🌀 nonlinear sciences

Nonlinear port-Hamiltonian system identification from input-state-output data (ISO-pHNN)

This paper presents ISO-pHNN, a framework that leverages neural networks to identify nonlinear port-Hamiltonian systems from input-state-output data, demonstrating that enforcing the system's physical structure preserves accuracy while enhancing long-term prediction capabilities.

Original authors: Karim Cherifi, Achraf El Messaoudi, Hannes Gernandt, Marco Roschkowski

Published 2026-08-24

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Karim Cherifi, Achraf El Messaoudi, Hannes Gernandt, Marco Roschkowski

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine trying to understand how a complex machine works, like a car engine or a weather system, without ever seeing the blueprints. Scientists often rely on mathematical models to predict how these systems will behave, but creating those models from scratch can be incredibly difficult, especially when the rules governing the system are complicated or the system itself is a proprietary secret. For decades, researchers have used two main approaches to solve this puzzle. One relies on pure physics, building models based on fundamental laws of nature, which are reliable but hard to derive for messy, real-world objects. The other relies on data, using computer programs to find patterns in recorded measurements, which is flexible but often fails to predict what happens outside the specific conditions it was trained on. The challenge has been to combine the best of both worlds: a model that learns from data but still respects the unbreakable laws of physics that govern energy and motion.

A team of researchers has developed a new method called ISO-pHNN to bridge this gap. They focused on a specific type of physical system known as a port-Hamiltonian system. In simple terms, these are systems where energy flows in, moves around, and dissipates, much like water flowing through a network of pipes, pumps, and leaks. The "port" part refers to how the system connects to the outside world, allowing energy to enter or leave. The "Hamiltonian" part is a mathematical way of describing the total energy stored within the system. These models are prized because they naturally preserve important physical properties, such as the conservation of energy, making them stable and reliable for long-term predictions. The researchers wanted to see if they could use artificial intelligence to figure out the hidden rules of these systems just by watching how they react to inputs, without needing to know the equations beforehand.

To do this, the team created a framework that treats the system as a collection of four distinct parts: the connections that move energy around, the parts that lose energy through friction or resistance, the total energy stored, and the way outside forces push into the system. Instead of trying to guess the entire system at once, they used neural networks, which are computer programs designed to learn complex patterns, to figure out each of these four parts separately. They fed the computer streams of data showing how the system's state changed over time in response to different inputs. The computer then adjusted its internal settings to match the observed behavior while strictly adhering to the rules of the port-Hamiltonian structure. This ensured that the resulting model didn't just mimic the data but actually behaved like a physical system that obeys the laws of thermodynamics.

The researchers tested their method on several different scenarios, including a system of masses connected by springs and dampers, a ball levitated by a magnetic field, and a complex electric motor. In each case, they compared their new approach against standard data-driven models that ignored the physical structure. The results showed that their method was just as accurate, if not better, at predicting the system's future behavior. Crucially, they found that adding what they call "prior information" made the models even more powerful. If the researchers already knew that a specific part of the system, like the friction or the way energy enters, was constant or followed a simple rule, they could lock that part into the model and let the computer learn only the unknown parts. This not only improved the accuracy of the predictions but also made the models more reliable over long periods of time, reducing errors that often creep in when machines try to guess too much.

One of the most interesting findings was that the type of neural network used mattered depending on the system. For some systems, the standard networks worked best, while for others, a newer type of network designed to handle complex mathematical functions performed significantly better. This suggests there is no single "best" tool for every job; instead, the right choice depends on the specific nature of the system being studied. The team also demonstrated that their method could handle noisy data, where the measurements contained small errors, without losing its ability to predict the system's behavior correctly. This robustness is vital for real-world applications where sensors are never perfect.

Ultimately, this work proves that it is possible to teach artificial intelligence to learn the deep, structural rules of physical systems without sacrificing accuracy for the sake of structure. By combining the flexibility of data-driven learning with the reliability of physical laws, the researchers have created a tool that can uncover the hidden dynamics of complex machines. This approach offers a promising path forward for engineers and scientists who need to model systems that are too complicated to write down by hand but too critical to get wrong. The ability to learn these models from input and output data means that even systems with no available blueprints can be understood, simulated, and controlled with a high degree of confidence.

Technical Summary: ISO-pHNN for Nonlinear Port-Hamiltonian System Identification

Problem Statement

Physics-based models, particularly those grounded in fundamental laws, offer superior interpretability and stability guarantees for dynamical systems. However, deriving such models for complex, proprietary, or high-dimensional systems is often intractable. While purely data-driven methods offer flexibility, they frequently suffer from poor generalizability and a lack of physical consistency (e.g., energy conservation) outside their training scope.

Existing approaches to identifying Port-Hamiltonian (pH) systems have largely focused on linear dynamics or have only partially addressed nonlinearities by approximating specific components (like the Hamiltonian) while fixing others. Furthermore, many current learning-based methods are tailored to specific architectural choices or fixed structural assumptions, lacking a unified framework to systematically incorporate varying degrees of prior physical knowledge. This paper addresses the gap in identifying nonlinear port-Hamiltonian systems from input-state-output (ISO) data, aiming to recover structure-preserving models that maintain physical properties like passivity and energy conservation while leveraging the approximation power of neural networks.

Methodology: The ISO-pHNN Framework

The authors propose ISO-pHNN, a unified framework that learns the constituent components of a nonlinear pH system simultaneously using neural networks. The target system is defined by the standard pH equations:
$\dot{x} = (J(x) - R(x)) \nabla H(x) + B(x)u, \quad y = B(x)^\top \nabla H(x)$
where $J(x)$ is skew-symmetric (interconnection), $R(x)$ is symmetric positive semi-definite (dissipation), $H(x)$ is the Hamiltonian (energy), and $B(x)$ is the input matrix.

1. Structure-Preserving Parametrization

To ensure the learned models strictly adhere to pH structural constraints, the framework employs specific reshaping operations and parametrization schemes:

Matrix Construction: The parameter functions $\theta_J, \theta_R, \theta_B, \theta_H$ $θ_{J}, θ_{R}, θ_{B}, θ_{H}$ are mapped to matrix-valued functions using a vector-to-matrix reshaping operation ($vtf$).
- $J(x)$ is constructed as the difference between a reshaped vector and its transpose, guaranteeing skew-symmetry.
- $R(x)$ is constructed via a product of a reshaped vector and its transpose (scaled by $1/\sqrt{n}$ ), guaranteeing symmetry and positive semi-definiteness.
- $B(x)$ and $H(x)$ are directly mapped from their parameter vectors.
Neural Architectures: The parameter functions are approximated using either Multilayer Perceptrons (MLPs) or Kolmogorov-Arnold Networks (KANs).
- MLPs: Standard fully connected networks with Swish activation functions (chosen for differentiability required by the Hamiltonian gradient).
- KANs: Networks where learnable univariate functions are placed on edges, inspired by the Kolmogorov-Arnold representation theorem. This allows for potentially more interpretable learned dynamics.

2. Incorporation of Prior Information

A key feature of ISO-pHNN is the flexible integration of prior physical knowledge. The framework allows users to specify "ansatz functions" for any subset of the system components ( $J, R, H, B$ ).

If a component is known (e.g., $J$ and $B$ are constant from first principles), it can be fixed or parametrized as a constant matrix.
If a component is known to be quadratic (e.g., a standard mechanical Hamiltonian), it can be constrained to a quadratic form.
This selective parametrization reduces model complexity, improves data efficiency, and constrains the learning process to physically meaningful solutions.

3. Training and Loss Function

The system is trained on input-state-output trajectories.

Data Generation: Trajectories are generated using various initial conditions and sinusoidal input signals.
Loss Function: The objective minimizes the normalized mean absolute error (NMAE) between the predicted and true state derivatives ( $\dot{x}$ ) and outputs ( $y$ ). The loss is weighted by the standard deviation of the targets within the minibatch to handle varying magnitudes. A weight $\gamma$ balances the importance of state derivative prediction versus output prediction, with $\gamma=0.25$ typically used to prioritize long-term forecasting accuracy.
Optimization: The AdamW optimizer with cosine annealing learning rate scheduling is used.

Key Contributions

Unified Framework: The proposal of ISO-pHNN, a flexible framework capable of learning all pH components ( $J, R, H, B$ ) simultaneously from ISO data.
Physics-Informed Priors: The introduction of a mechanism to incorporate partial or complete prior knowledge (e.g., constant matrices, quadratic Hamiltonians) into the neural network parametrization, enhancing training efficiency and accuracy.
Architectural Exploration: The systematic comparison of MLPs and KANs for pH system identification, assessing their respective strengths in modeling different types of nonlinearities.
Validation of Structure: Demonstration that enforcing pH structural constraints does not degrade identification accuracy and, when combined with priors, significantly reduces long-term prediction errors.

Experimental Results

The framework was validated on three distinct nonlinear systems:

Mass-Spring-Damper System:
- Features nonlinear damping ( $R$ ) and a quadratic Hamiltonian.
- Results: The pH model slightly outperformed a black-box baseline. Incorporating priors (fixing $J, B$ and assuming quadratic $H$ ) yielded the lowest error. In this low-frequency setting, MLPs outperformed KANs.
Magnetically Levitated Ball:
- Features a non-quadratic Hamiltonian and state-dependent dissipation.
- Results: The default pH model outperformed the baseline. Prior knowledge did not yield significant improvements in this specific case. MLPs and KANs achieved comparable accuracy.
Permanent Magnet Synchronous Motor (PMSM):
- Features state-dependent interconnection ( $J$ ) and a quadratic Hamiltonian.
- Results: The pH model significantly outperformed the baseline. Crucially, KANs significantly outperformed MLPs in this high-frequency setting, suggesting architecture suitability depends on the system's spectral characteristics.
- Prior Sensitivity: Using "wrong" priors (e.g., assuming linear $J$ when it is nonlinear) led to significant performance degradation, highlighting the importance of accurate prior knowledge.
- Robustness: The method demonstrated robustness to noise, with performance degrading gracefully as Signal-to-Noise Ratio (SNR) decreased.

Significance and Claims

The paper claims that ISO-pHNN provides a robust, structure-preserving approach to identifying nonlinear dynamical systems. The authors emphasize that:

Physical Integrity: Enforcing pH constraints does not lower accuracy compared to black-box models; rather, it ensures the model respects physical laws (passivity, energy conservation).
Data Efficiency: Incorporating prior information (when available) significantly reduces identification error and improves long-term prediction capabilities.
Flexibility: The framework is not tied to a single neural architecture; the choice between MLPs and KANs should be guided by the specific characteristics of the system (e.g., frequency content).
Practicality: The method successfully recovers system components and can handle noisy data, making it suitable for applications where explicit equations are unavailable but simulation or measurement data exists.

The authors conclude that while the method currently relies on state derivatives (often estimated from data), future work will focus on removing this dependency and applying the framework to systems derived from partial differential equations and input-output-only data.