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

▪ CI-NODEs [4] combine NODEs with hypernetworks to learn parameterized dynamics: ▪ Here, θ c captures the shared information across all trajectories, while ξ e serves as an environment-specific context, analogous to the model parameter μ in physical systems. ▪ In our paper, we employed CI-NODEs based on the Low-Rank Adaptation (LoRA) following [4]: ▪ There are many variants that can play a similar role with the LoRA-based CI-NODEs. ▪ Anyway, all of them are capable of forecasting physical systems under varying parameters by modulating the context vector ξ, either through adaptation or exploration.