Energy-Preserving Operator Inference

This is a modified version of Operator Inference. Namely, it solves a constrained optimization for the reduced operators.

Energy Preserving Quadratic Nonlinearities in PDEs

We consider an $n$-dimensional ordinary differential equation (ODE) which is linear and quadratic in the state $\mathbf x$. Such an ODE often arises from spatially discretizing a PDE and is given by

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

where $\mathbf x(t) \in \mathbb{R}^n$ is the system state vector over $t \in [0, T_{\text{final}}]$, and $\otimes$ denotes the Kronecker product. The operators $\mathbf A \in \mathbb{R}^{n\times n}$ and $\mathbf H \in \mathbb{R}^{n\times n^2}$ are the linear and quadratic operators, respectively. In our setting, $n$ is large, so simulating the system is computationally expensive.

The quadratic operator $\mathbf H$ is called `energy-preserving' if

\[ \langle \mathbf x, \mathbf H (\mathbf x \otimes \mathbf x)\rangle = \mathbf x^\top \mathbf H(\mathbf x \otimes \mathbf x) = 0, \qquad \text{ for all } \mathbf x \in \mathbb R^n.\]

This condition is derived by setting the quadratic term in the time derivative of the energy, $\frac12\|\mathbf{x}\|^2$, to zero.

Energy-Preserving Operator Inference (EP-OpInf)

To impose this energy-preserving structure on the operator, we propose EP-OpInf. For this method, we incorporate the constraint into the standard OpInf optimization and formulate a constrained minimization as follows:

\[ \textbf{EP-OpInf}: \qquad \min_{\mathbf{O}\in\mathbb R^{r\times(r+r^2)}} \|\mathbf{D}\mathbf{O}^\top - \dot{\hat{\mathbf{X}}}\|_F^2 \quad \text{ subject to } \quad \hat h_{ijk} + \hat h_{jik} + \hat h_{kij} = 0, \quad 1 \leq i,j,k \leq r ~ .\]

epopinf(X, Vn, options; U, Xdot, IG)