Index of Notation

In the documentation, we generally denote scalars in lower case, vectors in bold lower case, matrices in upper case, and indicate low-dimensional quantities with a hat. In the code, low-dimensional quantities ends with an underscore (e.g., state is high-dimensional and state_ is low-dimensional).

Dimensions

Symbol	Code	Description
\(n\)	full_state_dimension	Dimension of the full-order system (large)
\(r\)	reduced_state_dimension	Dimension of the reduced-order system (small)
\(m\)	input_dimension	Dimension of the input \(\u\)
\(k\)	k	Number of state snapshots, i.e., the number of training points
\(s\)	s	Number of parameter samples for parametric training
\(p\)	p	Dimension of the parameter space
\(d\)	d	Number of columns of an operator entries matrix \(\Ohat\)

Vectors

Symbol	Code	Size	Description
\(\q\)	state	\(n\)	Full-order state vector
\(\qhat\)	state_	\(r\)	Reduced-order state vector
\(\dot{\qhat}\)	ddt_	\(r\)	Reduced-order state time derivative vector
\(\q_{\text{ROM}}\)	q_ROM	\(n\)	Approximation to \(\q\) produced by ROM
\(\chat\)	c_	\(r\)	Learned constant term
\(\u\)	inputs	\(m\)	Input vector
\(\qhat\otimes\qhat\)	np.kron(q_,q_)	\(r^2\)	Full quadratic Kronecker product of reduced state
\(\qhat\,\widehat{\otimes}\,\qhat\)	utils.kron2c(q_)	\(\frac{r(r+1)}{2}\)	Compact quadratic Kronecker product of reduced state
\(\qhat\otimes\qhat\otimes\qhat\)	np.kron(q_,np.kron(q_,q_))	\(r^3\)	Full cubic Kronecker product of reduced state
\(\qhat\,\widehat{\otimes}\,\qhat\widehat{\otimes}\,\qhat\)	utils.kron3c(q_)	\(\frac{r(r+1)(r+2)}{6}\)	Compact cubic Kronecker product of reduced state
\(\v_{j}\)	vj	\(n\)	\(j\)th basis vector, i.e., column \(j\) of \(\Vr\)

Matrices

Symbol	Code	Shape	Description
\(\Vr\)	basis	\(n \times r\)	low-rank basis of rank r (usually the POD basis)
\(\Q\)	states	\(n \times k\)	Snapshot matrix
\(\dot{\Q}\)	ddts	\(n \times k\)	Snapshot time derivative matrix
\(\U\)	inputs	\(m \times k\)	Input matrix (inputs corresonding to the snapshots)
\(\widehat{\Q}\)	states_	\(r \times k\)	Projected snapshot matrix
\(\dot{\widehat{\Q}}\)	ddts_	\(r \times k\)	Projected snapshot time derivative matrix
\(\D\)	D	\(k \times d(r,m)\)	Data matrix
\(\Ohat\)	Ohat	\(r \times d(r,m)\)	Operator matrix
\(\mathbf{R}\)	R	\(r \times k\)	Right-hand side matrix
\(\boldsymbol{\Gamma}\)	regularizer	\(d(r,m) \times d(r,m)\)	Tikhonov regularization matrix
\(\Ahat\)	A_	\(r \times r\)	Reduced-order linear state matrix
\(\Hhat\)	H_	\(r \times \frac{r(r+1)}{2}\)	Compact reduced-order matricized quadratic state tensor
\(\Ghat\)	G_	\(r \times \frac{r(r+1)(r+2)}{6}\)	Compact reduced-order matricized quadratic state tensor
\(\Bhat\)	B_	\(r \times m\)	Reduced-order input matrix
\(\Nhat\)	N_	\(r \times rm\)	Bilinear state-input matrix