DiffEqFlux: High Level Pre-Built Architectures for Implicit Deep Learning
DiffEqFlux.jl is an implicit deep learning library built using the SciML ecosystem. It is a high-level interface that pulls together all the tools with heuristics and helper functions to make training such deep implicit layer models fast and easy.
DiffEqFlux.jl is only for pre-built architectures and utility functions for deep implicit learning, mixing differential equations with machine learning. For details on automatic differentiation of equation solvers and adjoint techniques, and using these methods for doing things like calibrating models to data, nonlinear optimal control, and PDE-constrained optimization, see SciMLSensitivity.jl.
Pre-Built Architectures
The approach of this package is the easy and efficient training of Neural Ordinary Differential Equations and its variants. DiffEqFlux.jl provides architectures which match the interfaces of machine learning libraries such as Flux.jl and Lux.jl to make it easy to build continuous-time machine learning layers into larger machine learning applications.
The following layer functions exist:
- Neural Ordinary Differential Equations (Neural ODEs)
- Collocation-Based Neural ODEs (Neural ODEs without a solver, by far the fastest way!)
- Multiple Shooting Neural Ordinary Differential Equations
- Neural Stochastic Differential Equations (Neural SDEs)
- Neural Differential-Algebraic Equations (Neural DAEs)
- Neural Delay Differential Equations (Neural DDEs)
- Augmented Neural ODEs
- Hamiltonian Neural Networks (with specialized second order and symplectic integrators)
- Continuous Normalizing Flows (CNF) and FFJORD
Examples of how to build architectures from scratch, with tutorials on things like Graph Neural ODEs, can be found in the SciMLSensitivity.jl documentation.
Flux.jl vs Lux.jl
Both Flux and Lux defined neural networks are supported by DiffEqFlux.jl. However, Lux.jl neural networks are greatly preferred for many correctness reasons. Particularly, a Flux Chain does not respect Julia's type promotion rules. This causes major problems in that the restructuring of a Flux neural network will not respect the chosen types from the solver. Demonstration:
using Flux, Tracker
x = [0.8; 0.8]
ann = Chain(Dense(2, 10, tanh), Dense(10, 1))
p, re = Flux.destructure(ann)
z = re(Float64.(p))While one may think this recreates the neural network to act in Float64 precision, it does not and instead its values will silently downgrade everything to Float32. This is only fixed by Chain(Dense(2, 10, tanh), Dense(10, 1)) |> f64. Similar cases will lead to dropped gradients with complex numbers. This is not an issue with the automatic differentiation library commonly associated with Flux (Zygote.jl) but rather due to choices in the neural network library's decision for how to approach type handling and precision. Thus when using DiffEqFlux.jl with Flux, the user must be very careful to ensure that the precision of the arguments are correct, and anything that requires alternative types (like TrackerAdjoint tracked values and ForwardDiffSensitivity dual numbers) are suspect.
Lux.jl has none of these issues, is simpler to work with due to the parameters in its function calls being explicit rather than implicit global references, and achieves higher performance. It is built on similar foundations as Flux.jl, such as Zygote and NNLib. The better performance comes from not having the overhead of restructure required and fusing common operations in LuxLib.jl. Thus we highly recommend people use Lux instead and only use the Flux fallbacks for legacy code.
Citation
If you use DiffEqFlux.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,
Status `~/_work/DiffEqFlux.jl/DiffEqFlux.jl/docs/Project.toml`
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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`and using this machine and Julia version.
Julia Version 1.12.7
Commit 6d172b025e4 (2026-08-15 08:05 UTC)
Build Info:
Official https://julialang.org release
Platform Info:
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JULIA_CPU_THREADS = 8
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JULIA_DEBUG = DocumenterA more complete overview of all dependencies and their versions is also provided.
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[1d63c593] LLVMOpenMP_jll v22.1.7+0
[ad6e5548] LibTracyClient_jll v0.13.1+0
⌅ [e9f186c6] Libffi_jll v3.4.7+0
[7e76a0d4] Libglvnd_jll v1.7.1+1
[94ce4f54] Libiconv_jll v1.18.0+0
[4b2f31a3] Libmount_jll v2.42.0+0
[89763e89] Libtiff_jll v4.7.3+0
[38a345b3] Libuuid_jll v2.42.0+0
[856f044c] MKL_jll v2025.2.0+0
[7cb0a576] MPICH_jll v5.0.1+0
[f1f71cc9] MPItrampoline_jll v5.5.6+0
[c8ffd9c3] MbedTLS_jll v2.28.1010+0
[9237b28f] MicrosoftMPI_jll v10.1.4+3
[e98f9f5b] NVTX_jll v3.2.2+0
[e7412a2a] Ogg_jll v1.3.6+0
⌅ [fe0851c0] OpenMPI_jll v4.1.9+0
[9bd350c2] OpenSSH_jll v10.5.1+0
[efe28fd5] OpenSpecFun_jll v0.5.6+0
[91d4177d] Opus_jll v1.6.1+0
[36c8627f] Pango_jll v1.58.2+0
[30392449] Pixman_jll v0.46.4+0
[c0090381] Qt6Base_jll v6.10.2+2
[629bc702] Qt6Declarative_jll v6.10.2+2
[ce943373] Qt6ShaderTools_jll v6.10.2+1
[6de9746b] Qt6Svg_jll v6.10.2+0
[e99dba38] Qt6Wayland_jll v6.10.2+1
[f50d1b31] Rmath_jll v0.5.2+0
[a44049a8] Vulkan_Loader_jll v1.3.243+0
[a2964d1f] Wayland_jll v1.24.0+0
⌅ [02c8fc9c] XML2_jll v2.13.9+0
[ffd25f8a] XZ_jll v5.8.3+0
[f67eecfb] Xorg_libICE_jll v1.1.2+0
[c834827a] Xorg_libSM_jll v1.2.6+0
[4f6342f7] Xorg_libX11_jll v1.8.13+0
[0c0b7dd1] Xorg_libXau_jll v1.0.13+0
[935fb764] Xorg_libXcursor_jll v1.2.4+0
[a3789734] Xorg_libXdmcp_jll v1.1.6+0
[1082639a] Xorg_libXext_jll v1.3.8+0
[d091e8ba] Xorg_libXfixes_jll v6.0.2+0
[a51aa0fd] Xorg_libXi_jll v1.8.4+0
[d1454406] Xorg_libXinerama_jll v1.1.7+0
[ec84b674] Xorg_libXrandr_jll v1.5.6+0
[ea2f1a96] Xorg_libXrender_jll v0.9.12+0
[a65dc6b1] Xorg_libpciaccess_jll v0.19.0+0
[c7cfdc94] Xorg_libxcb_jll v1.17.1+0
[cc61e674] Xorg_libxkbfile_jll v1.2.0+0
[e920d4aa] Xorg_xcb_util_cursor_jll v0.1.6+0
[12413925] Xorg_xcb_util_image_jll v0.4.1+0
[2def613f] Xorg_xcb_util_jll v0.4.1+0
[975044d2] Xorg_xcb_util_keysyms_jll v0.4.1+0
[0d47668e] Xorg_xcb_util_renderutil_jll v0.3.10+0
[c22f9ab0] Xorg_xcb_util_wm_jll v0.4.2+0
[35661453] Xorg_xkbcomp_jll v1.4.7+0
[33bec58e] Xorg_xkeyboard_config_jll v2.47.0+2
[c5fb5394] Xorg_xtrans_jll v1.6.0+0
[3161d3a3] Zstd_jll v1.5.7+1
[1e29f10c] demumble_jll v1.3.0+0
[35ca27e7] eudev_jll v3.2.14+0
⌅ [214eeab7] fzf_jll v0.61.1+0
[477f73a3] libaec_jll v1.1.7+0
[a4ae2306] libaom_jll v3.14.1+0
[0ac62f75] libass_jll v0.17.5+0
[1183f4f0] libdecor_jll v0.2.2+0
[8e53e030] libdrm_jll v2.4.134+0
[2db6ffa8] libevdev_jll v1.13.4+0
[f638f0a6] libfdk_aac_jll v2.0.4+0
[36db933b] libinput_jll v1.28.1+0
[b53b4c65] libpng_jll v1.6.58+0
[9a156e7d] libva_jll v2.23.0+0
[f27f6e37] libvorbis_jll v1.3.8+0
[009596ad] mtdev_jll v1.1.7+0
[1317d2d5] oneTBB_jll v2022.3.0+0
⌅ [1270edf5] x264_jll v10164.0.1+0
[dfaa095f] x265_jll v4.1.0+0
[d8fb68d0] xkbcommon_jll v1.13.0+0
[0dad84c5] ArgTools v1.1.2
[56f22d72] Artifacts v1.11.0
[2a0f44e3] Base64 v1.11.0
[ade2ca70] Dates v1.11.0
[8ba89e20] Distributed v1.11.0
[f43a241f] Downloads v1.7.0
[7b1f6079] FileWatching v1.11.0
[9fa8497b] Future v1.11.0
[b77e0a4c] InteractiveUtils v1.11.0
[ac6e5ff7] JuliaSyntaxHighlighting v1.12.0
[4af54fe1] LazyArtifacts v1.11.0
[b27032c2] LibCURL v0.6.4
[76f85450] LibGit2 v1.11.0
[8f399da3] Libdl v1.11.0
[37e2e46d] LinearAlgebra v1.12.0
[56ddb016] Logging v1.11.0
[d6f4376e] Markdown v1.11.0
[a63ad114] Mmap v1.11.0
[ca575930] NetworkOptions v1.3.0
[44cfe95a] Pkg v1.12.1
[de0858da] Printf v1.11.0
[3fa0cd96] REPL v1.11.0
[9a3f8284] Random v1.11.0
[ea8e919c] SHA v0.7.0
[9e88b42a] Serialization v1.11.0
[6462fe0b] Sockets v1.11.0
[2f01184e] SparseArrays v1.12.0
[f489334b] StyledStrings v1.11.0
[4607b0f0] SuiteSparse
[fa267f1f] TOML v1.0.3
[a4e569a6] Tar v1.10.0
[8dfed614] Test v1.11.0
[cf7118a7] UUIDs v1.11.0
[4ec0a83e] Unicode v1.11.0
[e66e0078] CompilerSupportLibraries_jll v1.3.1+2
[deac9b47] LibCURL_jll v8.15.0+0
[e37daf67] LibGit2_jll v1.9.0+0
[29816b5a] LibSSH2_jll v1.11.3+1
[14a3606d] MozillaCACerts_jll v2025.11.4
[4536629a] OpenBLAS_jll v0.3.29+0
[05823500] OpenLibm_jll v0.8.7+0
[458c3c95] OpenSSL_jll v3.5.6+0
[efcefdf7] PCRE2_jll v10.44.0+1
[bea87d4a] SuiteSparse_jll v7.8.3+2
[83775a58] Zlib_jll v1.3.1+2
[8e850b90] libblastrampoline_jll v5.15.0+0
[8e850ede] nghttp2_jll v1.64.0+1
[3f19e933] p7zip_jll v17.7.0+0
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.