Standard Operator Inference (OpInf)

Theory

If we consider a linear-quadratic system, the goal of Operator Inference is to non-intrusively obtain a reduced model of the form

\[ \dot{\hat{\mathbf x}}(t) = \hat{\mathbf A}\hat{\mathbf x}(t) + \hat{\mathbf{H}}(\hat{\mathbf{x}}(t) \otimes \hat{\mathbf{x}}(t)),\]

To do so, we will fit reduced operators $\hat{\mathbf{A}}$ and $\hat{\mathbf{H}}$ to the reduced data in a least-squares sense. In addition to the state trajectory data

\[ \mathbf{X} = \begin{bmatrix} | & | & & | \\ \boldsymbol{\mathbf x}(t_1) & \mathbf{x}(t_2) & \cdots & \mathbf{x}(t_K) \\ | & | & & | \\ \end{bmatrix} \in \mathbb{R}^{n\times K} ~,\]

we also require paired state time derivative data:

\[\begin{aligned} \dot{\mathbf{X}} = \begin{bmatrix} | & | & & | \\ \dot{\mathbf{x}}(t_1) & \dot{\mathbf{x}}(t_2) & \cdots & \dot{\mathbf{x}}(t_K) \\ | & | & & | \end{bmatrix}\in\mathbb{R}^{n\times K} \quad \text{ where ~~} \dot{\mathbf x}(t_i)=\mathbf A\mathbf x(t_i)+ \mathbf H (\mathbf x(t_i) \otimes \mathbf x(t_i))~. \end{aligned}\]

This time derivative data can come directly from the simulation of the high-dimensional ODE or can be approximated numerically from the snapshot data. We use the POD basis $\mathbf{V}_r$ to compute reduced state and time derivative data as follows: let $\hat{\mathbf{x}}_i = \mathbf{V}_r^\top\mathbf{x}(t_i)$ as an ansatz, and $\dot{\hat{\mathbf{x}}}_i = \mathbf{V}_r^\top\dot{\mathbf{x}}(t_i)$ for $i=1,\ldots,K$. Then, define

\[ \hat{\textbf X} = \begin{bmatrix} | & | & & | \\ \boldsymbol{\hat{\mathbf x}}_1 & \boldsymbol{ \hat{\mathbf x}}_2 & \cdots & \boldsymbol{\hat{\mathbf x}}_K \\ | & | & & | \\ \end{bmatrix} \in \mathbb R^{r\times K}, \qquad \text{and} \qquad \dot{\hat{\textbf X}} = \begin{bmatrix} | & | & & | \\ \dot{\hat{\mathbf x}}_1 & \dot{\hat{\mathbf x}}_2 & \cdots & \dot{\hat{\mathbf x}}_K \\ | & | & & | \\ \end{bmatrix} \in \mathbb R^{r\times K}.\]

Additionally, we define the matrix formed by the quadratic terms of the state data

\[ \hat{\textbf X}_\otimes = \begin{bmatrix} | & | & & | \\ (\hat{\mathbf x}_1 \otimes \hat{\mathbf x}_1) & (\hat{\mathbf x}_2 \otimes \hat{\mathbf x}_2) & \cdots & (\hat{\mathbf x}_K \otimes \hat{\mathbf x}_K) \\ | & | & & | \\ \end{bmatrix} \in \mathbb R^{r^2\times K}.\]

This allows us to formulate the following minimization for finding the reduced operators $\hat{\mathbf{A}}$ and $\hat{\mathbf{H}}$:

\[ \textbf{Standard OpInf}: \qquad \min_{\mathbf{\hat{A}}\in\mathbb R^{r\times r},~\hat{\mathbf{H}}\in\mathbb R^{r\times r^2}} \sum_{i=1}^K \left\| \dot{\hat{\mathbf x}}_i - \mathbf{\hat{A}}\hat{\mathbf{x}}_i - \hat{\mathbf H}(\hat{\mathbf x}_i \otimes \hat{\mathbf x}_i) \right \|^2_2 = \min_{\mathbf{O}\in\mathbb R^{r\times(r+r^2)}} \|\mathbf{D}\mathbf{O}^\top - \dot{\hat{\mathbf X}}\|_F^2,\]

where $\mathbf D = [\hat{\mathbf X}^\top, ~\hat{\mathbf X}_\otimes^\top] \in \mathbb R^{K\times(r+r^2)}$ and $\mathbf O = [\mathbf{\hat A}, ~\hat{\mathbf H}] \in \mathbb R^{r\times(r+r^2)}$.

Implementation

In this package we implement the standard OpInf along with Tikhonov regularized version with the function opinf.

There are many things going on under the hood when the function:

• constructing the data matrix
• construting the Tikhonov matrix
• reprojection

But all of those operations are taken care of automatically. For full details please see the source code.

opinf(X::AbstractArray, Vn::AbstractArray, options::AbstractOption; 
    U::AbstractArray=zeros(1,1), Y::AbstractArray=zeros(1,1),
    Xdot::AbstractArray=[]) → op::Operators

opinf(X::AbstractArray, Vn::AbstractArray, full_op::Operators, options::AbstractOption;
    U::AbstractArray=zeros(1,1), Y::AbstractArray=zeros(1,1)) → op::Operators

opinf(W::AbstractArray, Vn::AbstractArray, lm::lifting, full_op::Operators,
        options::AbstractOption; U::AbstractArray=zeros(1,1), 
        Y::AbstractArray=zeros(1,1), IG::Operators=Operators()) → op::Operators