Port-Hamiltonian Networks

• Port-Hamiltonian Networks are structured dynamical systems that encode energy flows, conservation, and dissipation using skew-symmetric interconnection matrices.
• They leverage Dirac structures and port variables to connect diverse physical domains, ensuring modular design and energy-preserving interconnections.
• Extensions incorporate data-driven, stochastic, and infinite-dimensional formulations, broadening applications to electrical, fluid, and cyber-physical networks.

Port-Hamiltonian Networks are a class of structured dynamical systems on graphs or networks, formulated to explicitly encode energy flows, conservation, and dissipation at each scale—from individual components to arbitrarily interconnected multi-physical networks. Central to the framework is the use of ports as interconnection interfaces, a skew-symmetric structure representing conservative power flows, and a Hamiltonian function capturing energy storage. This approach generalizes classical Hamiltonian systems to include open, controlled, and dissipative systems, supporting robust modular modeling, analysis, and simulation of complex, networked physical, engineered, and cyber-physical systems (Schaft et al., 2011, Bartel et al., 2023). The framework has been further extended to data-driven and neural network-based modeling, as well as to stochastic and infinite-dimensional (PDE) settings.

1. Mathematical Structure of Port-Hamiltonian Networks

A finite-dimensional port-Hamiltonian system (PHS) on a network is written in the generalized state-space form: $$\dot{x} = \bigl[J(x) - R(x)\bigr] \nabla_x H(x) + G(x)u,\qquad y = G(x)^T \nabla_x H(x)$$ where:

• $x \in \mathbb{R}^n$: state vector (e.g., edge or vertex variables),
• $H: \mathbb{R}^n \rightarrow \mathbb{R}$: Hamiltonian (total stored energy),
• $J(x) = -J(x)^T$: skew-symmetric structure matrix (encodes interconnection via Dirac structure or incidence matrix of the network),
• $R(x) = R(x)^T \succeq 0$: symmetric positive semi-definite dissipation matrix,
• $G(x)$: input (port) map, handling external actuation or coupling,
• $u$, $y$: input and output port variables (often effort and flow) (Schaft et al., 2011, Bartel et al., 2023).

The structure matrix $J(x)$ is determined by the network topology, particularly by the incidence matrix $B$ of the underlying (directed) graph, and for the lossless case (no internal dissipation), takes a block form (for flows and node potentials) as: $x \in \mathbb{R}^n$0 with block-diagonal augmentation for heterogeneous networks or multi-physical coupling.

The energy balance for the network is: $x \in \mathbb{R}^n$1 ensuring passivity, i.e., the network cannot generate energy, and all dissipation is explicit (Bartel et al., 2023, Schaft et al., 2011).

2. Network Construction and Dirac Structures

The Dirac structure formalism is foundational for port-Hamiltonian networks, providing the algebraic language for power-preserving interconnections:

• Flow variables (e.g., edge currents, mass flows) and effort variables (e.g., voltages, pressures) are dual spaces assigned to edges and nodes, respectively.
• The incidence matrix $x \in \mathbb{R}^n$2 induces a map between these variables, producing power-conserving constraints.
• Dirac structures define subspaces where the total power exchange $x \in \mathbb{R}^n$3 is conserved (Schaft et al., 2011).

Network interconnections are performed by composition of Dirac structures: subsystems are "snapped together" along shared boundary ports, ensuring the preservation of energy and passivity on all scales without introducing artificial algebraic variables or dummy elements. This modularity is critical: the composition of multiple subsystems yields a new valid PHS with inherited energy and dissipation structure (Bartel et al., 2023, Schaft et al., 2011).

Graph realizability of general boundary conditions is algebraically characterized for 1D hyperbolic systems via the line digraph condition on boundary matrices, allowing one to reconstruct the underlying metric graph and its adjacency solely from local coupling matrices (Banasiak et al., 2021).

3. Extensions: Dissipation, Stochasticity, and Infinite Dimensionality

Dissipative effects (leaks, friction, turbulent losses) are naturally incorporated as symmetric positive semi-definite components in $x \in \mathbb{R}^n$4, maintaining explicit passivity and the Lyapunov energy structure.

• For practical systems (e.g., flow networks, electrical grids, gas and thermal networks), this enables explicit modeling of losses, component-level dissipation, and arbitrary nonlinear resistive characteristics (Bartel et al., 2023, Hauschild et al., 2019, Bendokat et al., 2024).

Stochastic port-Hamiltonian networks generalize the deterministic case by including stochastic forcing through additional "noise ports." The resulting dynamics

$x \in \mathbb{R}^n$5

retain weak passivity in expectation under explicit generator conditions, and have measurable energy drift determined by a combination of dissipation and stochasticity (Persio et al., 10 Mar 2026, Persio et al., 8 Sep 2025). Universal approximation theorems guarantee any Itô SPHS can be represented to arbitrary accuracy with neural architectures preserving structure (Persio et al., 10 Mar 2026).

Infinite-dimensional port-Hamiltonian networks (PDEs on graphs/manifolds) extend the formalism to accommodate distributed parameter systems (e.g., pipes, strings). Well-posedness and stability are achieved by semigroup methods under maximal dissipativity at the boundaries. Network assembly is accomplished via contraction relations (boundary systems), ensuring the generation of contraction semigroups even on infinite or non-compact graphs (Augner, 2018, Waurick et al., 2019).

4. Computational and Data-driven Approaches

Modern port-Hamiltonian networks are now frequently modeled, identified, and simulated using neural network-augmented or data-driven strategies:

• Neural parameterizations of the Hamiltonian, structure, and dissipation matrices (pHNNs) are constructed to guarantee exact skew-symmetry and positive semi-definiteness by design.
• Training is performed by derivative-matching, trajectory simulation loss, or output-error minimization, often leveraging efficient gradient and projection schemes that exploit sparsity and locality in the network (Roth et al., 4 Feb 2025, Moradi et al., 20 Feb 2025, Desai et al., 2021, Linares et al., 12 May 2026).
• Recent advances have extended these techniques to PDE settings (musical acoustics, distributed damping), stochastic systems (Itô noise, process and measurement uncertainty), and multi-equilibria regimes (Linares et al., 12 May 2026, Persio et al., 8 Sep 2025, Nguyen et al., 14 Apr 2026).
• Structure-preserving integration, such as discrete-gradient or symplectic schemes, is key to ensuring long-term stability and accuracy (Linares et al., 17 Feb 2026).
• Theoretical results guarantee robust energy consistency and stability, with local or global Lyapunov guarantees depending on the convexity and dissipativity properties (Roth et al., 4 Feb 2025, Nguyen et al., 14 Apr 2026).

5. Applications Across Physical and Engineering Networks

Port-Hamiltonian networks have been systematically applied in multiple domains:

• Electrical networks: Modified Nodal Analysis (MNA) reveals that standard circuit equations (RLC, transformers, non-linear devices) are equivalent to port-Hamiltonian DAEs, enabling modular mixed-technology and multi-domain simulation (Bartel et al., 2023).
• Fluid and gas networks: PHS models for gas pipelines, compressor stations, and heating grids provide explicit energy, momentum, and mass balances, supporting structure-preserving model reduction and passivity-based control (Malan et al., 2023, Hauschild et al., 2019, Bendokat et al., 2024, Liljegren-Sailer et al., 2020).
• Mechanical and multi-physics systems: Consensus algorithms, mass-spring-damper arrays, constrained robotics, and more are directly represented by port-Hamiltonian networks, with explicit mapping between network topology and dynamic behavior (Schaft et al., 2011).
• Control, optimization, and inference: The port-Hamiltonian framework enables adjoint-based optimization for static and dynamic network flow problems, offering efficient alternatives to time-expanded combinatorial formulations and supporting generalizations to nonlinear and time-varying settings (Doganay et al., 2023).

6. Advantages, Limitations, and Generalization

Advantages

• Enforces exact conservation laws at all scales by construction through skew-symmetric interconnection structure.
• Facilitates modular, compositional design and analysis, scalable from individual components to full-scale networks.
• Passivity and Lyapunov properties are preserved under arbitrary port-based interconnections, supporting robust stability and control syntheses.
• Admits flexible inclusion of dissipation, nonlinearity, stochasticity, and dynamic port coupling.
• Enables efficient adjoint computation and projection methods, exploiting network sparsity.

Limitations and Open Challenges

• For static linear cost problems, specialized solvers like network simplex outperform pHS-based methods, but this gap reverses for dynamic, nonlinear, or time-dependent cases (Doganay et al., 2023).
• Algebraic characterization of network realizability is complete for linear (Kirchhoff-type) coupling but does not extend fully to nonlinear boundary conditions or partial Kirchhoff splitting (Banasiak et al., 2021).
• Identifiability of physical parameters in data-driven models relies on careful regularization and may fail when multiple energy terms are indistinguishable (Linares et al., 12 May 2026).
• Global region-of-attraction (non-local stability) and global convexity for general networks with many equilibria remain less well-understood (Nguyen et al., 14 Apr 2026).
• Extension to very large-scale, highly-structured, or infinite-dimensional settings may demand additional considerations regarding computational efficiency and well-posedness (Augner, 2018, Waurick et al., 2019).

7. Future Directions

Ongoing research in port-Hamiltonian networks targets:

• Fully scalable and distributed stochastic PHS, including learning-based inference on communication graphs with uncertainty (Persio et al., 8 Sep 2025, Persio et al., 10 Mar 2026).
• Integration with advanced deep learning architectures, symplectic autoencoders, and structure-preserving model reduction for complex, high-dimensional PDE and network systems (Linares et al., 12 May 2026, Moradi et al., 20 Feb 2025).
• Control co-design with data-driven models, leveraging passivity in learned or partially known networks (Nguyen et al., 14 Apr 2026).
• Unified frameworks for cyber-physical systems combining physical energy flows and information/communication dynamics under a single port-Hamiltonian formalism (Schaft et al., 2011).

Port-Hamiltonian networks, through their explicit encoding of energy, passivity, and modularity, continue to provide a principled, extensible foundation for the modeling, analysis, and control of modern, interconnected dynamical systems across scientific and engineering domains.