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 1.10.10
Commit 95f30e51f41 (2025-06-27 09:51 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
  LIBM: libopenlibm
  LLVM: libLLVM-15.0.7 (ORCJIT, znver3)
Threads: 2 default, 0 interactive, 1 GC (on 4 virtual cores)
Environment:
  JULIA_CI = true
  JULIA_CONDAPKG_OFFLINE = true
  LD_LIBRARY_PATH = /opt/hostedtoolcache/Python/3.14.3/x64/lib
  JULIA_PROJECT = /home/runner/work/jl-ude/jl-ude/Project.toml
  JULIA_DEPOT_PATH = /home/runner/.julia:/opt/hostedtoolcache/julia/1.10.10/x64/local/share/julia:/opt/hostedtoolcache/julia/1.10.10/x64/share/julia
  JULIA_CONDAPKG_BACKEND = Null
  JULIA_NUM_THREADS = 2
  JULIA_LOAD_PATH = @:@v#.#:@stdlib

using Pkg
Pkg.status()

Status `~/work/jl-ude/jl-ude/Project.toml`
  [b0b7db55] ComponentArrays v0.15.32
  [aae7a2af] DiffEqFlux v4.6.0
  [5b8099bc] DomainSets v0.7.16
  [7da242da] Enzyme v0.13.129
  [f6369f11] ForwardDiff v1.3.2
  [d3d80556] LineSearches v7.6.0
  [b2108857] Lux v1.31.3
  [961ee093] ModelingToolkit v11.14.0
  [315f7962] NeuralPDE v5.22.0
  [8913a72c] NonlinearSolve v4.16.0
  [7f7a1694] Optimization v5.5.0
  [36348300] OptimizationOptimJL v0.4.10
  [42dfb2eb] OptimizationOptimisers v0.3.16
  [500b13db] OptimizationPolyalgorithms v0.3.5
  [1dea7af3] OrdinaryDiffEq v6.108.0
  [91a5bcdd] Plots v1.41.6
  [ce78b400] SimpleUnPack v1.1.0
  [37e2e46d] LinearAlgebra
  [9a3f8284] Random

This notebook was generated using Literate.jl.