fit()

ParametricContinuousModel.fit(parameters, states, ddts, inputs=None)

Learn the model operators from data.

The operators are inferred by solving the regression problem

\[\min_{\Ohat^{(1)},\ldots,\Ohat^{(s)}} \sum_{i=1}^{s}\sum_{j=0}^{k_{i}-1}\left\| \Ophat(\qhat_{i,j}, \u_{i,j}; \bfmu_i) - \dot{\qhat}_{i,j} \right\|_2^2 = \min_{\Ohat^{(1)},\ldots,\Ohat^{(s)}} \sum_{i=1}^{s}\left\| \D^{(i)}(\Ohat^{(i)})\trp - [~\dot{\qhat}_{i,0}~~\cdots~~\dot{\qhat}_{i,k_{i}-1}~]\trp \right\|_F^2\]

where \(\ddt\qhat(t; \bfmu) = \Ophat(\qhat(t; \bfmu), \u(t); \bfmu) = \Ohat(\bfmu)\d(\qhat(t; \bfmu), \u(t))\) is the model and

• \(\qhat_{i,j}\in\RR^r\) is the \(j\)-th measurement of the state corresponding to training parameter value \(\bfmu_i\),

• \(\u_{i,j}\in\RR^m\) is the \(j\)-th measurement of the input corresponding to training parameter value \(\bfmu_i\),

• \(\dot{qhat}_{i,j}\in\RR^r\) is a measurement of the time derivative of the state \(\qhat_{i,j}\), i.e., \(\dot{\qhat}_{i,j} = \ddt\qhat(t; \bfmu_i)\big|_{t=t_{i,j}}\) where qhat_{i,j} = qhat(t_{i,j};bfmu_i),

• \(\Ohat^{(i)} = \Ohat(\bfmu_i)\) is the operator matrix evaluated at training parameter value \(\bfmu_i\), and

• \(\D^{(i)}\) is the data matrix for data corresponding to training parameter value \(\bfmu_i\), given by \([~\d(\qhat_{i,0},\u_{i,0})~~\cdots~~ \d(\qhat_{i,k_i-1},\u_{i,k_i-1})~]\trp\).

Because all operators in this model are interpolatory, the least-squares problem decouples into \(s\) individual regressions. That is, for \(i = 1, \ldots, s\), we solve (independently) the regressions

\[\min_{\Ohat^{(i)}}\left\| \D^{(i)}(\Ohat^{(i)})\trp - [~\dot{\qhat}_{i,0}~~\cdots~~\dot{\qhat}_{i,k_{i}-1}~]\trp \right\|_F^2\]

and define the full operator matrix via elementwise interpolation,

\[\Ohat(\bfmu) = \textrm{interpolate}( (\bfmu_1, \Ohat^{(i)}), \ldots, (\bfmu_s, \Ohat^{(s)}); \bfmu).\]

The strategy for solving the regression, as well as any additional regularization or constraints, are specified by the solver in the constructor.

Parameters:

parameterslist of s scalars or (p,) 1D ndarrays

Parameter values for which the operator entries are known or will be inferred from data.

stateslist of s (r, k_i) ndarrays

Snapshot training data. Each array states[i] is the data corresponding to parameter value parameters[i]; each column states[i][:, j] is a single snapshot.

ddtslist of s (r, k_i) ndarrays

Snapshot time derivative data. Each array ddts[i] is the data corresponding to parameter value parameters[i]; each column ddts[i][:, j] corresponds to the snapshot states[i][:, j].

inputslist of s (m, k_i) or (k_i,) ndarrays, or None

Input training data. Each array inputs[i] is the data corresponding to parameter value parameters[i]; each column inputs[i][:, j] corresponds to the snapshot states[i][:, j]. May be a two-dimensional array if \(m=1\) (scalar input). Only required if one or more model operators depend on inputs.

Returns:

self