Theory
Jacobian of the Unique Monomial Map
Given a vector $\hat{s} \in \mathbb{R}^r$, the unique monomial map of degree $i$ is the vector
\[p_i^u(\hat{s}) \in \mathbb{R}^{q_i}, \quad q_i := \binom{r + i - 1}{i},\]
whose entries are all unique monomials of total degree $i$ in the components of $\hat{s}$, ordered in non-decreasing index order. For example, with $r = 2$:
\[p_2^u(\hat{s}) = \begin{bmatrix} \hat{s}_1^2 \\ \hat{s}_1 \hat{s}_2 \\ \hat{s}_2^2 \end{bmatrix}, \qquad p_3^u(\hat{s}) = \begin{bmatrix} \hat{s}_1^3 \\ \hat{s}_1^2 \hat{s}_2 \\ \hat{s}_1 \hat{s}_2^2 \\ \hat{s}_2^3 \end{bmatrix}.\]
The Jacobian $\nabla_{\hat{s}} p_i^u(\hat{s}) \in \mathbb{R}^{q_i \times r}$ is the matrix of partial derivatives of these monomials with respect to $\hat{s}$.
Differentiation Matrices
Rather than computing the Jacobian via the full Kronecker product, we use a factored form involving sparse differentiation matrices $N_d^{(i)} \in \mathbb{R}^{q_i \times q_{i-1}}$ for $d = 1, \dots, r$.
Each differentiation matrix is defined entry-wise by
\[[N_d^{(i)}]_{\alpha,\beta} = \begin{cases} \alpha_d & \text{if } \beta = \alpha - e_d \text{ and } \alpha_d \ge 1, \\ 0 & \text{otherwise}, \end{cases}\]
where $\alpha \in \mathcal{I}_i$ and $\beta \in \mathcal{I}_{i-1}$ are multi-indices with $|\alpha| = i$ and $|\beta| = i-1$, and $e_d$ is the $d$-th coordinate unit multi-index.
- Each $N_d^{(i)}$ has at most one nonzero per row, making it extremely sparse.
- The total number of nonzeros across all $r$ blocks is $r \cdot \binom{r+i-2}{i-1}$.
- The sum of all values across all blocks equals $i \cdot q_i$.
- All matrices are constant (independent of $\hat{s}$), so they can be precomputed once and stored.
The Jacobian formula then reads:
\[\nabla_{\hat{s}} p_i^u(\hat{s}) = N^{(i)} \bigl(I_r \otimes p_{i-1}^u(\hat{s})\bigr),\]
where $N^{(i)} = [N_1^{(i)}, \dots, N_r^{(i)}] \in \mathbb{R}^{q_i \times r \, q_{i-1}}$ is the horizontal concatenation. Equivalently, the $d$-th column of the Jacobian is
\[\frac{\partial p_i^u(\hat{s})}{\partial \hat{s}_d} = N_d^{(i)} \, p_{i-1}^u(\hat{s}),\]
which is simply a sparse matrix–vector product.
Worked Example ($r = 2$, $i = 2$)
The index sets are $\mathcal{I}_2 = \{(2,0), (1,1), (0,2)\}$ (rows) and $\mathcal{I}_1 = \{(1,0), (0,1)\}$ (columns).
Differentiation matrix $N_1^{(2)}$:
| Row $\alpha$ | $\alpha - e_1$ | $\alpha_1$ | Column |
|---|---|---|---|
| $(2,0)$ | $(1,0)$ | $2$ | $1$ |
| $(1,1)$ | $(0,1)$ | $1$ | $2$ |
| $(0,2)$ | — | $0$ | — |
\[N_1^{(2)} = \begin{bmatrix} 2 & 0 \\ 0 & 1 \\ 0 & 0 \end{bmatrix}\]
Differentiation matrix $N_2^{(2)}$:
| Row $\alpha$ | $\alpha - e_2$ | $\alpha_2$ | Column |
|---|---|---|---|
| $(2,0)$ | — | $0$ | — |
| $(1,1)$ | $(1,0)$ | $1$ | $1$ |
| $(0,2)$ | $(0,1)$ | $2$ | $2$ |
\[N_2^{(2)} = \begin{bmatrix} 0 & 0 \\ 1 & 0 \\ 0 & 2 \end{bmatrix}\]
Jacobian evaluation at $\hat{s} = [3, 5]^\top$:
\[\nabla_{\hat{s}} p_2^u(\hat{s}) = \begin{bmatrix} N_1^{(2)} p_1^u(\hat{s}) & N_2^{(2)} p_1^u(\hat{s}) \end{bmatrix} = \begin{bmatrix} 2 \cdot 3 & 0 \\ 5 & 3 \\ 0 & 2 \cdot 5 \end{bmatrix} = \begin{bmatrix} 6 & 0 \\ 5 & 3 \\ 0 & 10 \end{bmatrix}\]
which matches the expected derivative of $p_2^u(\hat{s}) = [\hat{s}_1^2, \hat{s}_1\hat{s}_2, \hat{s}_2^2]^\top$.
Coupling Matrix
In the ManTA-OpInf framework, the coupling matrix $C(\hat{s}) \in \mathbb{R}^{j \times r}$ links the standard and enriched operator inference problems:
\[C(\hat{s}) = \sum_{i=2}^{p} G^{(i)} \nabla_{\hat{s}} p_i^u(\hat{s}),\]
where $G^{(i)} = V_j^\top V^{(i)} \in \mathbb{R}^{j \times q_i}$ are precomputed projection matrices. The coupling matrix appears in both the mass matrix $M(\hat{s}) = I_r + C(\hat{s})^\top C(\hat{s})$ and the augmented right-hand side of the reduced-order model.
Relation to the Elimination Matrix
The differentiation matrices provide an alternative to the route via the elimination and symmetrizer matrix $D_{r,i}$. While the identity $\nabla_{\hat{s}} p_i^u(\hat{s}) = L_{r,i} S_{r,i} \nabla_{\hat{s}} (\hat{s}^{\otimes i})$ is mathematically equivalent, the approach via $N_d^{(i)}$ avoids forming the full $r^i$-dimensional Kronecker Jacobian entirely and exploits the fact that $p_{i-1}^u(\hat{s})$ is already computed during ROM feature evaluation.