MilesCranmer / DataDrivenDiffEq.jl

Data driven modeling and automated discovery of dynamical systems for the SciML Scientific Machine Learning organization

DataDrivenDiffEq.jl

DataDrivenDiffEq.jl is a package in the SciML ecosystem for data-driven differential equation structural estimation and identification. These tools include automatically discovering equations from data and using this to simulate perturbed dynamics.

For information on using the package, see the stable documentation. Use the in-development documentation for the version of the documentation which contains the un-released features.

Quick Demonstration

## Generate some data by solving a differential equation
########################################################
using DataDrivenDiffEq
using ModelingToolkit
using OrdinaryDiffEq

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, progress = true)


## Start the automatic discovery
ddprob = ContinuousDataDrivenProblem(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, normalize = true)
print(ddsol, Val{true})

Explicit Result
Solution with 3 equations and 7 parameters.
Returncode: success
Sparsity: 7.0
L2 Norm Error: 26.7343984476783
AICC: 1.0013570199499398

Model ##Basis#366 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)

Parameters:
   p₁ : -10.0
   p₂ : 10.0
   p₃ : 28.0
   p₄ : -1.0
   p₅ : -1.0
   p₆ : 1.0
   p₇ : -2.7