Title: Stable Port-Hamiltonian Neural Networks
Venue: NeurIPS 2025
Accepted: yes
Keywords: dynamic system identification

port-Hamiltonian systems

Lyapunov stability
Abstract: In recent years, nonlinear dynamic system identification using artificial neural networks has garnered attention due to its broad potential applications across science and engineering. 
However, purely data-driven approaches often struggle with extrapolation and may yield physically implausible forecasts. 
Furthermore, the learned dynamics can exhibit instabilities, making it difficult to apply such models safely and robustly. 
This article introduces stable port-Hamiltonian neural networks, a machine learning architecture that incorporates physical biases of energy conservation and dissipation while ensuring global Lyapunov stability of the learned dynamics. 
Through illustrative and real-world examples, we demonstrate that these strong inductive biases facilitate robust learning of stable dynamics from sparse data, while avoiding instability and surpassing purely data-driven approaches in accuracy and physically meaningful generalization. 
Furthermore, the model's applicability and potential for data-driven surrogate modeling are showcased on multi-physics simulation data.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper tackles the problem of learning stable dynamical systems from data using neural networks. Building on the framework of Lyapunov stability and port-Hamiltonian systems, the authors propose a novel architecture, stable port-Hamiltonian neural networks (sPHNN), in which the system dynamics are modeled using port-Hamiltonian structure. Neural networks are used to learn an energy function and structured matrices representing energy-conserving flows, dissipative effects, and control inputs. The architecture enforces Lyapunov conditions on the learned energy function, thereby guaranteeing asymptotic stability. The proposed model is evaluated on a set of experiments, including both synthetic and real-world datasets, and demonstrates improved stability and data efficiency compared to several baselines.\n\nSTRENGTHS AND WEAKNESSES:\nOverall, the paper is very pleasant to read. In spite of minor clairty issues, the method is soundly and clearly presented, and the results are strong.\n \n### Strengths\n\n- The paper is clearly written and mathematically well-grounded. The theoretical framework is well-motivated, and all necessary background is introduced with appropriate references.\n- The proposed approach addresses an important and practical problem: learning dynamical models that are both data-efficient and stable, which is particularly relevant in low-data regimes.\n- The architecture is original and benefits from strong theoretical foundations. Structuring the model around a learned energy function with interpretable terms (conservative and dissipative) adds physical meaning and interpretability.\n- The paper provides a clear and detailed exposition of the model, with Theorem 3.1 nicely summarizing the desirable properties of the architecture.\n- The baseline methods are well-documented and chosen appropriately, enabling a fair comparison.\n- Experimental results on real-world and high-dimensional synthetic systems show a clear advantage of the proposed method over alternatives.\n\n### Weaknesses\n\n  1. It would be insightful to provide early in the paper an example or figure of a physical system illustrating the need for stability, and what an unconstrained architecture might produce. It would also be nice to introduce an example of port-Hamiltonian system to illustrate the framework of section 2. Typically, a torque-actuated pendulum with friction, where all the terms have a clear physical interpretation, would help ground the abstract formulation of (3). Perhaps a slightly more complex system might also be interesting to expose. \n\n2.   Although the model enforces structure via an interpretable decomposition into conservative and dissipative terms, this structure is defined with respect to a learned energy function. This function is optimized jointly with other parameters to minimize prediction error, and does not necessarily align with any known physical energy.\nIn the case where the learned system is known to be close to be Hamiltonian, and some prior knowledge on some energy function of the system is available, I am understanding that it could be incorporated in function $f$, and (3) then mimics the system dynamics. When it is not the case though, I am wondering what the interpretation of modeling the dynamics with (3) is. Why should we expect this structure to learn more efficiently, compared to alternative Lyapounov-based learning architectures? See my questions 4. and 5..\n\n3. Evaluation methodology needs clarification.  The experimental setup, particularly how predictions are computed and evaluated, lacks clarity. See question 3.\n\n4.  The paper does not discuss potential downsides of the proposed approach, especially regarding computational cost. The gradient computations involved in the Hamiltonian structure may significantly increase training time.\n\nQUESTIONS:\n1. In Theorem 3.1, isn't coercity for a convex function admitting an uniquer minimizer a standard result in convex analysis? See Rockafeller's Convex Analysis book, Corollary 8.7.1. \n\n2. In line 93, the derivative $\\dot{V}(x)$ is scalar-valued. Should it not be referred to as “nonpositive” rather than “negative semi-definite”? Also, I found the second member of the equality chain (2) confusing as $\\dot{x}$  is a velocity, implying a trajectory, while the definition $\\dot{V}(x)$ is a function of the state and can be defined as a function of $f$, $V$ and $x$ regardless of any trajectory or dynamical system.\n\n3. How is the prediction task evaluated? Are the models used autoregressively, with known or fixed control inputs $u$? More details on this setup would help interpret the results.\n\n4.   If the system is known to admit a Lyapunov function vanishing at $x_0 = 0$, can we expect the learned Hamiltonian to converge to this function under the training dynamics?\n\n5.   What is the interpretation or expected benefit of applying a Hamiltonian structure to systems that are not inherently physical or do not follow Hamiltonian dynamics (e.g., temperature fields)? Could the same stability guarantees not be obtained more simply by enforcing a global Lyapunov function, as done in sNODEs?\n\n6.   The proposed architecture involves gradient computations of the learned Hamiltonian, which may incur additional cost. How does the computational complexity compare to standard NODEs or other baseline methods?\n\nLIMITATIONS:\nThe submission does not discuss some important potential limitations such as the computational cost. See my question 6.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-12"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper discusses a type of physics guided neural network based on port-Hamiltonian systems, and in particular how global Lyapunov stability can be enforced in such a network. Lyapunov stability is desirable for some problems, because it is often a realistic inductive bias for physical systems, can improve extrapolation, and makes model training more robust.  The model works by representing each of the constituent terms of a port-Hamiltonian system (Hamiltonian, structure matrix, dissipation matrix, and input matrix) with a neural network. The key contribution seems to be the choice of architecture for the Hamiltonian network; by ensuring it is convex (and has a minimum of value 0 at the origin) global stability is enforced. The proposed model is validated on a number of different examples, both with real and synthetic data, and is shown to perform better than alternatives.\n\nSTRENGTHS AND WEAKNESSES:\n## Strengths\n\n- The paper is technically sound to the best of my knowledge (Note: I have not checked the proofs in detail, and do not have a deep knowledge of the theory employed). The main claims made by the paper are adequately supported by the experiments. The experiments are standard for papers in this area.\n- The clarity of the paper is excellent. The writing is clear and easy to follow, and the mathematical exposition precise. The paper has a very polished feel, and the figures and plots are well designed and communicate the experimental results well. I enjoyed reading this paper!\n- While impact is difficult to judge, the proposed method seems to solve the problem in a more comprehensive way than past work (e.g OnsagerNet).\n- The paper is original in the sense that it offers a new combination of existing techniques.\n\n## Weaknesses \n\n- For me, the main weakness of this paper is the lack of clear explanation of the relationship of the proposed method to prior work. This makes it difficult for the reader to understand exactly what the contribution is. A number of related methods are discussed in the introduction, but it is unclear which of these proposed methods could be used to tackle the problems in the experiments section. The only existing method that seems to be used as comparison in the experiments is the sNODE, could other methods be applied here?\n- If I am correct in my understanding that the key contribution is constraining the form of the Hamiltonian network, then the contribution of the paper is modest in relation to prior work.\n\nQUESTIONS:\n- Please can you more clearly describe the contribution of the proposed model in relation to prior work? \n- Can you explain in more detail which real word problems this type of model can be applied to? How easy is it to determine if a real system is globally stable within the region of interest? How well does the model deal with e.g. noisy data?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-06-24"}

comments: SUMMARY:
This paper aims to incorporate physical prior into dynamical system learning. The authors introduce a framework named stable port-Hamiltonian neural networks. This framework aims to previous the physical biases while ensuring global Lyapunov stability. They also establish the requirement for stability of port-Hamiltonian systems and utilize this for model design. The authors evaluate the proposed method on different systems in comparison to NODEs, PHNNs, and bounded PHNNs (bPHNNs).

STRENGTHS AND WEAKNESSES:
Strengths

1. This paper is well organized and clearly written. 

2. The technical details are also easy to follow.

3. The authors evaluate the proposed method on different systems to validate the effectiveness of the proposed method.

Weaknesses

1. The major concern is that the compared methods are out-of-date and limited. I do suggest that authors should include more state-of-the-art and up-to-date methods in performance comparison given that there are so many works for dynamical system modeling.

2. It seems that all the data is generated by simulation. Are there any real-world datasets to validate the effectiveness of the proposed method? 

3. The difference between Stable Port-Hamiltonian Neural Networks and Stable NODEs should be carefully discussed to show the novelty of the proposed method. 

4. "Why port-Hamiltonian dynamics?" seems not to be the contribution of the paper. This part should move to the preliminary section instead.

QUESTIONS:
Please see weaknesses.

LIMITATIONS:
Yes
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 2
DATE: 2025-07-02

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis work concerns an introduction of new neural network architecture for data-driven dynamical system identification based on port-Hamiltonian systems. The architecture is based on the port-Hamiltonian system formulation, in which (1) the structure matrix and the dissipation matrix are each modeled by feed forward neural networks and their structure is enforced (skew-symmetry and symmetric positive semi-definite), (2) the port-Hamiltonian is modeled by a fully input convex neural network (FICNN). The authors show that this new formulation is asymptotically stable, based on a theorem developed in the work. The new model is tested on a few dissipative systems, and shows improved asymptotic behavior as well as superior overall accuracy.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths\n\n- The presentation is clear and straightforward. The new architecture as well as the main theorem is explained clearly.\n\n- The specific formulation of the port-Hamiltonian is new to this reviewer (although the use of input convex neural networks for Hamiltonian systems is not new).\n\nWeaknesses\n\n- The developed theorem requires a specification of a special point in the state space $\\mathbf{x}_0$. This requirement is a bit peculiar, given that generally stability conditions are often formulated without specifying special points. This point is not sufficiently elaborated upon in the paper; how do you select this point in general situations? Does one need to retrain the model if the point needs to be changed?\n\nQUESTIONS:\n- Can this framework be extended to parametrized systems?\n\n- Why is the Lyapunov function introduced, if it is not required in the Theorem? It appears the function is only necessary in the proof?\n\n- Is there another baseline comparison beyond deep learning models (that also incorporate classical methods like SINDy: Lee, Kookjin, Nathaniel Trask, and Panos Stinis. \"Structure-preserving sparse identification of nonlinear dynamics for data-driven modeling.\" Mathematical and Scientific Machine Learning. PMLR, 2022.)?\n\nLIMITATIONS:\n- The explicit specification using $\\mathbf{x}_0$ should be elaborated upon. More examples where equilibrium points are not known a priori would be enlightening.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 5, "CLARITY": 4, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-03"}