DataDrivenDiffEq.jl

DataDrivenDiffEq.jl is a package for finding systems of equations automatically from a dataset.

The methods in this package take in data and return the model which generated the data. A known model is not required as input. These methods can estimate equation-free and equation-based models for discrete, continuous differential equations or direct mappings.

There are two main types of estimation, depending on if you need the result to be human-understandable:

Structural identification - returns a human-readable result in symbolic form.
Structural estimation - returns a function that predicts the derivative and generates a correct time series, but is not necessarily human-readable.

A quick-start example:

using DataDrivenDiffEq
using ModelingToolkit
using OrdinaryDiffEq
using DataDrivenSparse
using LinearAlgebra

# Create a test problem
function lorenz(u, p, t)
    x, y, z = u

    ẋ = 10.0 * (y - x)
    ẏ = x * (28.0 - z) - y
    ż = x * y - (8 / 3) * z
    return [ẋ, ẏ, ż]
end

u0 = [1.0; 0.0; 0.0]
tspan = (0.0, 100.0)
dt = 0.1
prob = ODEProblem(lorenz, u0, tspan)
sol = solve(prob, Tsit5(), saveat = dt)

## Start the automatic discovery
ddprob = DataDrivenProblem(sol)

@variables t x(t) y(t) z(t)
u = [x; y; z]
basis = Basis(polynomial_basis(u, 5), u, iv = t)
opt = STLSQ(exp10.(-5:0.1:-1))
ddsol = solve(ddprob, basis, opt, options = DataDrivenCommonOptions(digits = 1))
println(get_basis(ddsol))

Model ##Basis#306 with 3 equations
States : x(t) y(t) z(t)
Parameters : 7
Independent variable: t
Equations
Differential(t, 1)(x(t)) = p₁*x(t) + p₂*y(t)
Differential(t, 1)(y(t)) = p₃*x(t) + p₄*y(t) + p₅*x(t)*z(t)
Differential(t, 1)(z(t)) = p₇*z(t) + p₆*x(t)*y(t)

Installation

To use DataDrivenDiffEq.jl, install via:

using Pkg
Pkg.add("DataDrivenDiffEq")

Package Overview

Several algorithms for structural estimation and identification are implemented in the following subpackages.

Koopman Based Inference

Uses Dynamic Mode Decomposition (DMD) and Extended Dynamic Mode Decomposition (EDMD) on discrete and continuous differential equations to infer an approximation of the corresponding Koopman operator (discrete case) or generator (continuous case).

To use this functionality, install DataDrivenDMD via:

using Pkg
Pkg.add("DataDrivenDMD")

Sparse Regression

Uses Sparse Regression algorithms to find a suitable and sparse combination of basis functions to approximate a system of (differential) equations.

To use this functionality, install DataDrivenSparse via:

using Pkg
Pkg.add("DataDrivenSparse")

Symbolic Regression

Uses SymbolicRegression.jl to find a suitable set of equations to match the data.

To use this functionality, install DataDrivenSR via:

using Pkg
Pkg.add("DataDrivenSR")

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. You can message @AlCap23 to start a discussion.
- 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/DataDrivenDiffEq.jl/DataDrivenDiffEq.jl/docs/Project.toml`
  [3c9adf31] DataDrivenDMD v0.1.4 `~/_work/DataDrivenDiffEq.jl/DataDrivenDiffEq.jl/lib/DataDrivenDMD`
  [2445eb08] DataDrivenDiffEq v1.15.1 `~/_work/DataDrivenDiffEq.jl/DataDrivenDiffEq.jl`
  [7fed8a53] DataDrivenSR v0.1.5 `~/_work/DataDrivenDiffEq.jl/DataDrivenDiffEq.jl/lib/DataDrivenSR`
  [5b588203] DataDrivenSparse v0.1.5 `~/_work/DataDrivenDiffEq.jl/DataDrivenDiffEq.jl/lib/DataDrivenSparse`
  [e30172f5] Documenter v1.17.0
  [98b081ad] Literate v2.21.0
  [961ee093] ModelingToolkit v11.29.0
  [1dea7af3] OrdinaryDiffEq v7.1.1
  [d3585ca7] OrdinaryDiffEqFunctionMap v2.0.1
  [91a5bcdd] Plots v1.41.6
  [0bca4576] SciMLBase v3.30.0
  [860ef19b] StableRNGs v1.0.4
  [2913bbd2] StatsBase v0.34.12
  [37e2e46d] LinearAlgebra v1.12.0

and using this machine and Julia version.

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: 128 × AMD EPYC 9354 32-Core Processor
  WORD_SIZE: 64
  LLVM: libLLVM-18.1.7 (ORCJIT, znver4)
  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/DataDrivenDiffEq.jl/DataDrivenDiffEq.jl/docs/Manifest.toml`
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  [35ca27e7] eudev_jll v3.2.14+0
  [214eeab7] fzf_jll v0.61.1+0
  [a4ae2306] libaom_jll v3.13.3+0
  [0ac62f75] libass_jll v0.17.4+0
  [1183f4f0] libdecor_jll v0.2.2+0
  [8e53e030] libdrm_jll v2.4.125+1
  [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
  [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 v1.3.0+1
  [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 v3.5.4+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 ⌅ have new versions available but compatibility constraints restrict them from upgrading. To see why use `status --outdated -m`

You can also download the manifest file and the project file.