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‎README.md

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[![Documentation and Deployment](https://badge.buildkite.com/b4720f5f3a8d2d1ac8bbaebf5e0e38aaae28b6c8fb436ba05e.svg)](https://buildkite.com/julialang/scimldocs?branch=main)
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Attempts at a single documentation for the full SciML interface.
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‎docs/make.jl

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using Plots
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mathengine = MathJax3(Dict(:loader => Dict("load" => ["[tex]/require", "[tex]/mathtools"]),
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:tex => Dict("inlineMath" => [["\$", "\$"], ["\\(", "\\)"]],
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"packages" => [
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"base",
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"ams",
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"autoload",
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"mathtools",
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"require",
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])))
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:tex => Dict("inlineMath" => [["\$", "\$"], ["\\(", "\\)"]],
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"packages" => [
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"base",
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"ams",
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"autoload",
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"mathtools",
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"require"
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])))
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makedocs(sitename = "Overview of Julia's SciML",
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authors = "Chris Rackauckas",
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modules = Module[],
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clean = true, doctest = false, linkcheck = true,
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linkcheck_ignore = ["https://twitter.com/ChrisRackauckas/status/1477274812460449793",
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"https://epubs.siam.org/doi/10.1137/0903023",
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"https://bkamins.github.io/julialang/2020/12/24/minilanguage.html",
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"https://arxiv.org/abs/2109.06786",
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"https://arxiv.org/abs/2001.04385",
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],
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format = Documenter.HTML(assets = ["assets/favicon.ico"],
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canonical = "https://docs.sciml.ai/stable/",
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mathengine = mathengine),
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pages = [
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"SciML: Open Source Software for Scientific Machine Learning with Julia" => "index.md",
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"Getting Started" => [
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"getting_started/getting_started.md",
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"New User Tutorials" => [
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"getting_started/installation.md",
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"getting_started/first_simulation.md",
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"getting_started/first_optimization.md",
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"getting_started/fit_simulation.md",
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"getting_started/find_root.md",
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],
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"Comparison With Other Tools" => [
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"comparisons/python.md",
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"comparisons/matlab.md",
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"comparisons/r.md",
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"comparisons/cppfortran.md",
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],
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],
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"Showcase of Cool Examples" => Any["showcase/showcase.md",
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"Automated Model Discovery" => Any["showcase/missing_physics.md",
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"showcase/bayesian_neural_ode.md",
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"showcase/blackhole.md"],
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"Solving Difficult Equations Efficiently" => Any["showcase/brusselator.md",
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"showcase/pinngpu.md",
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"showcase/massively_parallel_gpu.md",
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"showcase/gpu_spde.md"],
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"Useful Cool Wonky Things" => Any["showcase/ode_types.md",
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"showcase/symbolic_analysis.md",
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"showcase/optimization_under_uncertainty.md"]],
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"What is SciML?" => ["overview.md",
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"Solvers" => ["highlevels/equation_solvers.md",
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"highlevels/inverse_problems.md",
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"highlevels/partial_differential_equation_solvers.md"],
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"Modeling Tools" => ["highlevels/modeling_languages.md",
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"highlevels/model_libraries_and_importers.md",
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"highlevels/symbolic_tools.md",
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"highlevels/array_libraries.md"],
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"Simulation Analysis" => ["highlevels/parameter_analysis.md",
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"highlevels/uncertainty_quantification.md",
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"highlevels/plots_visualization.md"],
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"Machine Learning" => ["highlevels/function_approximation.md",
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"highlevels/implicit_layers.md",
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"highlevels/symbolic_learning.md"],
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"Developer Tools" => ["highlevels/numerical_utilities.md",
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"highlevels/interfaces.md",
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"highlevels/developer_documentation.md"],
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"Extra Learning Resources" => ["highlevels/learning_resources.md"],
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]])
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authors = "Chris Rackauckas",
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modules = Module[],
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clean = true, doctest = false, linkcheck = true,
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linkcheck_ignore = ["https://twitter.com/ChrisRackauckas/status/1477274812460449793",
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"https://epubs.siam.org/doi/10.1137/0903023",
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"https://bkamins.github.io/julialang/2020/12/24/minilanguage.html",
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"https://arxiv.org/abs/2109.06786",
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"https://arxiv.org/abs/2001.04385"],
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format = Documenter.HTML(assets = ["assets/favicon.ico"],
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canonical = "https://docs.sciml.ai/stable/",
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mathengine = mathengine),
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pages = [
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"SciML: Open Source Software for Scientific Machine Learning with Julia" => "index.md",
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"Getting Started" => [
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"getting_started/getting_started.md",
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"New User Tutorials" => [
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"getting_started/installation.md",
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"getting_started/first_simulation.md",
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"getting_started/first_optimization.md",
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"getting_started/fit_simulation.md",
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"getting_started/find_root.md"
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],
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"Comparison With Other Tools" => [
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"comparisons/python.md",
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"comparisons/matlab.md",
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"comparisons/r.md",
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"comparisons/cppfortran.md"
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]
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],
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"Showcase of Cool Examples" => Any["showcase/showcase.md",
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"Automated Model Discovery" => Any["showcase/missing_physics.md",
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"showcase/bayesian_neural_ode.md",
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"showcase/blackhole.md"],
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"Solving Difficult Equations Efficiently" => Any["showcase/brusselator.md",
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"showcase/pinngpu.md",
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"showcase/massively_parallel_gpu.md",
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"showcase/gpu_spde.md"],
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"Useful Cool Wonky Things" => Any["showcase/ode_types.md",
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"showcase/symbolic_analysis.md",
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"showcase/optimization_under_uncertainty.md"]],
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"What is SciML?" => ["overview.md",
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"Solvers" => ["highlevels/equation_solvers.md",
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"highlevels/inverse_problems.md",
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"highlevels/partial_differential_equation_solvers.md"],
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"Modeling Tools" => ["highlevels/modeling_languages.md",
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"highlevels/model_libraries_and_importers.md",
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"highlevels/symbolic_tools.md",
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"highlevels/array_libraries.md"],
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"Simulation Analysis" => ["highlevels/parameter_analysis.md",
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"highlevels/uncertainty_quantification.md",
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"highlevels/plots_visualization.md"],
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"Machine Learning" => ["highlevels/function_approximation.md",
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"highlevels/implicit_layers.md",
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"highlevels/symbolic_learning.md"],
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"Developer Tools" => ["highlevels/numerical_utilities.md",
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"highlevels/interfaces.md",
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"highlevels/developer_documentation.md"],
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"Extra Learning Resources" => ["highlevels/learning_resources.md"]
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]])
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deploydocs(repo = "github.com/SciML/SciMLDocs")

‎docs/src/comparisons/matlab.md

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|:------------------- |:------------------------------------------------------------------------------------- |
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| `plot` | [Plots](https://docs.juliaplots.org/stable/), [Makie](https://docs.makie.org/stable/) |
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| `sparse` | [SparseArrays](https://docs.julialang.org/en/v1/stdlib/SparseArrays/#Sparse-Arrays) |
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| `interp1` | [DataInterpolations](https://docs.sciml.ai/DataInterpolations/) |
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| `\`, `gmres`, `cg` | [LinearSolve](http://linearsolve.sciml.ai/dev/) |
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| `interp1` | [DataInterpolations](https://docs.sciml.ai/DataInterpolations/) |
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| `\`, `gmres`, `cg` | [LinearSolve](https://linearsolve.sciml.ai/dev/) |
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| `fsolve` | [NonlinearSolve](https://nonlinearsolve.sciml.ai/) |
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| `quad` | [Integrals](https://docs.sciml.ai/Integrals/stable/) |
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| `quad` | [Integrals](https://docs.sciml.ai/Integrals/stable/) |
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| `fmincon` | [Optimization](https://optimization.sciml.ai/) |
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| `odeXX` | [DifferentialEquations](https://diffeq.sciml.ai/latest/) |
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| `ode45` | `Tsit5` |

‎docs/src/comparisons/python.md

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|:---------------------------- |:----------------------------------------------------------------------------------------------------------------------------------------------------------------------- |
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| Matplotlib | [Plots](https://docs.juliaplots.org/stable/), [Makie](https://docs.makie.org/stable/) |
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| `scipy.special` | [SpecialFunctions](https://github.com/JuliaMath/SpecialFunctions.jl) |
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| `scipy.linalg.solve` | [LinearSolve](http://linearsolve.sciml.ai/dev/) |
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| `scipy.linalg.solve` | [LinearSolve](https://linearsolve.sciml.ai/dev/) |
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| `scipy.integrate` | [Integrals](https://docs.sciml.ai/Integrals/stable/) |
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| `scipy.optimize` | [Optimization](https://optimization.sciml.ai/) |
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| `scipy.optimize.fsolve` | [NonlinearSolve](https://nonlinearsolve.sciml.ai/) |
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| `scipy.sparse` | [SparseArrays](https://docs.julialang.org/en/v1/stdlib/SparseArrays/#Sparse-Arrays), [ARPACK](https://github.com/JuliaLinearAlgebra/Arpack.jl) |
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| `odeint`/`solve_ivp` | [DifferentialEquations](https://diffeq.sciml.ai/latest/) |
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| `scipy.integrate.solve_bvp` | [Boundary-value problem](https://diffeq.sciml.ai/latest/tutorials/bvp_example/#Boundary-Value-Problems) |
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| PyTorch | [Flux](https://fluxml.ai/), [Lux](http://lux.csail.mit.edu/stable/) |
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| PyTorch | [Flux](https://fluxml.ai/), [Lux](https://lux.csail.mit.edu/stable/) |
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| gillespy2 | [Catalyst](https://catalyst.sciml.ai/dev/), [JumpProcesses](https://github.com/SciML/JumpProcesses.jl) |
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| scipy.optimize.approx_fprime | [FiniteDiff](https://github.com/JuliaDiff/FiniteDiff.jl) |
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| autograd | [ForwardDiff\*](https://github.com/JuliaDiff/ForwardDiff.jl), [Enzyme\*](https://github.com/EnzymeAD/Enzyme.jl), [DiffEqSensitivity](https://sensitivity.sciml.ai/dev/) |

‎docs/src/getting_started/find_root.md

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sol = solve(prob, NewtonRaphson())
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# Analyze the solution
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@show sol[[x,y,z]], sol.resid
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@show sol[[x, y, z]], sol.resid
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```
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## Step-by-Step Solution
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```@example first_rootfind
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@show sol[[x, y, z]], sol.resid
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```

‎docs/src/getting_started/first_optimization.md

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The following parts of the SciML Ecosystem will be used in this tutorial:
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| Module | Description |
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|:---------------------------------------------------------------------------------------------- |:---------------------------------- |
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| [Optimization.jl](https://docs.sciml.ai/Optimization/stable/) | The numerical optimization package |
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| [OptimizationNLopt.jl](https://docs.sciml.ai/Optimization/stable/optimization_packages/nlopt/) | The NLopt optimizers we will use |
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| [ForwardDiff.jl](https://docs.sciml.ai/Optimization/stable/API/optimization_function/#Optimization.AutoForwardDiff) | The automatic differentiation library for gradients |
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| Module | Description |
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|:------------------------------------------------------------------------------------------------------------------- |:--------------------------------------------------- |
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| [Optimization.jl](https://docs.sciml.ai/Optimization/stable/) | The numerical optimization package |
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| [OptimizationNLopt.jl](https://docs.sciml.ai/Optimization/stable/optimization_packages/nlopt/) | The NLopt optimizers we will use |
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| [ForwardDiff.jl](https://docs.sciml.ai/Optimization/stable/API/optimization_function/#Optimization.AutoForwardDiff) | The automatic differentiation library for gradients |
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## Problem Setup
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- [Optimization.jl](https://docs.sciml.ai/Optimization/stable/), the optimization interface.
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- [OptimizationNLopt.jl](https://docs.sciml.ai/Optimization/stable/optimization_packages/nlopt/), the optimizers we will use.
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- [ForwardDiff.jl](https://docs.sciml.ai/Optimization/stable/API/optimization_function/#Optimization.AutoForwardDiff),
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- [ForwardDiff.jl](https://docs.sciml.ai/Optimization/stable/API/optimization_function/#Optimization.AutoForwardDiff),
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the automatic differentiation library for gradients
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Note that Optimization.jl is an interface for optimizers, and thus we always have to choose
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which optimizer we want to use. Here we choose to demonstrate `OptimizationNLopt` because
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of its efficiency and versatility. But there are many other possible choices. Check out
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package for creating the gradient and other derivatives required by the optimizer.
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‎docs/src/getting_started/first_simulation.md

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differential equation but an algebraic equation, and thus we call this set of equations a
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Differential-Algebraic Equation (DAE). The symbolic system of ModelingToolkit can eliminate
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Notice that in our equations we have an algebraic equation `z ~ x + y`. This is not a
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differential equation but an algebraic equation, and thus we call this set of equations a
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Differential-Algebraic Equation (DAE). The symbolic system of ModelingToolkit can eliminate
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such equations to return simpler forms to numerically approximate.
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Notice that what is returned is an `ODESystem`, but now with the simplified set of

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