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 1.12.6
Commit 15346901f00 (2026-04-09 19:20 UTC)
Build Info:
  Official https://julialang.org release
Platform Info:
  OS: Linux (x86_64-linux-gnu)
  CPU: 4 × AMD EPYC 7763 64-Core Processor
  WORD_SIZE: 64
  LLVM: libLLVM-18.1.7 (ORCJIT, znver3)
  GC: Built with stock GC
Threads: 2 default, 1 interactive, 2 GC (on 4 virtual cores)
Environment:
  JULIA_CPU_TARGET = generic;icelake-server,clone_all;znver3,clone_all
  JULIA_CONDAPKG_OFFLINE = true
  JULIA_CONDAPKG_BACKEND = Null
  JULIA_CI = true
  JULIA_NUM_THREADS = 2

using Pkg
Pkg.status()

Status `~/work/jl-ude/jl-ude/Project.toml`
  [0bf59076] AdvancedHMC v0.8.4
  [b0b7db55] ComponentArrays v0.15.38
  [aae7a2af] DiffEqFlux v4.6.0
  [31c24e10] Distributions v0.25.125
⌅ [5b8099bc] DomainSets v0.7.18
  [7da242da] Enzyme v0.13.147
  [f6369f11] ForwardDiff v1.3.3
  [682c06a0] JSON v1.6.0
  [d3d80556] LineSearches v7.6.2
  [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 v11.26.3
  [315f7962] NeuralPDE v6.0.0
  [8913a72c] NonlinearSolve v4.19.1
  [7f7a1694] Optimization v5.6.1
  [36348300] OptimizationOptimJL v0.4.14
  [42dfb2eb] OptimizationOptimisers v0.3.17
  [500b13db] OptimizationPolyalgorithms v0.3.6
⌃ [1dea7af3] OrdinaryDiffEq v6.111.0
  [91a5bcdd] Plots v1.41.6
  [ce78b400] SimpleUnPack v1.1.0
  [bd369af6] Tables v1.12.1
  [37e2e46d] LinearAlgebra v1.12.0
  [9a3f8284] Random v1.11.0
  [ea8e919c] SHA v0.7.0
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 To see why use `status --outdated`

This notebook was generated using Literate.jl.