Multilithic Models

[shanemcq18]

Monolithic Models

The philosophy of the package is to specify model structure as a list of state/input operators that can be calibrated with Operator Inference. Roughly speaking, we currently support "monolithic" models of the form

$$ \frac{d}{dt}\mathbf{q}(t) = \mathbf{f}(\mathbf{q}(t), \mathbf{u}(t)), $$

in which $\mathbf{f}$ is comprised of operators that act on the entire state and/or the entire input.

Multilithic Models

In many applications, it is desirable to have more specific block structure, for example

$$ \begin{align*} \frac{d}{dt}\mathbf{q}_{0}(t) &= \mathbf{A}_{0,1}\mathbf{q}_{1}(t) + \mathbf{H}_{0,0}[\mathbf{q}_{0}(t) \otimes \mathbf{q}_{0}(t)] \\ \frac{d}{dt} \mathbf{q}_{1} (t) &= \mathbf{A}_{1,0}\mathbf{q}_{0}(t). \end{align*} $$

In this scenario, the right-hand side is made up of operators that act on only parts of the states and/or inputs. This issue is a follow-up to #48 for tracking efforts to implement models with block structure.

Vision and Roadmap

To do this, we need

• multilithic operators, where the user specifies state/input subdimensions and indices, and
• multilithic models, which are initialized with multilithic operators and state/input dimensions. These will have decoupled Operator Inference problems, one per subequation.

These models should be compatible with the existing ROM class, though we should add a warning if a multilithic model is used with a monolithic basis. The end result might be something like this:

model = opinf.models.ContinuousModelMulti(
    operators=[
        opinf.operators.LinearOperatorMulti(0, 1),      # A_{0, 1}
        opinf.operators.QuadraticOperatorMulti(0, 0),   # H_{0, 0}
        opinf.operators.LinearOperatorMulti(0, 0),      # A_{1, 0}
    ],
    substate_dimensions=(4, 6),
)

The ROM class should also be able to pass the substate_dimensions from a basis.BasisMulti to the model.

I have some initial code for this, but @bgzastrow will take the lead on the effort.