UDEs - Julia UDE

Universal Differential Equations (UDEs) are hybrids of differential equations and neural networks.

https://github.com/SciML/DiffEqFlux.jl : fusing differential equations (DifferentialEquations.jl) and neural networks (Lux.jl).
https://github.com/SciML/NeuralPDE.jl : physics-Informed Neural Networks (PINN) Solvers, learning and building the equations from the ground up. NeuralPDE.jl is slower than DiffEqFlux.jl.

Runtime information¶

using InteractiveUtils
InteractiveUtils.versioninfo()

Julia Version 1.13.0
Commit d1c37793dd2 (2026-09-09 19:00 UTC)
Build Info:
  Official https://julialang.org release
Platform Info:
  OS: Linux (aarch64-linux-gnu)
  CPU: 4 × unknown
  WORD_SIZE: 64
  LLVM: libLLVM-20.1.8 (ORCJIT, neoverse-n2)
  GC: Built with stock GC
Threads: 1 default, 0 interactive, 1 GC (on 4 virtual cores)
Environment:
  JULIA_LOAD_PATH = @:@v#.#:@stdlib
  JULIA_DEPOT_PATH = /home/runner/.julia:/opt/hostedtoolcache/julia/1.13.0/aarch64/local/share/julia:/opt/hostedtoolcache/julia/1.13.0/aarch64/share/julia
  JULIA_CI = true
  JULIA_PROJECT = /home/runner/work/jl-ude/jl-ude/Project.toml
  JULIA_NUM_THREADS = 1

using Pkg
Pkg.status()

Status `~/work/jl-ude/jl-ude/Project.toml`
⌃ [0bf59076] AdvancedHMC v0.8.6
⌃ [b0b7db55] ComponentArrays v0.15.44
⌃ [aae7a2af] DiffEqFlux v4.8.0
⌃ [31c24e10] Distributions v0.25.130
⌃ [5b8099bc] DomainSets v0.8.1
⌃ [7da242da] Enzyme v0.13.199
⌃ [f6369f11] ForwardDiff v1.4.5
⌃ [682c06a0] JSON v1.7.1
⌃ [d3d80556] LineSearches v7.7.1
  [98b081ad] Literate v2.21.0
  [6fdf6af0] LogDensityProblems v2.2.0
  [b2108857] Lux v1.31.4
  [c7f686f2] MCMCChains v7.7.0
  [1862ce21] MarkdownTables v1.1.0
⌃ [961ee093] ModelingToolkit v11.39.0
⌃ [315f7962] NeuralPDE v6.2.1
⌃ [8913a72c] NonlinearSolve v4.27.0
⌃ [7f7a1694] Optimization v5.7.0
⌃ [36348300] OptimizationOptimJL v0.4.18
⌃ [42dfb2eb] OptimizationOptimisers v0.3.21
⌃ [500b13db] OptimizationPolyalgorithms v0.3.10
⌃ [1dea7af3] OrdinaryDiffEq v7.6.0
⌃ [91a5bcdd] Plots v1.41.6
  [ce78b400] SimpleUnPack v1.1.0
⌃ [bd369af6] Tables v1.13.0
  [37e2e46d] LinearAlgebra v1.13.0
  [9a3f8284] Random v1.11.0
  [ea8e919c] SHA v1.0.0
Info Packages marked with ⌃ have new versions available and may be upgradable.

This notebook was generated using Literate.jl.