SciMLSensitivity: Automatic Differentiation and Adjoints for (Differential) Equation Solvers

SciMLSensitivity.jl is the automatic differentiation and adjoints system for the SciML ecosystem. Also known as local sensitivity analysis, these methods allow for calculation of fast derivatives of SciML problem types which are commonly used to analyze model sensitivities, calibrate models to data, train neural ODEs, perform automated model discovery via universal differential equations, and more. SciMLSensitivity.jl is a high-level interface that pulls together all the tools with heuristics and helper functions to make solving inverse problems and inferring models as easy as possible without losing efficiency.

Thus, what SciMLSensitivity.jl provides is:

  • Automatic differentiation overloads for improving the performance and flexibility of AD calls over solve.
  • A lower level direct interface for defining forward sensitivity and adjoint problems to allow for minimal overhead and maximal performance.
  • A bunch of tutorials, documentation, and test cases for this combination with parameter estimation (data fitting / model calibration), neural network libraries and GPUs.
Note

This documentation assumes familiarity with the solver packages for the respective problem types. If one is not familiar with the solver packages, please consult the documentation for pieces like DifferentialEquations.jl, NonlinearSolve.jl, LinearSolve.jl, etc. first.

Installation

To install SciMLSensitivity.jl, use the Julia package manager:

using Pkg
Pkg.add("SciMLSensitivity")

High-Level Interface: sensealg

The highest level interface is provided by the function solve:

solve(prob, args...; sensealg = InterpolatingAdjoint(), checkpoints = sol.t, kwargs...)

solve is fully compatible with automatic differentiation libraries like:

and will automatically replace any calculations of the solution's derivative with a fast method. The keyword argument sensealg controls the dispatch to the AbstractSensitivityAlgorithm used for the sensitivity calculation. Note that solve in an AD context does not allow higher order interpolations unless sensealg=DiffEqBase.SensitivityADPassThrough() is used, i.e. going back to the AD mechanism.

Note

The behavior of ForwardDiff.jl is different from the other automatic differentiation libraries mentioned above. The sensealg keyword is ignored. Instead, the differential equations are solved using Dual numbers for u0 and p. If only p is perturbed in the sensitivity analysis, but not u0, the state is still implemented as a Dual number. ForwardDiff.jl will thus not dispatch into continuous forward nor adjoint sensitivity analysis even if a sensealg is provided.

Equation Scope

SciMLSensitivity.jl supports all the equation types of the SciML Common Interface, extending the problem types by adding overloads for automatic differentiation to improve the performance and flexibility of the differentiation system. This includes:

  • Linear systems (LinearProblem)

    • Direct methods for dense and sparse
    • Iterative solvers with preconditioning
  • Nonlinear Systems (NonlinearProblem)

    • Systems of nonlinear equations
    • Scalar bracketing systems
  • Integrals (quadrature) (QuadratureProblem)

  • Differential Equations

    • Discrete equations (function maps, discrete stochastic (Gillespie/Markov) simulations) (DiscreteProblem)
    • Ordinary differential equations (ODEs) (ODEProblem)
    • Split and Partitioned ODEs (Symplectic integrators, IMEX Methods) (SplitODEProblem)
    • Stochastic ordinary differential equations (SODEs or SDEs) (SDEProblem)
    • Stochastic differential-algebraic equations (SDAEs) (SDEProblem with mass matrices)
    • Random differential equations (RODEs or RDEs) (RODEProblem)
    • Differential algebraic equations (DAEs) (DAEProblem and ODEProblem with mass matrices)
    • Delay differential equations (DDEs) (DDEProblem)
    • Neutral, retarded, and algebraic delay differential equations (NDDEs, RDDEs, and DDAEs)
    • Stochastic delay differential equations (SDDEs) (SDDEProblem)
    • Experimental support for stochastic neutral, retarded, and algebraic delay differential equations (SNDDEs, SRDDEs, and SDDAEs)
    • Mixed discrete and continuous equations (Hybrid Equations, Jump Diffusions) (DEProblems with callbacks)
  • Optimization (OptimizationProblem)

    • Nonlinear (constrained) optimization
  • (Stochastic/Delay/Differential-Algebraic) Partial Differential Equations (PDESystem)

    • Finite difference and finite volume methods
    • Interfaces to finite element methods
    • Physics-Informed Neural Networks (PINNs)
    • Integro-Differential Equations
    • Fractional Differential Equations

SciMLSensitivity and Universal Differential Equations

SciMLSensitivity is for universal differential equations, where these can include delays, physical constraints, stochasticity, events, and all other kinds of interesting behavior that shows up in scientific simulations. Neural networks can be all or part of the model. They can be around the differential equation, in the cost function, or inside the differential equation. Neural networks representing unknown portions of the model or functions can go anywhere you have uncertainty in the form of the scientific simulator. Forward sensitivity and adjoint equations are automatically generated with checkpointing and stabilization to ensure it works for large stiff equations, while specializations on static objects allows for high efficiency on small equations. For an overview of the topic with applications, consult the paper Universal Differential Equations for Scientific Machine Learning.

You can efficiently use the package for:

  • Parameter estimation of scientific models (ODEs, SDEs, DDEs, DAEs, etc.)
  • Neural ODEs, Neural SDE, Neural DAEs, Neural DDEs, etc.
  • Nonlinear optimal control, including training neural controllers
  • (Stiff) universal ordinary differential equations (universal ODEs)
  • Universal stochastic differential equations (universal SDEs)
  • Universal delay differential equations (universal DDEs)
  • Universal partial differential equations (universal PDEs)
  • Universal jump stochastic differential equations (universal jump diffusions)
  • Hybrid universal differential equations (universal DEs with event handling)

with high order, adaptive, implicit, GPU-accelerated, Newton-Krylov, etc. methods. For examples, please refer to the DiffEqFlux release blog post (which we try to keep updated for changes to the libraries). Additional demonstrations, like neural PDEs and neural jump SDEs, can be found at this blog post (among many others!). All these features are only part of the advantage, as this library routinely benchmarks orders of magnitude faster than competing libraries like torchdiffeq. Use with GPUs is highly optimized by recompiling the solvers to GPUs to remove all CPU-GPU data transfers, while use with CPUs uses specialized kernels for accelerating differential equation solves.

Many training techniques are supported by this package, including:

all while mixing forward mode and reverse mode approaches as appropriate for the most speed. For more details on the adjoint sensitivity analysis methods for computing fast gradients, see the adjoints details page.

With this package, you can explore various ways to integrate the two methodologies:

  • Neural networks can be defined where the “activations” are nonlinear functions described by differential equations
  • Neural networks can be defined where some layers are ODE solves
  • ODEs can be defined where some terms are neural networks
  • Cost functions on ODEs can define neural networks

Note on Modularity and Composability with Solvers

Note that SciMLSensitivity.jl purely built on composable and modular infrastructure. SciMLSensitivity provides high-level helper functions and documentation for the user, but the code generation stack is modular and composes in many ways. For example, one can use and swap out the ODE solver between any common interface compatible library, like:

In addition, due to the composability of the system, none of the components are directly tied to the Lux.jl machine learning framework. For example, you can use SciMLSensitivity.jl to generate TensorFlow graphs and train the neural network with TensorFlow.jl, use PyTorch arrays via Torch.jl, and more all with single line code changes by utilizing the underlying code generation. The tutorials shown here are thus mostly a guide on how to use the ecosystem as a whole, only showing a small snippet of the possible ways to compose the thousands of differentiable libraries together! Swap out ODEs for SDEs, DDEs, DAEs, etc., put quadrature libraries or Tullio.jl in the loss function, the world is your oyster!

As a proof of composability, note that the implementation of Bayesian neural ODEs required zero code changes to the library, and instead just relied on the composability with other Julia packages.

Contributing

Citation

If you use SciMLSensitivity.jl or are influenced by its ideas, please cite:

@article{rackauckas2020universal,
  title={Universal differential equations for scientific machine learning},
  author={Rackauckas, Christopher and Ma, Yingbo and Martensen, Julius and Warner, Collin and Zubov, Kirill and Supekar, Rohit and Skinner, Dominic and Ramadhan, Ali},
  journal={arXiv preprint arXiv:2001.04385},
  year={2020}
}

Reproducibility

The documentation of this SciML package was built using these direct dependencies,
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Info Packages marked with  have new versions available but compatibility constraints restrict them from upgrading. To see why use `status --outdated`
and using this machine and Julia version.
Julia Version 1.10.2
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Build Info:
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  [d5829a12] TriangularSolve v0.1.20
  [410a4b4d] Tricks v0.1.8
  [781d530d] TruncatedStacktraces v1.4.0
  [5c2747f8] URIs v1.5.1
  [3a884ed6] UnPack v1.0.2
  [1cfade01] UnicodeFun v0.4.1
  [1986cc42] Unitful v1.19.0
  [45397f5d] UnitfulLatexify v1.6.3
  [013be700] UnsafeAtomics v0.2.1
  [d80eeb9a] UnsafeAtomicsLLVM v0.1.3
  [41fe7b60] Unzip v0.2.0
  [3d5dd08c] VectorizationBase v0.21.65
  [33b4df10] VectorizedRNG v0.2.25
  [19fa3120] VertexSafeGraphs v0.2.0
  [d49dbf32] WeightInitializers v0.1.7
  [e88e6eb3] Zygote v0.6.69
  [700de1a5] ZygoteRules v0.2.5
  [02a925ec] cuDNN v1.3.0
  [6e34b625] Bzip2_jll v1.0.8+1
 [4ee394cb] CUDA_Driver_jll v0.7.0+1
 [76a88914] CUDA_Runtime_jll v0.11.1+0
 [62b44479] CUDNN_jll v8.9.4+0
  [83423d85] Cairo_jll v1.18.0+1
  [7cc45869] Enzyme_jll v0.0.102+0
  [2702e6a9] EpollShim_jll v0.0.20230411+0
  [2e619515] Expat_jll v2.5.0+0
 [b22a6f82] FFMPEG_jll v4.4.4+1
  [a3f928ae] Fontconfig_jll v2.13.93+0
  [d7e528f0] FreeType2_jll v2.13.1+0
  [559328eb] FriBidi_jll v1.0.10+0
  [0656b61e] GLFW_jll v3.3.9+0
  [d2c73de3] GR_jll v0.73.3+0
  [78b55507] Gettext_jll v0.21.0+0
  [f8c6e375] Git_jll v2.44.0+1
  [7746bdde] Glib_jll v2.80.0+0
  [3b182d85] Graphite2_jll v1.3.14+0
  [2e76f6c2] HarfBuzz_jll v2.8.1+1
  [1d5cc7b8] IntelOpenMP_jll v2024.0.2+0
  [aacddb02] JpegTurbo_jll v3.0.2+0
  [9c1d0b0a] JuliaNVTXCallbacks_jll v0.2.1+0
  [c1c5ebd0] LAME_jll v3.100.1+0
  [88015f11] LERC_jll v3.0.0+1
  [dad2f222] LLVMExtra_jll v0.0.29+0
  [1d63c593] LLVMOpenMP_jll v15.0.7+0
  [dd4b983a] LZO_jll v2.10.1+0
 [e9f186c6] Libffi_jll v3.2.2+1
  [d4300ac3] Libgcrypt_jll v1.8.7+0
  [7e76a0d4] Libglvnd_jll v1.6.0+0
  [7add5ba3] Libgpg_error_jll v1.42.0+0
  [94ce4f54] Libiconv_jll v1.17.0+0
  [4b2f31a3] Libmount_jll v2.39.3+0
 [89763e89] Libtiff_jll v4.5.1+1
  [38a345b3] Libuuid_jll v2.39.3+1
  [856f044c] MKL_jll v2024.0.0+0
  [079eb43e] NLopt_jll v2.7.1+0
  [e98f9f5b] NVTX_jll v3.1.0+2
  [e7412a2a] Ogg_jll v1.3.5+1
  [458c3c95] OpenSSL_jll v3.0.13+0
  [efe28fd5] OpenSpecFun_jll v0.5.5+0
  [91d4177d] Opus_jll v1.3.2+0
  [30392449] Pixman_jll v0.42.2+0
  [c0090381] Qt6Base_jll v6.5.3+1
  [f50d1b31] Rmath_jll v0.4.0+0
 [fb77eaff] Sundials_jll v5.2.2+0
  [a44049a8] Vulkan_Loader_jll v1.3.243+0
  [a2964d1f] Wayland_jll v1.21.0+1
  [2381bf8a] Wayland_protocols_jll v1.31.0+0
  [02c8fc9c] XML2_jll v2.12.5+0
  [aed1982a] XSLT_jll v1.1.34+0
  [ffd25f8a] XZ_jll v5.6.1+0
  [f67eecfb] Xorg_libICE_jll v1.0.10+1
  [c834827a] Xorg_libSM_jll v1.2.3+0
  [4f6342f7] Xorg_libX11_jll v1.8.6+0
  [0c0b7dd1] Xorg_libXau_jll v1.0.11+0
  [935fb764] Xorg_libXcursor_jll v1.2.0+4
  [a3789734] Xorg_libXdmcp_jll v1.1.4+0
  [1082639a] Xorg_libXext_jll v1.3.4+4
  [d091e8ba] Xorg_libXfixes_jll v5.0.3+4
  [a51aa0fd] Xorg_libXi_jll v1.7.10+4
  [d1454406] Xorg_libXinerama_jll v1.1.4+4
  [ec84b674] Xorg_libXrandr_jll v1.5.2+4
  [ea2f1a96] Xorg_libXrender_jll v0.9.10+4
  [14d82f49] Xorg_libpthread_stubs_jll v0.1.1+0
  [c7cfdc94] Xorg_libxcb_jll v1.15.0+0
  [cc61e674] Xorg_libxkbfile_jll v1.1.2+0
  [e920d4aa] Xorg_xcb_util_cursor_jll v0.1.4+0
  [12413925] Xorg_xcb_util_image_jll v0.4.0+1
  [2def613f] Xorg_xcb_util_jll v0.4.0+1
  [975044d2] Xorg_xcb_util_keysyms_jll v0.4.0+1
  [0d47668e] Xorg_xcb_util_renderutil_jll v0.3.9+1
  [c22f9ab0] Xorg_xcb_util_wm_jll v0.4.1+1
  [35661453] Xorg_xkbcomp_jll v1.4.6+0
  [33bec58e] Xorg_xkeyboard_config_jll v2.39.0+0
  [c5fb5394] Xorg_xtrans_jll v1.5.0+0
  [3161d3a3] Zstd_jll v1.5.5+0
  [35ca27e7] eudev_jll v3.2.9+0
  [214eeab7] fzf_jll v0.43.0+0
  [1a1c6b14] gperf_jll v3.1.1+0
  [a4ae2306] libaom_jll v3.4.0+0
  [0ac62f75] libass_jll v0.15.1+0
  [2db6ffa8] libevdev_jll v1.11.0+0
  [f638f0a6] libfdk_aac_jll v2.0.2+0
  [36db933b] libinput_jll v1.18.0+0
  [b53b4c65] libpng_jll v1.6.43+1
  [f27f6e37] libvorbis_jll v1.3.7+1
  [009596ad] mtdev_jll v1.1.6+0
  [1270edf5] x264_jll v2021.5.5+0
  [dfaa095f] x265_jll v3.5.0+0
  [d8fb68d0] xkbcommon_jll v1.4.1+1
  [0dad84c5] ArgTools v1.1.1
  [56f22d72] Artifacts
  [2a0f44e3] Base64
  [ade2ca70] Dates
  [8ba89e20] Distributed
  [f43a241f] Downloads v1.6.0
  [7b1f6079] FileWatching
  [9fa8497b] Future
  [b77e0a4c] InteractiveUtils
  [4af54fe1] LazyArtifacts
  [b27032c2] LibCURL v0.6.4
  [76f85450] LibGit2
  [8f399da3] Libdl
  [37e2e46d] LinearAlgebra
  [56ddb016] Logging
  [d6f4376e] Markdown
  [a63ad114] Mmap
  [ca575930] NetworkOptions v1.2.0
  [44cfe95a] Pkg v1.10.0
  [de0858da] Printf
  [9abbd945] Profile
  [3fa0cd96] REPL
  [9a3f8284] Random
  [ea8e919c] SHA v0.7.0
  [9e88b42a] Serialization
  [1a1011a3] SharedArrays
  [6462fe0b] Sockets
  [2f01184e] SparseArrays v1.10.0
  [10745b16] Statistics v1.10.0
  [4607b0f0] SuiteSparse
  [fa267f1f] TOML v1.0.3
  [a4e569a6] Tar v1.10.0
  [8dfed614] Test
  [cf7118a7] UUIDs
  [4ec0a83e] Unicode
  [e66e0078] CompilerSupportLibraries_jll v1.1.0+0
  [deac9b47] LibCURL_jll v8.4.0+0
  [e37daf67] LibGit2_jll v1.6.4+0
  [29816b5a] LibSSH2_jll v1.11.0+1
  [c8ffd9c3] MbedTLS_jll v2.28.2+1
  [14a3606d] MozillaCACerts_jll v2023.1.10
  [4536629a] OpenBLAS_jll v0.3.23+4
  [05823500] OpenLibm_jll v0.8.1+2
  [efcefdf7] PCRE2_jll v10.42.0+1
  [bea87d4a] SuiteSparse_jll v7.2.1+1
  [83775a58] Zlib_jll v1.2.13+1
  [8e850b90] libblastrampoline_jll v5.8.0+1
  [8e850ede] nghttp2_jll v1.52.0+1
  [3f19e933] p7zip_jll v17.4.0+2
Info Packages marked with  and  have new versions available. Those with  may be upgradable, but those with  are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m`

You can also download the manifest file and the project file.