Context-Informed Neural ODEs Unexpectedly Identify Broken Symmetries: Insights from the Poincaré–Hopf Theorem

We sincerely appreciate the reviewer’s thoughtful and constructive feedback. Below, we address each comment carefully. Additional experimental results are available in the README at https://github.com/anonymous-account123/icml2025-7637 and will be thoroughly incorporated into our revised paper.

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Q1. Reliability of the zero-shot identification.

We fully agree with your comment that evaluating the reliability of bifurcation identification is of critical importance. To statistically validate the robustness of zero-shot bifurcation identification, we conducted repeated experiments (five runs) using different random seeds and initial conditions. Consistently, the model reliably identified post-bifurcation behaviors, demonstrating statistical robustness and reproducibility (Figure 1(a) at the provided link).

In addition, we performed additional robustness tests under realistic scenarios, including noisy observations and limited training data. These challenging settings revealed that the model remains capable of accurately identifying symmetry-breaking bifurcations, albeit with slightly increased variance compared to the ideal noise-free scenario (Figure 1(b-c) at the provided link).

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Q2. Hallucinated bifurcations and empirical validation of the theorem.

The hallucinated symmetry-breaking scenario in Section 4.3 is purposefully designed to illustrate the predictive capability of Proposition 4.1. In the analyzed linear system, a simple center-to-saddle bifurcation is expected if the model merely approximates the functional form. However, our empirical results demonstrate a spurious symmetry-breaking bifurcation, as predicted by Proposition 4.1, confirming that the model genuinely leverages topological constraints rather than simple functional approximation. We will clarify this critical interpretation in the revised paper as you suggested.

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Q3. How to detect and avoid hallucinated bifurcations.

We completely agree that identifying whether a model hallucinates bifurcations is vital. We propose a straightforward yet robust criterion: evaluating the variance in bifurcation diagrams generated by multiple independent training runs. Correctly identified bifurcations show minimal variance, indicating stable and consistent identification. Conversely, hallucinated bifurcations yield significantly higher variance due to their structural instability. Figure 2 at the provided link clearly demonstrates this differentiation approach.

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Q4. Other type of bifurcations or dynamical systems.

Following your suggestion, we investigated a non-Hamiltonian dynamical system exhibiting a codimension-2 cusp bifurcation: $(\dot{x}, \dot{y}) = (y, b + a x - x^3 – d y)$, with variable parameters $(a, b)$ and fixed $d = 0.5$. This system serves as a canonical model for capturing catastrophic transitions and hysteresis phenomena, which are fundamental in science and engineering. Remarkably, despite being trained only on the simple, monostable pre-bifurcation regime $(a,b) \in \[-2.0, -1.5, -1.0, -0.5 ]^2$, the model successfully identified the cusp bifurcation surface, validating the model's broader applicability. Detailed visualizations are available in Figure 3 at the provided link.

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Q5. Comparison with PINNs.

We acknowledge that PINNs effectively utilize explicit physical priors. However, they become inapplicable if physical laws are unknown or incorrectly assumed. Our proposed approach circumvents this limitation by leveraging general topological invariants. For instance, models representing vector fields on a sphere (e.g., climate models) naturally adopt a global index constraint of $\chi(S^2) = +2$. Such universally applicable prior knowledge positions our approach as complementary and potentially advantageous in scenarios where explicit physical laws are unavailable. This point will be discussed in the revised manuscript.

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Q6. Computational complexity.

Modeling $N$ environments using vanilla NODEs with $M$ parameters requires $N \times M$ parameters. Conversely, the CI-NODEs reduce this requirement to approximately $M(1 + K)$, where $K$ represents the context dimensionality, showing substantial parameter efficiency, particularly when $N > K$.

Regarding computational overhead from topological regularization, our experiments confirm a moderate increase in computation time (from 53.5 ms/epoch for vanilla models to 80.9 ms/epoch for regularized models under the same environment). Nevertheless, the regularized model benefits from improved learning stability and accordingly demonstrates faster empirical convergence, as shown in Figure 4 at the provided link. We will clearly address this trade-off in the revision.

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Q7. Clarity of the paper.

We appreciate your suggestions for clarity improvement. We will refine the main text and Appendix to highlight essential details clearly, and will include the additional discussions and experiments made during the rebuttal.

We sincerely appreciate the reviewer’s thoughtful and constructive feedback. Below, we address each comment carefully. Additional experimental results are available in the README at https://github.com/anonymous-account123/icml2025-7637 and will be thoroughly incorporated into our revised paper.

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Q1. Brenner et al.

Thank you for pointing out the relevant reference, Brenner et al. (2024), which also enables both interpolation and extrapolation of model parameters via linear modulation of subject-specific features. However, as you noted, it does not address the bifurcation-driven OOD, specifically the case where training is conducted solely in the pre-bifurcation regime while the post-bifurcation regime is explored at test time. We will cite this work and include a discussion in the revised paper.

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Q2. How to detect and avoid hallucinated bifurcations.

The hallucinated symmetry-breaking experiment in Section 4.3 is intentionally designed as a pathological case to illustrate how the model mistakenly classifies a simple center-to-saddle bifurcation as a symmetry-breaking one, as predicted by Proposition 4.1. In the linear system considered in Section 4.3, if the NODE model is merely approximating a known functional form, it should undergo a simple center-to-saddle bifurcation. However, the experimental results show that the NODE model instead undergoes a spurious symmetry-breaking bifurcation. This indicates that the model is leveraging (or in this case misusing) topological constraints. Therefore, its behavior should be interpreted using the topological arguments provided by Poincaré–Hopf theorem and Proposition 4.1.

We completely agree that identifying whether a model hallucinates bifurcations is vital. We propose a straightforward yet robust criterion: evaluating the variance in bifurcation diagrams generated by multiple independent training runs. Correctly identified bifurcations show minimal variance, indicating stable and consistent identification. Conversely, hallucinated bifurcations yield significantly higher variance due to their structural instability. Figure 2 at the provided link illustrates this distinction clearly, highlighting how increased variance serves as an indicator of hallucinated behavior. We will include this result in the revised version, along with appropriate discussion.

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Q3. Non-Hamiltonian DS.

Following your suggestion, we tested the context-informed NODEs on a non-Hamiltonian, codimension-2 cusp bifurcating system: $(\dot{x}, \dot{y}) = (y, b + a x - x^3 – d y)$, with variable parameters $(a,b)$ and fixed $d=0.5$. This system serves as a canonical model for capturing catastrophic transitions and hysteresis phenomena, which are fundamental in science and engineering. Despite its dissipative, non-Hamiltonian nature, its bounded dynamics imply a preserved total Poincaré index of +1. Training was conducted exclusively on simple, monostable pre-bifurcation conditions within the parameter range $(a,b) \in [-2.0, -1.5, -1.0, -0.5]^2$, yet the model successfully identified the cusp bifurcation surface, confirming broader applicability (Figure 3 at the provided link). This result will be included in our revised paper.

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Q4. Regularization without physical priors.

We agree with your comment that topological regularization requires a certain level of prior knowledge about the system. In particular, the amount of required knowledge depends on whether we apply only a global constraint or both global and local ones. Local regularization typically requires more detailed, system-specific information. In contrast, using only global constraints demands relatively less prior knowledge. As stated in the Poincaré–Hopf theorem, it depends only on coarse information about the phase manifold on which the ODE is defined. For example, in a 2D closed-orbit system, the total Poincaré index is fixed at +1. Thus, as long as we know the system is non-divergent, this global constraint can often be applied without detailed knowledge of the vector field itself. In other setting—such as when the phase manifold is a sphere—the total index is determined by the Euler characteristic. For instance, when modeling vector fields on Earth (as in climate models), it is typically reasonable to assume a global index of $\chi(S^2) = +2$.

This type of global information can be especially useful when the available training data is confined to a restricted domain of initial conditions. To assess the effectiveness of topological regularization under minimal prior assumptions, we revisited the experiment in Section 5 using only the global constraint. In this additional study, we limited the training domain to $[-1.0, 1.0]^2$ to mimic a restricted coverage scenario. Figure 5 at the provided link illustrates that the topologically regularized model exhibits improved performance, even when relying solely on global information. We will include this result in the revised paper.

We sincerely appreciate the reviewer’s thoughtful and constructive feedback. Below, we address each comment carefully. Additional experimental results are available in the README at https://github.com/anonymous-account123/icml2025-7637 and will be thoroughly incorporated into our revised paper.

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Q1. As acknowledged by the authors, the scope of the investigation is limited.

Our analysis primarily targets closed-orbit systems (e.g., Hamiltonian flows), but the underlying principle of Proposition 4.1—the topological invariance of the total index from the Poincaré–Hopf theorem—has broader applicability. Indeed, Proposition 4.1 follows directly from Theorem 4.1 under closed orbit assumptions. Yet, the essence of the theorem lies in global topological invariants, specifically the Euler characteristic of the phase space $\mathcal{M}$ (e.g., $\chi(S^{2n}) = 2$, $\chi(S^{2n+1}) =0$, $\chi(T^{n}) = 0$, ...), allowing its application even when the associated ODE defined on $\mathcal{M}$ does not exhibit closed orbits or conserved quantities. We will explicitly address this generalization potential in our revision.

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Q2. The regularization term seems to require detailed knowledge of the dynamical system beforehand, such as the global and local contour. This may limit the applicability of the regularization.

We agree with your comment that topological regularization requires a certain level of prior knowledge about the system. In particular, the amount of required knowledge depends on whether we apply only a global constraint or both global and local ones. Local regularization typically requires more detailed, system-specific information. In contrast, using only global constraints demands relatively less prior knowledge. As stated in the Poincaré–Hopf theorem, it depends only on coarse information about the phase manifold on which the ODE is defined. For example, in a 2D closed-orbit system, the total Poincaré index is fixed at +1. Thus, as long as we know the system is non-divergent, this global constraint can often be applied without detailed knowledge of the vector field itself. In other setting—such as when the phase manifold is a sphere—the total index is determined by the Euler characteristic. For instance, when modeling vector fields on Earth (as in climate models), it is typically reasonable to assume a global index of $\chi(S^2) = +2$.

This type of global information can be especially useful when the available training data is confined to a restricted domain of initial conditions. To assess the effectiveness of topological regularization under minimal prior assumptions, we revisited the experiment in Section 5 using only the global constraint. In this additional study, we limited the training domain to $[-1.0, 1.0]^2$ to mimic a restricted coverage scenario. Figure 5 at the provided link illustrates that the topologically regularized model exhibits improved performance, even when relying solely on global information. We will include this result in the revised paper.

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Q3. On page 3 line 160, the left column, the definition of OOD condition for parameters in not clear for n-dimensional parameters.

Following your suggestion, we formally describe the definition of parameter OOD conditions with codimension-$k$ bifurcation in $n$-dimensional parameter space:

Let $\mathcal{P} \subset \mathbb{R}^n$ be a parameter space and let $\mathcal{B} \subset \mathcal{P}$ be a bifurcation set. The complement $\mathcal{P} \setminus \mathcal{B}$ admits a decomposition into $L$ disjoint connected components: $\mathcal{P} \setminus \mathcal{B} = \bigcup_{i=1}^L \mathcal{P}_i$, where each $\mathcal{P}_i$ is a maximal connected subdomain. Intuitively, each $\mathcal{P}_i$ is one qualitatively uniform parameter regime. Then, a parameter OOD condition in learning dynamics arises when the support of training distribution is $\mathrm{supp}(p^\mathrm{tr}_e(\mu)) \subseteq \mathcal{P}_l$, but the support of the test distribution is $\mathrm{supp}(p^\mathrm{test}_e(\mu)) \subseteq \mathcal{P}_i$ for some $i \neq l$. Equivalently, there exists no continuous path $\gamma: [0, 1] \to \mathcal{P} \setminus \mathcal{B}$ connecting $\mu^\mathrm{tr}_e$ and $\mu^\mathrm{test}_e$ without crossing $\mathcal{B}$.

This generalized notion of parameter OOD parallels the concept of initial condition OOD, enabling a unified perspective on OOD in learning dynamics. Accordingly, we will revise our paper to correctly describe this more general setting beyond the one-dimensional case.

We sincerely appreciate the reviewer’s thoughtful and constructive feedback. Below, we address each comment carefully. Additional experimental results are available in the README at https://github.com/anonymous-account123/icml2025-7637 and will be thoroughly incorporated into our revised paper.

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Q1. Do NODEs genuinely utilize topology? Is the hallucinated broken symmetry general?

You highlight a critical point about distinguishing whether context-informed NODEs genuinely leverage topological constraints or merely approximate functions that happen to exhibit topological properties. Indeed, the double-well scenario poses a dilemma: better approximations naturally reflect correct symmetry-breaking behavior. Our hallucinated bifurcation scenario in Section 4.3, grounded explicitly in Proposition 4.1, is intentionally designed to address this challenge. If NODEs were merely approximating a functional form, a simple center-to-saddle bifurcation would emerge. Instead, the model exhibits an incorrect symmetry-breaking bifurcation, confirming the model genuinely utilizes (in this scenario, misuses) topological constraints, precisely as Proposition 4.1 predicts. Hence, our findings demonstrate that NODEs indeed exploit topological invariants. In addition, the hallucinated symmetry-breaking can arise whenever Proposition 4.1's conditions hold. We will highlight this more clearly in the revised manuscript.

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Q2. Robustness of bifurcation identification under noisy conditions.

We acknowledge that approximation errors may increase under noisy perturbations or with limited training samples, as you pointed out. To address this, we conducted further experiments under these realistic conditions. Despite the increased variance, the model consistently identified the symmetry-breaking bifurcation, demonstrating its robustness. The results are illustrated in Figure 1(b-c) at the provided link, which will be included in the revised manuscript.

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Q3. Non-Hamiltonian and codimension-higher bifurcations.

Following your suggestion, we tested the context-informed NODEs on a non-Hamiltonian, codimension-2 cusp bifurcating system: $(\dot{x}, \dot{y}) = (y, b + a x - x^3 – d y)$, with variable parameters $(a,b)$ and fixed $d=0.5$. This system serves as a canonical model for capturing catastrophic transitions and hysteresis phenomena. Despite its dissipative, non-Hamiltonian nature, its bounded dynamics imply a preserved total Poincaré index of +1. Training was conducted exclusively on simple, monostable pre-bifurcation conditions within the parameter range $(a,b) \in [-2.0, -1.5, -1.0, -0.5]^2$, yet the model successfully identified the cusp bifurcation surface, confirming broader applicability (Figure 3 at the provided link). This result will be included in our revised paper.

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Q4. Generalization of Proposition 4.1.

Our analysis primarily targets closed-orbit systems (e.g., Hamiltonian flows), but the underlying principle of Proposition 4.1—the topological invariance of the total index from the Poincaré–Hopf theorem—has broader applicability. Indeed, Proposition 4.1 follows directly from Theorem 4.1 under closed orbit assumptions. Yet, the essence of the theorem lies in global topological invariants, specifically the Euler characteristic of the phase space $\mathcal{M}$ (e.g., $\chi(S^{2n}) = 2$, $\chi(S^{2n+1}) = 0$, $\chi(T^{n}) = 0$, ...), allowing its application even when the associated ODE defined on $\mathcal{M}$ does not exhibit closed orbits or conserved quantities. We will explicitly address this generalization potential in our revision.

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Q5. Experimental designs.

We have provided a detailed description and implementation of our proposed topological regularization approach in Appendix Section G. Additionally, Figure 17 in Appendix illustrates consistency scores across various threshold values ($\sigma$) ranging from 0.2 to 1.0, for the results of Section 5. The regularized model consistently outperforms vanilla models, maintaining scores close to 1.0 regardless of $\sigma$. Furthermore, Appendix Section H thoroughly summarizes our experimental setup (e.g., 16 training and 441 testing parameter combinations over an $(\alpha,\beta)$ grid).

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Q6. Why four training samples and one for adaptation?

Our experimental setup closely mirrors Kirchmeyer et al. (2022), who utilized four initial conditions per parameter for training on the Lotka–Volterra system—another 2D closed-orbit system analogous to ours. For adaptation, a single confined trajectory was selected intentionally to simulate an extreme symmetry-breaking scenario. This deliberately restricts the model’s exposure to the global phase space structure, effectively testing simultaneous parameter and initial condition OOD challenges.

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Q7. Minor comments.

We will clarify the manuscript as recommended. For the clarified definition of OOD in parameters, please refer to our response to Question 3 from Reviewer rQs3.