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#334 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
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 v0.1.1 `~/work/DataDrivenDiffEq.jl/DataDrivenDiffEq.jl/lib/DataDrivenDMD`
[2445eb08] DataDrivenDiffEq v1.3.0 `~/work/DataDrivenDiffEq.jl/DataDrivenDiffEq.jl`
[7fed8a53] DataDrivenSR v0.1.2 `~/work/DataDrivenDiffEq.jl/DataDrivenDiffEq.jl/lib/DataDrivenSR`
[5b588203] DataDrivenSparse v0.1.2 `~/work/DataDrivenDiffEq.jl/DataDrivenDiffEq.jl/lib/DataDrivenSparse`
⌅ [e30172f5] Documenter v0.27.25
[98b081ad] Literate v2.15.0
[961ee093] ModelingToolkit v8.69.1
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[91a5bcdd] Plots v1.39.0
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⌅ [2913bbd2] StatsBase v0.33.21
[37e2e46d] LinearAlgebra
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 1.9.3
Commit bed2cd540a1 (2023-08-24 14:43 UTC)
Build Info:
Official https://julialang.org/ release
Platform Info:
OS: Linux (x86_64-linux-gnu)
CPU: 2 × Intel(R) Xeon(R) Platinum 8370C CPU @ 2.80GHz
WORD_SIZE: 64
LIBM: libopenlibm
LLVM: libLLVM-14.0.6 (ORCJIT, icelake-server)
Threads: 1 on 2 virtual coresA more complete overview of all dependencies and their versions is also provided.
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[9a3f8284] Random
[ea8e919c] SHA v0.7.0
[9e88b42a] Serialization
[1a1011a3] SharedArrays
[6462fe0b] Sockets
[2f01184e] SparseArrays
[10745b16] Statistics v1.9.0
[4607b0f0] SuiteSparse
[fa267f1f] TOML v1.0.3
[a4e569a6] Tar v1.10.0
[8dfed614] Test
[cf7118a7] UUIDs
[4ec0a83e] Unicode
[e66e0078] CompilerSupportLibraries_jll v1.0.5+0
[deac9b47] LibCURL_jll v7.84.0+0
[29816b5a] LibSSH2_jll v1.10.2+0
[c8ffd9c3] MbedTLS_jll v2.28.2+0
[14a3606d] MozillaCACerts_jll v2022.10.11
[4536629a] OpenBLAS_jll v0.3.21+4
[05823500] OpenLibm_jll v0.8.1+0
[efcefdf7] PCRE2_jll v10.42.0+0
[bea87d4a] SuiteSparse_jll v5.10.1+6
[83775a58] Zlib_jll v1.2.13+0
[8e850b90] libblastrampoline_jll v5.8.0+0
[8e850ede] nghttp2_jll v1.48.0+0
[3f19e933] p7zip_jll v17.4.0+0
Info Packages marked with ⌃ and ⌅ have new versions available, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m`