DeepEquilibriumNetworks.jl

DeepEquilibriumNetworks.jl is a framework built on top of DifferentialEquations.jl and Lux.jl, enabling the efficient training and inference for Deep Equilibrium Networks (Infinitely Deep Neural Networks).

Installation

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

using Pkg
Pkg.add("DeepEquilibriumNetworks")

Quick-start

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

DeepEquilibriumSolution
 * Initial Guess: Float32[0.0 0.0 0.0; 0.0 0.0 0.0]
 * Steady State: Float32[-0.219407 0.637479 -0.744241; 1.50412 -0.962679 3.12426]
 * Residual: Float32[-2.98023f-8 0.0 0.0; -1.19209f-7 -5.96046f-8 -2.38419f-7]
 * Jacobian Loss: 0.0
 * NFE: 1

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

(layer_1 = (weight = Float32[-4.7019806 -8.232476; -13.165341 -20.534285], bias = Float32[-11.845193, -31.843906]), layer_2 = (model = (layer_1 = (weight = Float32[15.302854 -58.90247; -21.836525 81.743744],), layer_2 = (weight = Float32[17.714966 3.570962; -23.407288 -5.476556],)), init = nothing))

Citation

If you are using this project for research or other academic purposes, consider citing our paper:

@article{pal2022continuous,
  title={Continuous Deep Equilibrium Models: Training Neural ODEs Faster by Integrating Them to Infinity},
  author={Pal, Avik and Edelman, Alan and Rackauckas, Christopher},
  booktitle={2023 IEEE High Performance Extreme Computing Conference (HPEC)}, 
  year={2023}
}

For specific algorithms, check the respective documentations and cite the corresponding papers.

Contributing

Please refer to the SciML ColPrac: Contributor's Guide on Collaborative Practices for Community Packages for guidance on PRs, issues, and other matters relating to contributing to SciML.
See the SciML Style Guide for common coding practices and other style decisions.
There are a few community forums:
- The #diffeq-bridged and #sciml-bridged channels in the Julia Slack
- The #diffeq-bridged and #sciml-bridged channels in the Julia Zulip
- On the Julia Discourse forums
- See also SciML Community page

Reproducibility

The documentation of this SciML package was built using these direct dependencies,

Status `~/_work/DeepEquilibriumNetworks.jl/DeepEquilibriumNetworks.jl/docs/Project.toml`
  [6748aba7] DeepEquilibriumNetworks v2.8.0 `~/_work/DeepEquilibriumNetworks.jl/DeepEquilibriumNetworks.jl`
  [e30172f5] Documenter v1.17.0
  [daee34ce] DocumenterCitations v1.5.0
  [7ed4a6bd] LinearSolve v5.14.1
  [b2108857] Lux v1.31.4
  [d0bbae9a] LuxCUDA v0.3.6
  [eb30cadb] MLDatasets v0.7.21
  [f1d291b0] MLUtils v0.4.13
  [8913a72c] NonlinearSolve v4.28.1
  [0b1bfda6] OneHotArrays v0.2.11
  [3bd65402] Optimisers v0.4.9
  [1dea7af3] OrdinaryDiffEq v7.8.1
  [89bda076] OrdinaryDiffEqAdamsBashforthMoulton v2.2.0
  [1ed8b502] SciMLSensitivity v7.119.1
  [efcf1570] Setfield v1.1.2
  [9672c7b4] SteadyStateDiffEq v2.16.0
  [e88e6eb3] Zygote v0.7.12
  [ade2ca70] Dates v1.11.0
  [de0858da] Printf v1.11.0
  [9a3f8284] Random v1.11.0

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:
  OS: Linux (x86_64-linux-gnu)
  CPU: 112 × AMD EPYC 7663 56-Core Processor
  WORD_SIZE: 64
  LLVM: libLLVM-18.1.7 (ORCJIT, znver3)
  GC: Built with stock GC
Threads: 1 default, 1 interactive, 1 GC (on 4 virtual cores)
Environment:
  JULIA_CPU_THREADS = 4
  JULIA_NUM_PRECOMPILE_TASKS = 4

A more complete overview of all dependencies and their versions is also provided.

Status `~/_work/DeepEquilibriumNetworks.jl/DeepEquilibriumNetworks.jl/docs/Manifest.toml`
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  [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.