Poisson-Dirac Neural Networks for Modeling Coupled Dynamical Systems across Domains

Thank you for your thoughtful feedback and constructive suggestions. We appreciate the recognition of our paper's solid theoretical foundation and extensive experiments to demonstrate the effectiveness. Below, we address each of the points raised. > W1: enhance its accessibility to a broader audience Thank you for this valuable suggestion. To the best of our knowledge, there is no comprehensive textbook specifically dedicated to analytical mechanics using Dirac structures. For an accessible introduction to differential geometry, we will newly cite Abraham et al., Manifolds, Tensor Analysis, and Applications, Springer, 1988. Regarding the relationship between differential geometry and analytical mechanics, we will refer readers to the already-cited Marsden & Ratiu (1999). The port-Hamiltonian formulation, which closely relates to our Poisson-Dirac framework, is thoroughly documented in van der Schaft & Jeltsema (2014). Finally, for a rigorous treatment of the Poisson-Dirac formulation itself, Courant (1990) is a key reference. We will ensure that these references, along with their explanations, are included at the beginning of Section 3 to better guide readers. > Q1: it would be interesting to compare its performance with state-space model (SSM) based neural operators on the datasets used in this paper. To our best knowledge, neural operators are primarily designed to learn mappings from functions to functions, and hence they are particularly suited for PDEs. In contrast, our work focuses on learning ODEs as mappings from vectors to vectors. Thus, the objectives differ significantly. Additionally, SSMs generally incorporate hidden states, whereas our study assumes that all states are directly observable. This further differentiates the objectives of the two approaches. If one were to construct SSMs without neural operators or hidden states, they would effectively reduce to a variant of NODEs. Therefore, a direct comparison with such an approach would not provide meaningful insights. That said, we agree that extending PoDiNNs to handle PDEs and hidden states is an exciting and important avenue for future work. Combining PoDiNNs with neural operators could be particularly promising in such contexts. We will elaborate on this direction in the final version of the paper. > Q2: the authors could consider adding an appendix to provide additional explanations and visualizations of the key concepts and techniques. Thank you for your guidance. We have updated the submitted PDF to replace Fig. 1 and add a new subsection at the end of the manuscript. In the updated Fig. 1, we have clarified that each arrow represents a corresponding flow or effort variable and that the Dirac structure $D$ couples these variables. Additionally, the newly added subsection provides a matrix-based representation of PoDiNNs and demonstrates how PoDiNNs reduce to HNNs under specific conditions. We believe this will help clarify how PoDiNNs extend the basic HNNs and make them more accessible to a broader audience.

Thank you for the thoughtful and constructive review. We deeply appreciate your recognition of our paper as well-written and concise, and for finding the identification of coupling patterns fascinating. Given your deep understanding, we are curious about the reason why the confidence level was 2. Below, we address each of your points in detail, and we hope that these clarifications will help improve your confidence in our work. > W1: I think it would be good to expand this diagram and clearly indicate what parts of this system the method proposes to replace with learned functions (NNs) and relate it to e.g. eq 5, def 3. I think this would greatly help with readability. Thank you for this valuable suggestion. We have updated the submitted PDF and replaced Fig. 1. In the new figure, we clarified that each arrow represents a part of flows and efforts, and the Dirac structure $D$ couples these variables, as in Definition 3. We believe this revision provides a clearer correspondence between the components and equations. > W2: how well this generalized method works in the settings that the originally proposed HNN and LNN are validated on. Thank you for raising this important point. In the final section of the updated PDF, we have clarified that PoDiNNs reduce to HNNs under conditions where HNNs are applicable--namely, when applied to systems in Darboux coordinates without degeneracy, energy dissipation, or external forces. Consequently, a direct comparison between PoDiNNs and HNNs is not meaningful. > W3: compare your method against the baselines also in terms of computational complexity / overhead We sincerely acknowledge that PoDiNNs come with a higher computational overhead compared to the baselines. HNNs and LNNs use a single neural network, and dissipative SymODENs are implemented with three neural networks for $H$, $D$, and $G$. In contrast, PoDiNNs require a separate neural network for each nonlinear component due to their decomposition approach, along with the added computation for the bivector. For example, for system (a) in the relative coordinate system, PoDiNNs used five neural networks for three springs and two dampers, leading to approximately double the computational complexity compared to dissipative SymODENs. Nonetheless, we view this computational overhead as a reasonable trade-off for the improved accuracy and interpretability provided by PoDiNNs. It is worth noting that if explicit decomposition were to be omitted and multiple components were consolidated into a single neural network, the computational cost would align with that of the comparative methods, but the modeling performance would also be comparable. > W4: the reliance on specifying problem-specific constraints, i.e. the number of components is a model hyperparameter You are correct. Assuming a larger number of components, the computational overhead does indeed increase, but there is no degradation in performance. As noted earlier, we view this trade-off as justified, given the significant gains in interpretability and modeling performance that PoDiNNs provide.

Thank you for reviewing our manuscript and for recognizing that PoDiNNs achieve lower errors and provide stable predictions over longer time horizons. > W1: The content might be difficult to digest for the audience of ICLR, since a specific mathematical and physics background is required to understand the work. We aimed to emphasize the theoretical foundations and broad applicability of PoDiNNs, which led us to dedicate significant space to abstract discussions of differential geometry. However, the implementation itself is quite accessible. When appropriate local coordinates are introduced on the state space, a bivector is represented by a matrix, and PoDiNNs are implemented using matrix-vector multiplications. To aid in understanding, we have updated the submitted PDF file and presented the matrix-based representation in the final section, as well as its relation to prior works. We believe this will help clarify how PoDiNNs extend the basic Hamiltonian Neural Networks and make them more intuitive for a broader audience.

Thank you for the thoughtful review of our paper. We sincerely appreciate your recognition of the novelty, clarity, and solid theoretical analysis of our work, as well as the comprehensive evaluation we conducted. We are grateful for your constructive feedback and are pleased to address your concerns: > W1: through iterative refinement of two (or more) simulations of a single domain or system? Multiphysics simulations with iterative refinement are referred to as "weak coupling." In this work, we aim to achieve "strong coupling," where all domains are computed simultaneously rather than iteratively. In Poisson-Dirac formulation, the bivector element represents the flow of physical quantities--such as force, velocity, current, or voltage--between components. Its skew-symmetric property ensures that energy inflows and outflows between domains remain balanced, with energy loss or supply occurring only within individual components. This constraint defines the interactions across domains, making "strong coupling" feasible. By leveraging this framework, PoDiNNs enable the data-driven learning of inter-domain interactions. > W2: How to ensure the PoDiNNs capture the correct physics without causing overfitting problem or continuous good performance in extrapolation? As you pointed out, unlike PINN-alike works, PoDiNNs do not rely on predefined physical symbolics. Instead, methods like Hamiltonian Neural Networks leverage the equation frameworks (e.g., Hamilton's equations, Euler-Lagrange equations, etc.), which ensure the correct physics to be captured. Furthermore, PoDiNNs introduce assumptions that are weaker than those in PINNs but stronger than those in Hamiltonian Neural Networks: the target system can be decomposed into interacting components. For example, in a coupled system with two masses and two springs in a 1D space, the full state space is four-dimensional (two positions and two velocities), which requires a mapping $R^4\to R^4$ to define the vector field. When decomposing the system into four separate components, we can represent it using four separate mappings $R\to R$. In this way, we can mitigate the curse of dimensionality and enhance generalization. We will add this explanation to clarify our strategy for achieving robust extrapolation. > W3: PoDiNNs focus on behaviors that may not be captured by generic models, e.g., NODE, which seem to need intensive resources. You are correct. For instance, if our target system includes four components, we need to compute four neural networks, along with the bivector. This increases the computational cost to at least four times that of a single NODE. We view this computational overhead as a reasonable trade-off for the improved accuracy and interpretability provided by PoDiNNs. We will include this clarification in the revised manuscript. > W4: Could the reviewer further explain with more details: why ODEs or using NODE alone cannot capture the current flow and balanced voltage level? Thank you for this insightful question. While neural networks used in NODEs are universal approximators capable of learning the current flows and balanced voltage levels, they inherently introduce approximation errors. In practice, this means that Kirchhoff's laws are only approximately satisfied, and over time, these errors tend to accumulate, becoming significant. In contrast, PoDiNNs explicitly encode the current flows and balanced voltage levels using bivector elements, leading to Kirchhoff's laws, independently of the component-wise properties. This explicit separation enhances robustness. We will include this explanation in the revised manuscript.

Thank you for taking the time to review our manuscript and for recognizing our research as a very nice contribution to a very important topic of the integration of physics principles into neural networks. We appreciate your thoughtful feedback and would like to address the concerns you raised, some of which may stem from a lack of clarity in our explanations. > W1: The models used seem to be all linear: > > Q1: The models considered all seem to be linear. > > Q2: Can this generalize to nonlinear models? Or simply be applied to nonlinear models with success? *Our proposed method, PoDiNNs, is indeed nonlinear.* As illustrated in Fig. 4, our model successfully learns the nonlinear characteristics of Chua's diode, as well as other springs, dampers, and resistors. While the Dirac structure, symplectic forms, and Poisson bivectors are linear, this linearity is well-supported by theoretical foundations in analytical mechanics. Indeed, the significance of PoDiNNs lies in decomposing the coupled system into linear interactions and nonlinear component characteristics. This separation is essential for achieving both high modeling accuracy and interpretability. > W1: time-delay embedded DMD which models missing/coupled physics Our model identifies the characteristics of energy-dissipating components, such as diodes and dampers, whose inputs and outputs are unobservable. It is important to note that these components do not have states, and we did not attempt to estimate missing (unobservable) states. Instead, our model focuses on identifying the unknown relationships between observable states (of masses, springs, etc.) determined by unobservable components (like dampers and diodes). We acknowledge the importance of estimating unobservable states, and methods like time-delay embeddings could indeed be useful for that purpose. We will clarify in the revised manuscript that this topic is beyond the scope of our current work and is a promising direction for future research. > W2-W3: There are statements that are simply not true: The statements you mentioned were intended to reflect limitations specifically within the domain of Hamiltonian neural networks and similar approaches that integrate neural networks with analytical mechanics, as summarized in Table 1. We believe that, within this context, our claims are accurate. However, you are right that there are numerous alternative methods, such as DMD, Koopman operator theory, state-space models (including mamba/S4), and SINDy, which address a broader spectrum of systems, including non-mechanical and coupled systems. We will revise Section 2 of our manuscript to clarify our focus and provide a more comprehensive overview of related approaches to better situate our work within the broader landscape. > Q3: What have you not considered non-neural network methods like DMD... or system ID methods like mamba/S4 not been compared? General modeling methods like DMD or general system ID methods like mamba/S4 are indeed broadly applicable. However, because these methods typically do not ensure physical laws (such as laws of energy conservation and dissipation or Kirchhoff's laws). As confirmed in the prior studies summarized in Table 1, these general methods often face challenges in producing accurate long-term predictions and providing the physical interpretability. In contrast, as demonstrated in Section 4, PoDiNNs identify systems in the form of coupled components with individual characteristics (e.g., energy-storing or dissipating), which offers more interpretable and physically meaningful insights. This distinction arises from the difference in objectives, rather than a matter of superiority. Consequently, we believe it is appropriate to acknowledge these methods in the related work section, without making further direct comparisons.