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

In Huh, Changwook Jeong, Muhammad Alam
Proceedings of the 42nd International Conference on Machine Learning, PMLR 267:26311-26337, 2025.

Abstract

Out-Of-Domain (OOD) generalization is a significant challenge in learning dynamical systems, especially when they exhibit bifurcation, a sudden topological transition triggered by a model parameter crossing a critical threshold. A prevailing belief is that machine learning models, unless equipped with strong priors, struggle to generalize across bifurcations due to the abrupt changes in data characteristics. Contrary to this belief, we demonstrate that context-dependent Neural Ordinary Differential Equations (NODEs), trained solely on localized, pre-bifurcation, symmetric data and without physics-based priors, can still identify post-bifurcation, symmetry-breaking behaviors, even in a zero-shot manner. We interpret this capability to the model’s implicit utilization of topological invariants, particularly the Poincaré index, and offer a formal explanation based on the Poincaré–Hopf theorem. We derive the conditions under which NODEs can recover—or erroneously hallucinate—broken symmetries without explicit training. Building on this insight, we showcase a topological regularizer inspired by the Poincaré–Hopf theorem and validate it empirically on phase transitions of systems described by the Landau–Khalatnikov equation.

Cite this Paper

BibTeX
@InProceedings{pmlr-v267-huh25a, title = {Context-Informed Neural {ODE}s Unexpectedly Identify Broken Symmetries: Insights from the Poincaré–Hopf Theorem}, author = {Huh, In and Jeong, Changwook and Alam, Muhammad}, booktitle = {Proceedings of the 42nd International Conference on Machine Learning}, pages = {26311--26337}, year = {2025}, editor = {Singh, Aarti and Fazel, Maryam and Hsu, Daniel and Lacoste-Julien, Simon and Berkenkamp, Felix and Maharaj, Tegan and Wagstaff, Kiri and Zhu, Jerry}, volume = {267}, series = {Proceedings of Machine Learning Research}, month = {13--19 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v267/main/assets/huh25a/huh25a.pdf}, url = {https://proceedings.mlr.press/v267/huh25a.html}, abstract = {Out-Of-Domain (OOD) generalization is a significant challenge in learning dynamical systems, especially when they exhibit bifurcation, a sudden topological transition triggered by a model parameter crossing a critical threshold. A prevailing belief is that machine learning models, unless equipped with strong priors, struggle to generalize across bifurcations due to the abrupt changes in data characteristics. Contrary to this belief, we demonstrate that context-dependent Neural Ordinary Differential Equations (NODEs), trained solely on localized, pre-bifurcation, symmetric data and without physics-based priors, can still identify post-bifurcation, symmetry-breaking behaviors, even in a zero-shot manner. We interpret this capability to the model’s implicit utilization of topological invariants, particularly the Poincaré index, and offer a formal explanation based on the Poincaré–Hopf theorem. We derive the conditions under which NODEs can recover—or erroneously hallucinate—broken symmetries without explicit training. Building on this insight, we showcase a topological regularizer inspired by the Poincaré–Hopf theorem and validate it empirically on phase transitions of systems described by the Landau–Khalatnikov equation.} }
Endnote
%0 Conference Paper %T Context-Informed Neural ODEs Unexpectedly Identify Broken Symmetries: Insights from the Poincaré–Hopf Theorem %A In Huh %A Changwook Jeong %A Muhammad Alam %B Proceedings of the 42nd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2025 %E Aarti Singh %E Maryam Fazel %E Daniel Hsu %E Simon Lacoste-Julien %E Felix Berkenkamp %E Tegan Maharaj %E Kiri Wagstaff %E Jerry Zhu %F pmlr-v267-huh25a %I PMLR %P 26311--26337 %U https://proceedings.mlr.press/v267/huh25a.html %V 267 %X Out-Of-Domain (OOD) generalization is a significant challenge in learning dynamical systems, especially when they exhibit bifurcation, a sudden topological transition triggered by a model parameter crossing a critical threshold. A prevailing belief is that machine learning models, unless equipped with strong priors, struggle to generalize across bifurcations due to the abrupt changes in data characteristics. Contrary to this belief, we demonstrate that context-dependent Neural Ordinary Differential Equations (NODEs), trained solely on localized, pre-bifurcation, symmetric data and without physics-based priors, can still identify post-bifurcation, symmetry-breaking behaviors, even in a zero-shot manner. We interpret this capability to the model’s implicit utilization of topological invariants, particularly the Poincaré index, and offer a formal explanation based on the Poincaré–Hopf theorem. We derive the conditions under which NODEs can recover—or erroneously hallucinate—broken symmetries without explicit training. Building on this insight, we showcase a topological regularizer inspired by the Poincaré–Hopf theorem and validate it empirically on phase transitions of systems described by the Landau–Khalatnikov equation.
APA
Huh, I., Jeong, C. & Alam, M.. (2025). Context-Informed Neural ODEs Unexpectedly Identify Broken Symmetries: Insights from the Poincaré–Hopf Theorem. Proceedings of the 42nd International Conference on Machine Learning, in Proceedings of Machine Learning Research 267:26311-26337 Available from https://proceedings.mlr.press/v267/huh25a.html.

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