Defining library features that operate on vectors instead of their components.

[JGSuw]

Hello pysindy community,

The sparse regularization optimizer(s) you have implemented look very promising. I am interested in using PDELibrary to apply the Kansa Method for identifying unknown scalar functions of vector arguments appearing in a Lagrangian framework.

The Kansa Method works with radial basis functions. For instance, functions of the form $$\rho: \mathbb{R}^n \to \mathbb{R}$$, where $$\mathbb{R}^n$$ may describe the configuration space of a mechanical system, or even $$\rho: \mathbb{R}^{2n} \to \mathbb{R}$$ that operate on the phase space.

However, as best I can tell, the classes defined in the submodule pysindy.feature_library are designed to work with scalar functions of scalar variables e.g. $$f: \mathbb{R} \to \mathbb{R}$$. That is, PDELibrary seems to expect a library of functions that operate separately on the components of features. However, for reasons of parsimony and interpret-ability I would much prefer to treat configuration and momenta as whole "features" in their own right.

Here are some concrete details about this application:

• I possess an $$(M,2N)$$ data matrix $$X$$ containing $$M$$ measurements of the system phase.
• I have a function library where each function is of the form lambda q: np.exp(-rho(q)) and/or lambda q, qdot: f(q,qdot) * np.exp(-rho(q)) that operates on the phase measurements.
• I have a PDE in an unknown function $$u : \mathbb{R}^N \times \mathbb{R}^N \to \mathbb{R}$$ which appears in a Lagrangian.
• For a given candidate function $u$ I can compute the residuals of the PDE governing equation.
• In the Kansa method, these residuals are computed by a Matrix/Vector product where the matrix applies some linear operators to the basis functions appearing in $$u$$ on the data in $$X$$.

It is straight-forward to formulate this problem as a linear regression on the basis coefficients. I desire to leverage the sparse optimization procedures in pysindy to (a) broaden the space of basis functions and (b) regularize the fits to encourage sparsity.

Is it possible to present this data in such a way that it can be consumed by PDELibrary, and in turn, that a model can be fit by one of pysindy's optimization routines?

Best Regards,
Joseph Sullivan