Port-Hamiltonian Neural Networks with Output Error Noise Models

TL;DR

OE-pHNNs fuse port-Hamiltonian structure with an output-error model to learn nonlinear dynamic systems from noisy input-output data. The framework enforces energy-based interconnections through $J(x)$, $R(x)$, and $H(x)$ while absorbing measurement noise with an OE formulation, and uses SUBNET to efficiently identify initial states from short data windows. The paper provides a consistency analysis linking discrete-time learning to continuous-time dynamics and validates the approach on coupled mass-spring-damper systems and cascaded-tanks benchmarks, demonstrating robustness to noise and accurate recovery of continuous-time behavior across sampling rates. This work offers a scalable, reliable tool for data-driven identification of real-world engineering systems with inputs, dissipation, and measurement noise. $\dot{x}(t) = [J(x(t))-R(x(t))] \frac{\partial H}{\partial x}(x(t)) + G(x(t))u(t)$ and $y_0(t) = G^\top(x(t))\frac{\partial H}{\partial x}(x(t))$, with the learning objective formulated under an OE loss and optimized via SUBNET-based subsequences.$

Abstract

Hamiltonian neural networks (HNNs) represent a promising class of physics-informed deep learning methods that utilize Hamiltonian theory as foundational knowledge within neural networks. However, their direct application to engineering systems is often challenged by practical issues, including the presence of external inputs, dissipation, and noisy measurements. This paper introduces a novel framework that enhances the capabilities of HNNs to address these real-life factors. We integrate port-Hamiltonian theory into the neural network structure, allowing for the inclusion of external inputs and dissipation, while mitigating the impact of measurement noise through an output-error (OE) model structure. The resulting output error port-Hamiltonian neural networks (OE-pHNNs) can be adapted to tackle modeling complex engineering systems with noisy measurements. Furthermore, we propose the identification of OE-pHNNs based on the subspace encoder approach (SUBNET), which efficiently approximates the complete simulation loss using subsections of the data and uses an encoder function to predict initial states. By integrating SUBNET with OE-pHNNs, we achieve consistent models of complex engineering systems under noisy measurements. In addition, we perform a consistency analysis to ensure the reliability of the proposed data-driven model learning method. We demonstrate the effectiveness of our approach on system identification benchmarks, showing its potential as a powerful tool for modeling dynamic systems in real-world applications.

Figures (9)

• Figure 1: Output error port-Hamiltonian neural network (OE-pHNN).
• Figure 2: OE-pHNN and SUBNET: the subspace encoder $\psi_\eta$ estimates the initial state using past inputs and outputs. Then, the state is iteratively propagated through the framework until it reaches the truncation length $T$.
• Figure 3: Schematic representation of two body-based connected mass-spring-damper systems.
• Figure 4: Sampled inputs and outputs with sampling time $T_\mathrm{s}$, $(u_k,y_k)$ vs sampling time $\frac{T_\mathrm{s}}{5}$, $(\bar{u}_k,\bar{y}_k)$.
• Figure 5: NRMS of the simulated model response on the test dataset for the MSD system, when the model is estimated under different training sample sizes, $N$, and truncation lengths $T$, while SNR=50dB.