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#268 with 3 equations
States : x(t) y(t) z(t)
Parameters : 7
Independent variable: t
Equations
Differential(t)(x(t)) = p₁*x(t) + p₂*y(t)
Differential(t)(y(t)) = p₃*x(t) + p₄*y(t) + p₅*x(t)*z(t)
Differential(t)(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

Reproducibility

The documentation of this SciML package was built using these direct dependencies,
Status `~/work/DataDrivenDiffEq.jl/DataDrivenDiffEq.jl/docs/Project.toml`
  [3c9adf31] DataDrivenDMD v0.1.1 `~/work/DataDrivenDiffEq.jl/DataDrivenDiffEq.jl/lib/DataDrivenDMD`
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  [e30172f5] Documenter v1.2.1
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⌅ [2913bbd2] StatsBase v0.33.21
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Info Packages marked with ⌅ have new versions available but compatibility constraints restrict them from upgrading. To see why use `status --outdated`
and using this machine and Julia version.
Julia Version 1.10.1
Commit 7790d6f0641 (2024-02-13 20:41 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
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Threads: 1 default, 0 interactive, 1 GC (on 4 virtual cores)
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  [35ca27e7] eudev_jll v3.2.9+0
  [214eeab7] fzf_jll v0.43.0+0
  [1a1c6b14] gperf_jll v3.1.1+0
  [a4ae2306] libaom_jll v3.4.0+0
  [0ac62f75] libass_jll v0.15.1+0
  [2db6ffa8] libevdev_jll v1.11.0+0
  [f638f0a6] libfdk_aac_jll v2.0.2+0
  [36db933b] libinput_jll v1.18.0+0
  [b53b4c65] libpng_jll v1.6.40+0
  [f27f6e37] libvorbis_jll v1.3.7+1
  [009596ad] mtdev_jll v1.1.6+0
  [1270edf5] x264_jll v2021.5.5+0
  [dfaa095f] x265_jll v3.5.0+0
  [d8fb68d0] xkbcommon_jll v1.4.1+1
  [0dad84c5] ArgTools v1.1.1
  [56f22d72] Artifacts
  [2a0f44e3] Base64
  [ade2ca70] Dates
  [8ba89e20] Distributed
  [f43a241f] Downloads v1.6.0
  [7b1f6079] FileWatching
  [9fa8497b] Future
  [b77e0a4c] InteractiveUtils
  [4af54fe1] LazyArtifacts
  [b27032c2] LibCURL v0.6.4
  [76f85450] LibGit2
  [8f399da3] Libdl
  [37e2e46d] LinearAlgebra
  [56ddb016] Logging
  [d6f4376e] Markdown
  [a63ad114] Mmap
  [ca575930] NetworkOptions v1.2.0
  [44cfe95a] Pkg v1.10.0
  [de0858da] Printf
  [9abbd945] Profile
  [3fa0cd96] REPL
  [9a3f8284] Random
  [ea8e919c] SHA v0.7.0
  [9e88b42a] Serialization
  [1a1011a3] SharedArrays
  [6462fe0b] Sockets
  [2f01184e] SparseArrays v1.10.0
  [10745b16] Statistics v1.10.0
  [4607b0f0] SuiteSparse
  [fa267f1f] TOML v1.0.3
  [a4e569a6] Tar v1.10.0
  [8dfed614] Test
  [cf7118a7] UUIDs
  [4ec0a83e] Unicode
  [e66e0078] CompilerSupportLibraries_jll v1.1.0+0
  [deac9b47] LibCURL_jll v8.4.0+0
  [e37daf67] LibGit2_jll v1.6.4+0
  [29816b5a] LibSSH2_jll v1.11.0+1
  [c8ffd9c3] MbedTLS_jll v2.28.2+1
  [14a3606d] MozillaCACerts_jll v2023.1.10
  [4536629a] OpenBLAS_jll v0.3.23+4
  [05823500] OpenLibm_jll v0.8.1+2
  [efcefdf7] PCRE2_jll v10.42.0+1
  [bea87d4a] SuiteSparse_jll v7.2.1+1
  [83775a58] Zlib_jll v1.2.13+1
  [8e850b90] libblastrampoline_jll v5.8.0+1
  [8e850ede] nghttp2_jll v1.52.0+1
  [3f19e933] p7zip_jll v17.4.0+2
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.