SparseIdentification.jl

Data-driven discovery of governing equations, with an extension that preserves the symplectic structure of Hamiltonian systems.

Given measurements of a dynamical system's state, SparseIdentification recovers the equations that generated them, by selecting a small number of terms from a library of candidate functions. For Hamiltonian systems it goes further: rather than fitting the vector field, it identifies the scalar Hamiltonian, so that the discovered dynamics is Hamiltonian by construction rather than by approximation.

Why structure preservation

A general-purpose fit knows nothing about the physics. Applied to a Hamiltonian system it produces a model that conserves energy only approximately and drifts under long integration. Identifying $H$ and deriving $\dot z = J\nabla H$ makes conservation exact for every candidate model considered, including wrong ones — the constraint is structural, not a penalty.

The payoff is measurable. On the Toda lattice, the structure-preserving method reaches a maximum coefficient residual of 0.006 where classical SINDy on the same data reaches 0.5 — roughly two orders of magnitude, at half the optimiser iterations. Those two figures are reported from the thesis rather than recomputed here; see Toda Lattice, which says which of its numbers are quoted and which are run at build time.

Two lines to try it

using SparseIdentification

A = [-0.1 2.0; -2.0 -0.1]          # the system to rediscover
x = randn(2, 500); ẋ = A * x

result = identify(TrainingData(x, ẋ), SINDy(CompoundBasis(polyorder = 3); λ = 0.05))
parameters(result)[2:3, :]         # recovered: A transposed

2×2 Matrix{Float64}:
 -0.1  -2.0
  2.0  -0.1

Where to go next


Getting Started	installation, the shape of a run, both methods end to end
Hamiltonian Systems	notation, the symplectic form, why structure matters
Sparse Identification	the SINDy formulation, STLSQ, what is actually guaranteed
Hamiltonian SINDy	the extension, and why the problem is linear in the coefficients
Choosing λ	the one parameter that matters, with measured guidance
When It Fails	the failure modes, with evidence — read this before trusting a result

Provenance

The Hamiltonian method implemented here is due to Nigel Bruce Khan's master's thesis at the Technische Universität München, supervised by Michael Kraus Khan2023.

Claims taken from that thesis are verified rather than transcribed. The script scripts/verify_thesis_examples.jl checks each one and reports what holds and what does not; three of its equations need correcting, and the corrections are stated wherever this documentation touches them (Nonlinear Oscillator, When It Fails).

Status

Working and tested: classical SINDy with sequentially thresholded least squares, Hamiltonian SINDy in its flow-map formulation, and the polynomial, trigonometric, exponential, logarithmic and rational bases — the last three applied to differences of state components, which is what the Toda lattice, the point vortex and the N-body problem need. See Basis Libraries.

Not yet implemented, and tracked in the release notes: the linear vector-field formulation of Hamiltonian SINDy, the autoencoder variant that discovers canonical coordinates, and the weak formulation for noisy data. A norm of a difference of position vectors is not expressible either, so three-dimensional N-body is out of reach.