Neural method tracks how local conservation laws break down globally
Original title: From Flow-Box Invariants to Global Conservation Laws: Machine Learning-Assisted Detection of Singularities and Near-Integrability
Expanding-domain neural deflation recovers the expected local invariants, then their successive breakdown.
The paper formulates continuation of learned conservation laws from a sampled region to larger domains of phase space. An expanding-domain neural deflation method tracks how many independent laws persist as the domain grows. It recovers the expected local count and their breakdown toward the global limit, and relates extra surviving laws to proximity to Liouville integrability.
- Locally, a d-dimensional system generically has d-1 independent conservation laws.
- Pointwise losses can localize isolated obstructions.
- In complicated dynamics, loss breakdown is gradual and spatially diffuse.
From the original article
1 min read at arxiv.orgConservation laws reveal invariant structure in dynamical systems and constrain their evolution to lower-dimensional level sets. Locally, however, per the flow-box theorem, every $d$-dimensional system generically admits $d-1$ functionally independent conservation laws near a regular point even though, generically, only a few such laws persist globally.
By Jimmie Adriazola · arxiv.org
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