Modeling Dynamic Systems with Physics-Informed Networks

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Author: Denis Avetisyan

A new framework, PHAST, leverages the principles of port-Hamiltonian systems to create stable and interpretable models of complex temporal dynamics from limited observations.

PHAST unifies knowledge regimes and guarantees passivity through structured architectures and parameterizations for improved forecasting of dissipative dynamics.

Forecasting the long-term behavior of dissipative physical systems from limited observations remains a central challenge in scientific machine learning. To address this, we introduce \text{PHAST} (Port-Hamiltonian Architecture for Structured Temporal dynamics), a novel framework that learns stable and interpretable models of these dynamics by explicitly representing conservative and dissipative forces. \text{PHAST} unifies different levels of prior knowledge via structured architectures and low-rank parameterizations, guaranteeing passivity and enabling physically meaningful parameter recovery across diverse benchmarks-spanning mechanical, electrical, and thermal systems. However, can sufficiently strong, yet practical, anchors be identified to overcome the fundamental ill-posedness of identifying system parameters from partial observations and unlock truly robust, physics-informed forecasting?

Unveiling System Behavior: Beyond Energy as a Secondary Effect

Conventional system modeling frequently encounters limitations when attempting to represent the complex relationship between energy storage and dissipation, often treating these processes as secondary effects rather than fundamental drivers of system behavior. This approach can lead to inaccurate predictions, particularly in scenarios involving non-equilibrium dynamics or significant energy exchange. Traditional methods often rely on simplified assumptions that fail to capture the nuanced interplay between these opposing forces, hindering the ability to model phenomena like oscillations, instabilities, and energy transfer with sufficient fidelity. Consequently, the resulting models may lack the capacity to fully describe the system’s response to external stimuli or its long-term evolution, necessitating more robust frameworks capable of explicitly accounting for both energy storage and dissipation as integral components of the system’s dynamics.

Accurate depiction of real-world systems necessitates a clear understanding of both conservative and dissipative dynamics. Conservative systems, like a simple pendulum in a vacuum, exchange energy without loss – energy is continually transformed between kinetic and potential forms, resulting in perpetual motion if undisturbed. Conversely, dissipative systems, prevalent in most natural phenomena, experience energy loss through mechanisms like friction or resistance, ultimately leading to a system settling into equilibrium. Ignoring dissipation can lead to drastically inaccurate predictions; for example, modeling a mechanical oscillator without accounting for damping would predict oscillations continuing indefinitely, a clear departure from observed behavior. Therefore, a robust system model must explicitly account for both energy storage – characteristic of conservative elements – and energy dissipation, providing a more realistic and predictive representation of complex physical processes.

The Port-Hamiltonian Framework provides a robust methodology for modeling complex systems by distinctly categorizing elements as either energy storing (like inductors and capacitors) or energy dissipating (such as resistors and dampers). This separation isn’t merely organizational; it’s fundamental to guaranteeing the system’s inherent stability. Unlike traditional approaches that can inadvertently introduce instabilities through modeling choices, the Port-Hamiltonian formulation mathematically ensures that any modeled system will not spontaneously become unstable, regardless of external inputs or internal complexities. This is achieved through a specific mathematical structure that conserves energy or allows for predictable dissipation, preventing uncontrolled growth of internal states. Consequently, the framework is particularly valuable in fields requiring precise and reliable simulations, including power systems analysis, control design, and the modeling of biological networks, offering a predictable and physically consistent representation of dynamic behavior.

From Theory to Practice: Modeling Physical Reality

The Port-Hamiltonian framework builds upon Hamiltonian dynamics by generalizing the concept of energy storage and dissipation. Traditional Hamiltonian mechanics describes conservative systems using generalized coordinates and momenta, defined by a Hamiltonian function H(q,p). The Port-Hamiltonian approach extends this by introducing “ports,” which represent external connections to the system and allow for the modeling of energy and momentum flow between the system and its environment. This is achieved through the definition of a “port-Hamiltonian” function and associated “port variables” that describe these interactions. Consequently, the framework facilitates the modeling of open, dissipative systems-those exchanging energy with their surroundings-while retaining the geometric structure and beneficial properties of Hamiltonian mechanics, such as symplectic invariance, enabling a more complete and realistic description of system evolution.

The Port-Hamiltonian framework is particularly effective when modeling mechanical systems dominated by energy transfer, such as the double pendulum and cart-pole systems. These systems exhibit complex behaviors arising from the interplay of kinetic and potential energies, and the framework’s structure naturally accommodates their representation. Specifically, the double pendulum involves gravitational potential energy converted to kinetic energy as the pendulum swings, alongside frictional dissipation; the Port-Hamiltonian approach allows explicit modeling of these energy flows. Similarly, the cart-pole system’s dynamics are dictated by the kinetic energy of the cart and pole, potential energy due to the pole’s height, and energy lost to friction at the joint and wheels. By explicitly defining energy storage and dissipation elements, the Port-Hamiltonian formulation provides a robust and accurate method for simulating and analyzing the behavior of these and similar energy-centric systems, offering advantages over traditional approaches that may obscure these crucial energy interactions.

Separating a system’s dynamics into conservative and dissipative elements allows for a more nuanced understanding of its energy flow and stability. Conservative elements, described by \frac{dE}{dt} = 0 , represent energy storage without loss, while dissipative elements account for energy loss through mechanisms like friction or damping. This decomposition facilitates analysis by isolating the portions of the system responsible for oscillatory behavior from those governing stability and settling time. Consequently, control strategies can be specifically targeted; for instance, feedback control can be designed to counteract dissipative forces, stabilize the system, or shape the energy exchange between conservative and dissipative components, leading to improved performance and robustness.

PHAST: Reconstructing Dynamics from Limited Observations

The PHAST architecture addresses the challenge of learning the dynamics of dissipative systems when only position (q) data is available for observation. Utilizing the Port-Hamiltonian framework, PHAST models system behavior through energy conservation and dissipation principles, even with incomplete state information. This approach contrasts with traditional methods reliant on full state observation or complex estimations. By framing the learning problem within the Port-Hamiltonian formulation, PHAST effectively reconstructs system dynamics from partial observations, specifically leveraging the observed position data q to infer the complete state and predict future behavior without requiring direct velocity measurements.

Low-rank parameterizations are central to the computational scalability and stability of the PHAST architecture. By restricting the dimensionality of the damping G and mass M matrices to a significantly lower rank than their full dimensional counterparts, the number of trainable parameters is drastically reduced. This reduction in parameters directly translates to lower computational costs during both training and inference. Critically, these low-rank parameterizations are not merely an optimization technique; they also enforce positive semi-definiteness of the G and M matrices, a necessary condition for ensuring the physical plausibility and stability of the learned dynamics. This constraint prevents the model from learning unphysical behaviors and guarantees a well-conditioned system, improving the robustness and reliability of the learned control policies.

Strang Splitting is a numerical integration method utilized within the PHAST architecture to discretize continuous-time dynamics while maintaining key structural properties of the underlying system. This scheme achieves second-order accuracy by interleaving half-step applications of multiple operators, specifically those governing the system’s inertial and damping components. Formally, given a system described by \dot{x} = f(x) , a single Strang split step involves applying half of an operator A , then a full step of operator B , and finally another half-step of operator A . This approach is particularly beneficial for Port-Hamiltonian systems as it helps preserve properties like passivity and stability during discretization, preventing the introduction of spurious numerical artifacts and ensuring a more accurate representation of the physical dynamics.

The PHAST architecture incorporates an observer to estimate velocity information when only position (q) data is available, addressing the challenge of learning dynamics from partial observations. This velocity estimation is critical for accurate state reconstruction, as the system requires both position and velocity to model dynamics effectively. Performance benchmarks demonstrate a 9x reduction in control effort when utilizing this observer-based q-only feedback loop compared to traditional finite difference methods. This improvement stems from the observer’s ability to provide a more accurate approximation of the system’s state, leading to more precise control signals and reduced energy expenditure.

Validating Performance: A Rigorous Assessment of Forecasting and Recovery

A comprehensive assessment of PHAST’s capabilities necessitates a two-axis evaluation, dissecting its performance into distinct but interconnected facets: forecasting stability and physical parameter recovery. This approach moves beyond simple error metrics by explicitly verifying not only the system’s ability to predict future states, but also its fidelity in reconstructing the underlying physical properties governing the simulated environment. Separating these evaluations is crucial because a system might achieve short-term predictive accuracy through memorization or exploitation of specific initial conditions, without genuinely understanding the dynamics; conversely, accurate parameter recovery ensures the model has learned a robust representation of the system’s physics, enabling reliable long-term forecasts and generalization to unseen scenarios. This dual assessment provides a more nuanced and trustworthy validation of PHAST’s performance, revealing its strengths and limitations with greater clarity.

Performance benchmarking centered on the Windy Pendulum revealed the system’s robust forecasting capabilities. The approach achieved a mean squared error (MSE) of 0.092 over a 100-step rollout, representing a significant advancement over existing methods. This result constitutes a 4.76x improvement when contrasted against the highest-performing baseline model under identical conditions, indicating substantial gains in predictive accuracy and stability. Such a marked difference highlights the efficacy of the developed system in modeling and predicting the behavior of complex dynamical systems, paving the way for applications requiring precise and reliable forecasting.

The evaluation framework, initially demonstrated with the Windy Pendulum, was further tested on the markedly different dynamical systems of the Double Pendulum and Cart-Pole. This expansion was critical to establish whether the observed performance gains were specific to the initial test case or indicative of a more broadly applicable solution. Results from these additional systems consistently showcased the approach’s ability to maintain forecasting stability and accurately recover physical parameters, even amidst the increased complexity of these environments. The consistent performance across diverse systems underscores the generalizability of the methodology, suggesting it is not simply tailored to a specific scenario but rather represents a robust solution for state estimation and predictive modeling in a wider range of physical systems.

Analysis of the Windy Pendulum system reveals a high degree of fidelity in recovering the true damping coefficient, a critical parameter governing energy dissipation within the simulated environment. The methodology achieved an R-squared value of 0.996, indicating that 99.6% of the variance in the true damping is explained by the system’s estimations. This precise recovery of damping isn’t merely a numerical success; it signifies the system’s ability to accurately model the underlying physics governing the pendulum’s motion, extending beyond simple trajectory prediction to a deeper understanding of the system’s dynamic characteristics. Such accuracy is foundational for applications requiring reliable physical parameter identification, such as robotics, control systems, and biomechanical modeling, where even small errors in damping estimation can lead to instability or inaccurate performance.

The pursuit of robust and interpretable models, as demonstrated by PHAST, echoes a fundamental principle of system design. A system’s stability isn’t merely a feature, but a consequence of its inherent structure. Grace Hopper famously stated, “It’s easier to ask forgiveness than it is to get permission.” This resonates with the PHAST framework’s approach to learning from partial observations; rather than demanding complete information upfront, the model adapts and refines its understanding through interaction with the available data. Just as Hopper advocated for pragmatic action, PHAST embraces a flexible approach to modeling dissipative dynamics, prioritizing passivity and identifiability through its carefully constructed architecture and parameterizations. This allows for prediction even in the face of incomplete knowledge, preventing failures along invisible boundaries.

What’s Next?

The appeal of PHAST lies in its insistence on structure. If the system looks clever, it’s probably fragile. This work offers a valuable, if preliminary, demonstration that enforcing physical constraints – here, passivity – can yield more robust and interpretable models. However, the architecture itself is not a free lunch. The inherent limitations of port-Hamiltonian representations-their suitability for certain classes of systems, the difficulty in identifying appropriate coordinates-remain significant hurdles. Future efforts must address these constraints, perhaps by exploring hybrid approaches that gracefully relax the strict requirements of Hamiltonian or dissipative formalisms when necessary.

A crucial direction involves expanding beyond single, monolithic architectures. The world is rarely conveniently described by a single dynamical system. Developing methods for composing PHAST modules, or integrating them with other modeling paradigms, would unlock far greater expressive power. Furthermore, the question of identifiability looms large. Guaranteeing stability is insufficient if the learned parameters are unconstrained and lack meaningful physical interpretation.

Ultimately, this line of inquiry forces a difficult, but necessary, realization: architecture is the art of choosing what to sacrifice. There is no universal model. The challenge is not simply to build more complex systems, but to build simpler ones that capture the essential dynamics while remaining resilient to noise and uncertainty. That, and to be honest about what has been left out.

Original article: https://arxiv.org/pdf/2602.17998.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/

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2026-02-23 22:51