Using Symplectic Autoencoders for Port-Hamiltonian Systems

Symplectic autoencoders can also be used to reduce port-Hamiltonian systems [73]. Here we focus on linear port-Hamiltonian systems^[1]. These are of the form:

\[\Sigma_\mathrm{lpH}(\mathbb{R}^{2N}) = \Sigma_\mathrm{lpH} : \begin{cases} \dot{\hat{z}}(t) & = (\mathbb{J}_{2N} - \hat{R})\nabla{}H(\hat{z}(t)) + \hat{B}u(t) \\ y(t) & = \hat{B}^T\nabla{}H(\hat{z}(t)), \end{cases}\]

where $\mathbb{J}_{2N}$ is the Poisson tensor and $\hat{R}\in\mathbb{R}^{2N\times{}2N}$ is symmetric semi-positive definite (i.e. all its eigenvalues are non-negative). $\hat{z}\in\mathbb{R}^{2N}$ is called the state of the system, $u\in\mathbb{R}^m$ are the system inputs, $y\in\mathbb{R}^m$ are the system outputs, and $\hat{B}\in\mathbb{R}^{2N\times{}m}$ connects the inputs to the state. We also refer to linear port-Hamiltonian systems as lpH systems.

Similar to energy conservation of standard Hamiltonian systems, lpH systems have an associated energy balance equation:

The analogue to the Poisson tensor for lpH systems are so-called Dirac structures:

In numerically solving lpH systems the Dirac structure takes a similar role to the symplectic structure of canonical Hamiltonian systems [77] and the energy-balance equation takes a similar role to energy conservation for canonical Hamiltonian systems. Structure-preserving discretization of lpH systems discretize the Dirac structure with collocation methods [78]. A one-step update can be written as

\[\begin{aligned} z_f & = z_0 + h\sum_{i=1}^s\beta_if_i, \\ z_i & = z_0 + h\sum_{j=1}^s\alpha_{ij}f_j, \\ e_i & = \bar{e}_i = \nabla_{z_i}H, \\ \bar{\bar{e}}_i & = -Re_i, \\ f_i & = \mathbb{J}_{2N}^T\bar{e}_i - Bu_i - \bar{\bar{e}}_i, \\ y_i & = B^T\bar{e}_i. \end{aligned}\]

Model order reduction of port-Hamiltonian systems can be divided into two approaches: projection-based methods and interpolations of the transfer function [79]. The first approach is equivalent to Galerkin projection and we limit the discussion here to this approach. Similar to the case of canonical Hamiltonian systems, we reduce the system with a symplectic autoencoder.

When discussing symplectic model order reduciton we showed that a Hamiltonian system on the reduced space $\mathbb{R}^{2n}$ is equivalent to a Hamiltonian system on $\mathcal{M} = \mathcal{R}(\mathbb{R}^{2n}),$ where $\mathcal{R}$ is the reconstruction in a reduced order modeling framework. Similar statements are also true for lpH systems.

We will now demonstrate how to obtain a reduced-order lpH system from a full-order lpH system and vice-versa:

The figure above indicates that we can derive a full system $\tilde{\Sigma}_\mathrm{lpH}(\mathbb{R}^{2N}) := \Sigma_\mathrm{lpH}(\mathcal{M})$ from a reduced one $\Sigma_\mathrm{lpH}(\mathbb{R}^{2n}).$ If we have $R = 0,$ i.e. if the dissipative part of the system is zero, then we can also derive a reduced system $\Sigma^{R=0}_\mathrm{lpH}(\mathbb{R}^{2n})$ from a full one $\tilde{\Sigma}^{R=0}_\mathrm{lpH}(\mathbb{R}^{2N}) = \Sigma^{R=0}_\mathrm{lpH}(\mathcal{M}).$ If and when this is true for $R\neq0$ is an open question^[2]. We now proceed with showing this equivalence, first for the special case $R = 0.$

The Special Case $R = 0$

We first focus on the case where $R = 0.$ This case was also discussed in [77].

$(\nabla\Psi^d)^+ = \mathbb{J}_{2n}^T(\nabla\Psi^d)^T\mathbb{J}_{2N}$ is the symplectic inverse.

From the Reduced Space to the Full Space for $R\neq0$

Here we show that we can construct a lpH system on the full space from a lpH system on the reduced space; this holds for both $R=0$ and $R\neq0.$ The corresponding proof was already introduced in similar form by [80].

As was already discussed in the section on Hamiltonian model order reduction the encoder $\Psi^e$ can be constructed such that it is exactly the local inverse $\psi.$ This was done in e.g. [81]. Enforcing this for symplectic autoencoders is also straightforward.

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