Unifying SINDy and DMD

The SINDy and DMD algorithms have been combined under a single interface to match the interface of other SciML packages. Thus, you can simply define a Problem, and then seamlessly switch between solvers.

See the individual descriptions below for how to call the traditional SINDy and DMD solvers within the unified interface.

DMD

For dynamic mode decomposition, use DMDSVD() without a basis:

res = solve(problem, DMDSVD(), kwargs...)

Extended DMD

For extended dynamic mode decomposition, use DMDSVD() with a basis:

res = solve(problem, basis, DMDSVD(), kwargs...)

DMD Optional Arguments

If control signals are present, they get processed according to this paper for dynamic mode decomposition and as described here for extended dynamic mode decomposition, assuming a linear relationship on the operator.

Possible keyword arguments include:

  • B a linear mapping known a priori which maps the control signals onto the lifted states.
  • digits controls the digits / rounding used for deriving the system equations (digits = 1 would round 10.02 to 10.0).
  • operator_only returns a NamedTuple containing the operator, input and output mapping, and matrices used for updating the operator, as described here.
Info

If eval_expression is set to true, the returning result of the Koopman based inference will not contain a parameterized equation, but rather use the numeric values of the operator/generator.

SINDy

For Sparse Identification of Nonlinear Dynamics, use STLQS():

res = solve(problem, basis, STLQS(), kwargs...)

Implicit SINDy

For Sparse Identification of Nonlinear Dynamics, use ImplicitOptimizer():

res = solve(problem, basis, ImplicitOptimizer(), kwargs...)

Where control signals are included in the candidate basis.

For implicit optimizers, you can pass an additional Vector{Num} of variables corresponding to $u_t$ for implicitly defined equations $f(u_t, u, p, t) = 0$:

res = solve(problem, basis, ImplicitOptimizer(), implicits, kwargs...)

This indicates that we do not have coupling between two implicitly defined variables unless coupling is explicitly included. To elaborate, consider the following:

basis = Basis([x, y, z], [x,y,z])
res = solve(problem, basis, ImplicitOptimizer())

Allows solutions of the form x = y + z.

basis = Basis([x, y, z], [x,y,z])
res = solve(problem, basis, ImplicitOptimizer(), [x,y])

Excludes solutions of the form x = z or y = z since we declared x and y as implicit variables assuming they don't interact. However, defining:

basis = Basis([x, y, z, x*y], [x,y,z])
res = solve(problem, basis, ImplicitOptimizer(), [x,y])

Allows solutions of the form x = y*x + z or y = y*x + z, while suppressing x = y + z. This is because y*x includes both x and y, so the function will get included in the evaluation.

SINDy Optional Arguments

Possible keyword arguments include:

  • normalize -> normalizes the data matrix $\\Theta$ such that each row (corresponding to candidate functions) has a 2-norm of 1.0
  • denoise -> applies optimal shrinking to the matrix $\\Theta$ to remove the influence of noise
  • maxiter -> maximum iterations of the used optimizer
  • round -> rounds according to the currently used threshold of the optimizer