, and StochasticDiffEq.

```julia
using Pkg
Pkg.activate(; temp = true) # Creates a temporary environment, which is deleted when the Julia session ends.
Pkg.add("Catalyst")
Pkg.add("Plots")
Pkg.add("StochasticDiffEq")
```

--------------------------------

### Setup for Demonstrating Extrapolation Methods

Source: https://docs.sciml.ai/DataInterpolations/stable/extrapolation_methods

Initializes data and an interpolation object for demonstrating extrapolation. This setup is used across multiple examples.

```julia
using DataInterpolations, Plots

u = [0.86, 0.65, 0.44, 0.76, 0.73]
t = [0.0022, 0.68, 1.41, 2.22, 2.46]
t_eval_left = range(-1, first(t), length = 25)
t_eval_right = range(last(t), 3.5, length = 25)
A = QuadraticSpline(u, t)
plot(A)
```

--------------------------------

### Environment Setup and Package Installation

Source: https://docs.sciml.ai/Catalyst/stable/model_simulation/hybrid_simulations

Sets up a temporary Julia environment and installs necessary packages for running hybrid simulations. This includes Catalyst, JumpProcesses, DiffEqNoiseProcess, OrdinaryDiffEqTsit5, Plots, and StochasticDiffEq.

```julia
using Pkg
Pkg.activate(; temp = true) # Creates a temporary environment, which is deleted when the Julia session ends.
Pkg.add("Catalyst")
Pkg.add("JumpProcesses")
Pkg.add("DiffEqNoiseProcess")
Pkg.add("OrdinaryDiffEqTsit5")
Pkg.add("Plots")
Pkg.add("StochasticDiffEq")
```

--------------------------------

### Setup Environment and Install Packages

Source: https://docs.sciml.ai/Catalyst/stable/inverse_problems/structural_identifiability

Sets up a temporary Julia environment and installs the necessary Catalyst and StructuralIdentifiability packages. This environment is deleted when the Julia session ends.

```julia
using Pkg
Pkg.activate(; temp = true)
Pkg.add("Catalyst")
Pkg.add("StructuralIdentifiability")
```

--------------------------------

### Setup Environment and Install Packages

Source: https://docs.sciml.ai/Catalyst/stable/model_creation/examples/programmatic_generative_linear_pathway

Sets up a temporary Julia environment and installs necessary packages: Catalyst, OrdinaryDiffEqDefault, and Plots. This environment is deleted when the Julia session ends.

```julia
using Pkg
Pkg.activate(; temp = true) # Creates a temporary environment, which is deleted when the Julia session ends.
Pkg.add("Catalyst")
Pkg.add("OrdinaryDiffEqDefault")
Pkg.add("Plots")
```

--------------------------------

### Setup Environment and Install Packages

Source: https://docs.sciml.ai/Catalyst/stable/model_creation/dsl_basics

Sets up a temporary Julia environment and installs the Catalyst and Latexify packages. This environment is deleted when the Julia session ends.

```julia
using Pkg
Pkg.activate(; temp = true)
Pkg.add("Catalyst")
Pkg.add("Latexify")
```

--------------------------------

### Example: System Definition with Parameters

Source: https://docs.sciml.ai/ModelingToolkit/stable/basics/FAQ

Defines a system with scalar and array parameters, then completes the system. This setup is used to demonstrate parameter ordering.

```julia
using ModelingToolkit

@parameters p q[1:3] r[1:2, 1:2]

@named sys = System(Equation[], ModelingToolkit.t_nounits, [], [p, q, r])
sys = complete(sys)
```

--------------------------------

### Example: DeepSplitting Algorithm Setup

Source: https://docs.sciml.ai/HighDimPDE/stable/DeepSplitting.html

Illustrates how to set up the DeepSplitting algorithm by defining a neural network architecture and configuring the algorithm's parameters including Monte Carlo sampling.

```julia
hls = d + 50 # hidden layer size
d = 10 # size of the sample

# Neural network used by the scheme
nn = Flux.Chain(Dense(d, hls, tanh),
                Dense(hls,hls,tanh),
                Dense(hls, 1, x->x^2))

alg = DeepSplitting(nn, K=10, opt = ADAM(), λs = [5e-3,1e-3],
                    mc_sample = UniformSampling(zeros(d), ones(d)) )
```

--------------------------------

### Environment Setup and Package Installation

Source: https://docs.sciml.ai/Catalyst/stable/model_simulation/simulation_introduction

Sets up a temporary Julia environment and adds necessary packages for Catalyst simulations. This environment is deleted when the Julia session ends.

```julia
using Pkg
Pkg.activate(; temp = true) # Creates a temporary environment, which is deleted when the Julia session ends.
Pkg.add("Catalyst")
Pkg.add("JumpProcesses")
Pkg.add("OrdinaryDiffEqDefault")
Pkg.add("OrdinaryDiffEqRosenbrock")
Pkg.add("OrdinaryDiffEqTsit5")
Pkg.add("Plots")
Pkg.add("StochasticDiffEq")
```

--------------------------------

### Setup Julia Environment for Simulations

Source: https://docs.sciml.ai/Catalyst/stable/model_simulation/ensemble_simulations

Installs necessary packages for running simulations. Creates a temporary environment that is deleted upon session end.

```julia
using Pkg
Pkg.activate(; temp = true) 
Pkg.add("Catalyst")
Pkg.add("OrdinaryDiffEqTsit5")
Pkg.add("Plots")
Pkg.add("StochasticDiffEq")
```

--------------------------------

### Basic Nonlinear Problem Setup and Solve

Source: https://docs.sciml.ai/NonlinearSolve/stable/basics/diagnostics_api

Sets up a simple nonlinear problem and solves it using NonlinearSolve.jl. This is a fundamental example to get started.

```julia
import NonlinearSolve as NLS

function nlfunc(resid, u0, p)
    resid[1] = u0[1] * u0[1] - p
    resid[2] = u0[2] * u0[2] - p
    resid[3] = u0[3] * u0[3] - p
    nothing
end

prob = NLS.NonlinearProblem(nlfunc, [1.0, 3.0, 5.0], 2.0)

NLS.solve(prob)
```

--------------------------------

### Setup and Function Definition

Source: https://docs.sciml.ai/SciMLOperators/stable/tutorials/fftw

Initializes packages, defines grid parameters, and creates a sample function `v` and its analytical derivative `w` for testing.

```julia
using SciMLOperators
using LinearAlgebra, FFTW

L = 2π
n = 256
dx = L / n
x = range(start = -L / 2, stop = L / 2 - dx, length = n) |> Array

v = @. sin(5x)cos(7x);
w = @. 5cos(5x)cos(7x) - 7sin(5x)sin(7x);
```

--------------------------------

### Configure Optimization Algorithm Setup

Source: https://docs.sciml.ai/SciMLBenchmarksOutput/stable/GlobalOptimization/blackbox_global_optimizers

Creates a dictionary `setup` to configure various global optimization algorithms for benchmarking. It includes algorithms from NLopt, OptimizationEvolutionary, Optim.jl, BBO, OptimizationMetaheuristics, Optimisers, and SciPy, specifying their parameters and whether they are boxed.

```julia
setup = Dict(
    "NelderMead" => NelderMead(),
    #Optim.BFGS(),
    #"NLopt.GN_MLSL_LDS" => chain(NLopt.GN_MLSL_LDS(), isboxed=true), # gives me errors
    "NLopt.GN_CRS2_LM()" => chain(NLopt.GN_CRS2_LM(), isboxed = true),
    "NLopt.GN_DIRECT()" => chain(NLopt.GN_DIRECT(), isboxed = true),
    "NLopt.GN_ESCH()" => chain(NLopt.GN_ESCH(), isboxed = true),
    "OptimizationEvolutionary.GA()" => chain(OptimizationEvolutionary.GA(), isboxed = true),
    "OptimizationEvolutionary.DE()" => chain(OptimizationEvolutionary.DE(), isboxed = true),
    "OptimizationEvolutionary.ES()" => chain(OptimizationEvolutionary.ES(), isboxed = true),
    "Optim.SAMIN" => chain(SAMIN(verbosity = 0), isboxed = true),
    "BBO_adaptive_de_rand_1_bin" => chain(BBO_adaptive_de_rand_1_bin(), isboxed = true),
    "BBO_adaptive_de_rand_1_bin_radiuslimited" => chain(
        BBO_adaptive_de_rand_1_bin_radiuslimited(), isboxed = true), # same as BBO_adaptive_de_rand_1_bin
    "BBO_separable_nes" => chain(BBO_separable_nes(), isboxed = true),
    "BBO_de_rand_2_bin" => chain(BBO_de_rand_2_bin(), isboxed = true),
    #"BBO_xnes" => chain(BBO_xnes(), isboxed=true), # good but slow
    #"BBO_dxnes" => chain(BBO_dxnes(), isboxed=true), 
    "OptimizationMetaheuristics.ECA" => chain(OptimizationMetaheuristics.ECA(), isboxed = true),
    #"OptimizationMetaheuristics.CGSA" => () -> chain(OptimizationMetaheuristics.CGSA(), isboxed=true), #give me strange results
    "OptimizationMetaheuristics.DE" => chain(OptimizationMetaheuristics.DE(), isboxed=true),
    "Optimisers.AdamW" => chain(Optimisers.AdamW(), isboxed=false),
    "Optimisers.RMSProp" => chain(Optimisers.RMSProp(), isboxed=false),
    # SciPy global optimizers
    "ScipyDifferentialEvolution" => chain(ScipyDifferentialEvolution(), isboxed=true),
    #"ScipyBasinhopping" => chain(ScipyBasinhopping(), isboxed=true),
    #"ScipyDualAnnealing" => chain(ScipyDualAnnealing(), isboxed=true), # taking long time
    "ScipyShgo" => chain(ScipyShgo(), isboxed=true),
    "ScipyDirect" => chain(ScipyDirect(), isboxed=true),
    "ScipyBrute" => chain(ScipyBrute(), isboxed=true),
    # "NOMADOpt" => chain(NOMADOpt()), too much printing
    # "OptimizationPRIMA.UOBYQA()" => chain(OptimizationPRIMA.UOBYQA()), :StackOverflowError?
    # "OptimizationPRIMA.NEWUOA()" => OptimizationPRIMA.UOBYQA(),
    #
)
```

--------------------------------

### Install and Open SciML Tutorials

Source: https://docs.sciml.ai/SciMLTutorialsOutput/stable

Installs the SciMLTutorials package, activates its environment, and opens the interactive notebooks. Ensure you are in the correct directory when running.

```julia
]
add SciMLTutorials#master
]activate SciMLTutorials
]instantiate
using SciMLTutorials
SciMLTutorials.open_notebooks()
```

--------------------------------

### Setup: Fetch Required Packages

Source: https://docs.sciml.ai/SciMLBenchmarksOutput/stable/NonlinearProblem/nonlinear_battery_problem

Imports necessary packages for nonlinear solving, linear algebra, automatic differentiation, and benchmarking.

```julia
using NonlinearSolve, LinearSolve, StaticArrays, Sundials, SpeedMapping,
      BenchmarkTools, LinearAlgebra, DiffEqDevTools, PolyesterForwardDiff, CairoMakie,
      RecursiveFactorization, Enzyme
import MINPACK, NLsolve, PETSc
import LineSearches

const RUS = RadiusUpdateSchemes;
```

--------------------------------

### Load and Solve a Linear ODE Problem

Source: https://docs.sciml.ai/DiffEqDocs/latest/types/ode_types

Demonstrates how to add the DiffEqProblemLibrary, import ODE problems, and solve a linear ODE using DifferentialEquations.jl. This is a basic example for getting started.

```julia
#] add DiffEqProblemLibrary
import DiffEqProblemLibrary.ODEProblemLibrary
import DifferentialEquations as DE
# load problems
ODEProblemLibrary.importodeproblems()
prob = ODEProblemLibrary.prob_ode_linear
sol = DE.solve(prob)
```

--------------------------------

### Setting up the computation device

Source: https://docs.sciml.ai/NeuralOperators/stable/models/fno

Initializes the computation device using Reactant.jl for potential hardware acceleration.

```julia
xdev = reactant_device()

```

--------------------------------

### Complete example for precompilation-friendly components

Source: https://docs.sciml.ai/ModelingToolkit/stable/basics/PrecompileComponents

This example demonstrates a complete structure for a precompilation-friendly component, including variables, parameters, initial conditions, equations, and subsystems, adhering to best practices for performance.

```julia
using ModelingToolkit
using PrecompileTools

@precompile_all ModelingToolkit begin
    @parameters t k
    @variables x(t)
    vars = SymbolicT[]
    params = SymbolicT[]
    push!(vars, @variables x(t))
    push!(params, @parameters k)
    initial_conditions = Dict{SymbolicT, SymbolicT}()
    guesses = Dict{SymbolicT, SymbolicT}()
    initial_conditions[x] = 0.0
    guesses[x] = 0.0
    eqs = Equation[]
    push!(eqs, D(x) ~ k*x)
    subsystems = Vector{System}()
    # Example of creating a simple system
    sys = @system begin
        t
        k
        x(t)
        D(x) ~ k*x
    end
    push!(subsystems, sys)

    # Example of creating a model from components
    # This part would typically involve composing systems
    # For demonstration, we'll just show the structure
    # mtkmodel = @mtkmodel begin ... end
end
```

--------------------------------

### Setup and PDE Definition

Source: https://docs.sciml.ai/SciMLSensitivity/stable/examples/pde/pde_constrained

Imports necessary libraries and defines the parameters, initial conditions, and the differential equations for the Heat Equation. Includes auxiliary functions for spatial derivatives.

```julia
import SciMLSensitivity as SMS
import DelimitedFiles, Plots
import OrdinaryDiffEq as ODE, Optimization as OPT, OptimizationPolyalgorithms as OPA, Zygote

# Problem setup parameters:
Lx = 10.0
x = 0.0:0.01:Lx
dx = x[2] - x[1]
Nx = size(x)

u0 = exp.(-(x .- 3.0) .^ 2) # I.C

## Problem Parameters
p = [1.0, 1.0]    # True solution parameters
const xtrs = [dx, Nx]      # Extra parameters
dt = 0.40 * dx^2    # CFL condition
t0, tMax = 0.0, 1000 * dt
tspan = (t0, tMax)
t = t0:dt:tMax;

## Definition of Auxiliary functions
function ddx(u, dx)
    """
    2nd order Central difference for 1st degree derivative
    """
    return [[zero(eltype(u))]; (u[3:end] - u[1:(end - 2)]) ./ (2.0 * dx); [zero(eltype(u))]]
end

function d2dx(u, dx)
    """
    2nd order Central difference for 2nd degree derivative
    """
    return [zero(eltype(u));
            (@view(u[3:end]) .- 2.0 .* @view(u[2:(end - 1)]) .+ @view(u[1:(end - 2)])) ./
            (dx^2)
            zero(eltype(u))]
end

## ODE description of the Physics:
function heat(u, p, t, xtrs)
    # Model parameters
    a0, a1 = p
    dx, Nx = xtrs #[1.0,3.0,0.125,100]
    return 2.0 * a0 .* u + a1 .* d2dx(u, dx)
end
heat_closure(u, p, t) = heat(u, p, t, xtrs)

# Testing Solver on linear PDE
prob = ODE.ODEProblem(heat_closure, u0, tspan, p)
sol = ODE.solve(prob, ODE.Tsit5(); dt, saveat = t);
arr_sol = Array(sol)

Plots.plot(x, sol.u[1], lw = 3, label = "t0", size = (800, 500))
Plots.plot!(x, sol.u[end], lw = 3, ls = :dash, label = "tMax")
```

--------------------------------

### Install and Import IRKGaussLegendre

Source: https://docs.sciml.ai/DiffEqDocs/latest/solvers/ode_solve

To use the IRKGL16 solver, you need to install and import the IRKGaussLegendre.jl package. This setup is not automatically included with DifferentialEquations.jl.

```julia
import Pkg
Pkg.add("IRKGaussLegendre")
import IRKGaussLegendre
```

--------------------------------

### Install and Import LSODA

Source: https://docs.sciml.ai/DiffEqDocs/latest/solvers/ode_solve

To use the LSODA solver, you must install and import the LSODA.jl package. This setup is not automatically included with DifferentialEquations.jl.

```julia
import Pkg
Pkg.add("LSODA")
import LSODA
```

--------------------------------

### Install and Import ODEInterfaceDiffEq

Source: https://docs.sciml.ai/DiffEqDocs/latest/solvers/ode_solve

To use the ODEInterface algorithms, you need to install and import the ODEInterfaceDiffEq package. This setup is not included by default with DifferentialEquations.jl.

```julia
import Pkg
Pkg.add("ODEInterfaceDiffEq")
import ODEInterfaceDiffEq
```

--------------------------------

### Initial Solver Setup and First Simulation

Source: https://docs.sciml.ai/SciMLBenchmarksOutput/stable/ComplicatedPDE/SpringBlockNonLinearResistance

Configures the solver with KLU factorization and sets absolute and relative tolerances. It then defines the initial conditions and time span, creates an ODEProblem, and solves it using the specified solver and callbacks. The solution is then used to create a TestSolution object.

```julia
solver = KenCarp47(linsolve=KLUFactorization())
abstol = 1e-12
reltol = 1e-8
u0 = get_IC()
tspan = (0.0,1e9)
prob1 = ODEProblem(f,u0,tspan,nothing)
sys = modelingtoolkitize(prob1)
sys_simplified = structural_simplify(sys)
prob_mtk1 = ODEProblem(sys_simplified,[],tspan,jac=true,sparse=true)
global sol,tcpu, bytes, gctime, memallocs = @timed solve(prob_mtk1,solver,reltol = reltol,abstol = abstol,maxiters = Int(1e12),save_everystep = false,dtmin = 1e-20,callback = cb1)
u1 = sol.u[end]
t1 = sol.t[end]
test_sol1 = TestSolution(sol)
```

--------------------------------

### Parameter Estimation Setup

Source: https://docs.sciml.ai/ModelingToolkit/stable/tutorials/dynamic_optimization

Initializes a dynamic optimization problem for parameter estimation with `tune_parameters=true`. Requires a defined system `sys_pe′`, initial conditions `u0map_pe`, and time span `tspan_pe`.

```julia
iprob_pe = InfiniteOptDynamicOptProblem(sys_pe′, u0map_pe, tspan_pe; dt=1/50, tune_parameters=true)
isol_pe = solve(iprob_pe, InfiniteOptCollocation(Ipopt.Optimizer, InfiniteOpt.OrthogonalCollocation(3)))
```

--------------------------------

### Setup and Problem Definition for qmax Determination

Source: https://docs.sciml.ai/SciMLBenchmarksOutput/stable/AdaptiveSDE/qmaxDetermination

Initializes parameters, loads necessary packages, and defines the SDE problem for benchmarking. Includes a compilation run to ensure accurate timing.

```julia
qs = 1.0 .+ 2.0 .^ (-5:2)
times = Array{Float64}(undef, length(qs), 4)
means = Array{Float64}(undef, length(qs), 4)

using StochasticDiffEq, SDEProblemLibrary, Random,
      Plots, ParallelDataTransfer, DiffEqMonteCarlo, Distributed
Random.seed!(99)

full_prob = SDEProblemLibrary.oval2ModelExample(largeFluctuations = true, useBigs = false)
import SDEProblemLibrary: prob_sde_additivesystem,
                          prob_sde_additive, prob_sde_2Dlinear, prob_sde_linear,
                          prob_sde_wave
prob = remake(full_prob, tspan = (0.0, 1.0))

println("Solve once to compile.")
sol = solve(prob, EM(), dt = 1/2^(18))
Int(sol.u[end][1]!=NaN)
println("Compilation complete.")
num_runs = 10000

probs = Vector{SDEProblem}(undef, 3)
p1 = Vector{Any}(undef, 3)
p2 = Vector{Any}(undef, 3)
p3 = Vector{Any}(undef, 3)
## Problem 1
probs[1] = prob_sde_linear
## Problem 2
probs[2] = prob_sde_wave
## Problem 3
probs[3] = prob_sde_additive

println("Setup Complete")
```

--------------------------------

### Full Distributed Batching Example

Source: https://docs.sciml.ai/SciMLSensitivity/stable/tutorials/data_parallel

A complete example of distributed batching for an optimization problem. This setup requires `Distributed.addprocs` and defining necessary functions on all worker processes.

```julia
import Distributed
Distributed.addprocs(4)

Distributed.@everywhere begin
    import OrdinaryDiffEq as ODE
    import Optimization as OPT
    import OptimizationOptimisers as OPO
    function f(u, p, t)
        1.01u .* p
    end
end

pa = [1.0]
u0 = [3.0]
θ = [u0; pa]

function model1(θ, ensemble)
    prob = ODE.ODEProblem(f, [θ[1]], (0.0, 1.0), [θ[2]])

    function prob_func(prob, i, repeat)
        ODE.remake(prob, u0 = 0.5 .+ i / 100 .* prob.u0)
    end

    ensemble_prob = ODE.EnsembleProblem(prob; prob_func)
    sim = ODE.solve(ensemble_prob, ODE.Tsit5(), ensemble, saveat = 0.1, trajectories = 100)
end

callback = function (θ, l) # callback function to observe training
    @show l
    false
end

opt = OPO.Adam(0.1)
loss_distributed(θ) = sum(abs2, 1.0 .- Array(model1(θ, ODE.EnsembleDistributed())))
l1 = loss_distributed(θ)

adtype = OPT.AutoZygote()
optf = OPT.OptimizationFunction((x, p) -> loss_distributed(x), adtype)
optprob = OPT.OptimizationProblem(optf, θ)

res_distributed = OPT.solve(optprob, opt; callback, maxiters = 100)
```

--------------------------------

### Benchmark Setup and Execution

Source: https://docs.sciml.ai/SciMLBenchmarksOutput/stable/OptimizationCUTEst/CUTEst_unconstrained

This code sets up the necessary packages, defines optimizers, and implements functions to run benchmarks on CUTEst problems. It includes logic for handling large problems, timing solutions, and collecting results.

```julia
using Optimization
using OptimizationNLPModels
using CUTEst
using OptimizationOptimJL
using OptimizationOptimisers
using OptimizationOptimJL: LBFGS, ConjugateGradient, NelderMead, SimulatedAnnealing, ParticleSwarm
using Ipopt
using OptimizationMOI
using OptimizationMOI: MOI as MOI
using DataFrames
using Plots
using StatsPlots
using StatsBase: countmap


optimizers = [
    ("LBFGS", LBFGS()),
    ("ConjugateGradient", ConjugateGradient()),
    ("NelderMead", NelderMead()),
    ("SimulatedAnnealing", SimulatedAnnealing()),
    ("ParticleSwarm", ParticleSwarm()),
]

function get_stats(sol, optimizer_name)
    if hasfield(typeof(sol), :stats) && hasfield(typeof(sol.stats), :time)
        solve_time = sol.stats.time
    else
        solve_time = NaN
    end
    return (length(sol.u), solve_time, optimizer_name, Symbol(sol.retcode))
end

function run_benchmarks(problems, optimizers; chunk_size=1)
    problem = String[]
    n_vars = Int64[]
    secs = Float64[]
    solver = String[]
    retcode = Symbol[]
    optz = length(optimizers)
    n = length(problems)
    @info "Processing $(n) problems with $(optz) optimizers in chunks of $(chunk_size)"
    broadcast(c -> sizehint!(c, optz * n), [problem, n_vars, secs, solver, retcode])
    for chunk_start in 1:chunk_size:n
        chunk_end = min(chunk_start + chunk_size - 1, n)
        chunk_problems = problems[chunk_start:chunk_end]
        @info "Processing chunk $(div(chunk_start-1, chunk_size)+1)/$(div(n-1, chunk_size)+1): problems $(chunk_start)-$(chunk_end)"
        for (idx, prob_name) in enumerate(chunk_problems)
            current_problem = chunk_start + idx - 1
            @info "Problem $(current_problem)/$(n): $(prob_name)"
            nlp_prob = nothing
            try
                nlp_prob = CUTEstModel(prob_name)
                if nlp_prob.meta.nvar > 10000
                    @info "  Skipping $(prob_name) (too large: $(nlp_prob.meta.nvar) variables)"
                    finalize(nlp_prob)
                    continue
                end
                prob = OptimizationNLPModels.OptimizationProblem(nlp_prob, Optimization.AutoFiniteDiff())
                for (optimizer_name, optimizer) in optimizers
                    try
                        sol = solve(prob, optimizer; maxiters = 1000, maxtime = 30.0)
                        @info "✓ Solved $(prob_name) with $(optimizer_name) - Status: $(sol.retcode)"
                        vars, time, alg, code = get_stats(sol, optimizer_name)
                        push!(problem, prob_name)
                        push!(n_vars, vars)
                        push!(secs, time)
                        push!(solver, alg)
                        push!(retcode, code)
                    catch e
                        push!(problem, prob_name)
                        push!(n_vars, nlp_prob !== nothing ? nlp_prob.meta.nvar : -1)
                        push!(secs, NaN)
                        push!(solver, optimizer_name)
                        push!(retcode, :FAILED)
                    end
                end
            catch e
                for (optimizer_name, optimizer) in optimizers
                    push!(problem, prob_name)
                    push!(n_vars, -1)
                    push!(secs, NaN)
                    push!(solver, optimizer_name)
                    push!(retcode, :LOAD_FAILED)
                end
            finally
                if nlp_prob !== nothing
                    try
                        finalize(nlp_prob)
                    catch e
                    end
                end
            end
        end
        GC.gc()
        @info "Completed chunk, memory usage cleaned up"
    end
    return DataFrame(problem = problem, n_vars = n_vars, secs = secs, solver = solver, retcode = retcode)
end

unc_problems = collect(CUTEst.select_sif_problems(contype="unc"))
println("Number of problems: ", length(unc_problems))
println("First 5 problems: ", unc_problems[1:min(5, end)])
unc_problems = unc_problems[1:min(50, length(unc_problems))]
println("Limited to ", length(unc_problems), " problems for comprehensive testing")
unc_results =  run_benchmarks(unc_problems, optimizers)
@show unc_results
successful_codes = [:Success, :MaxIters, :MaxTime, :FirstOrderOptimal]
successful_results = filter(row -> row.retcode in successful_codes, unc_results)
total_attempts = nrow(unc_results)

```

--------------------------------

### Get and Set Variable Metadata

Source: https://docs.sciml.ai/ModelingToolkit/stable/basics/ContextualVariables

Demonstrates how to get and set metadata for a variable using `hasmetadata`, `get_unit`, and `setmetadata`. This example shows setting and retrieving the unit for a variable.

```julia
@variables x [unit = u"m^3/s"];

hasmetadata(x, VariableUnit)

ModelingToolkit.get_unit(x)

x = setmetadata(x, VariableUnit, u"m/s")

ModelingToolkit.get_unit(x)
```
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