arXiv 论文提出扩张域神经消解框架，追踪守恒律从局部到全局的失效
From Flow-Box Invariants to Global Conservation Laws: Machine Learning-Assisted Detection of Singularities and Near-Integrability
一篇把机器学习和动力系统理论结合的论文，用扩张域训练追踪守恒律在哪里失效，还能判断系统离 Liouville 可积有多近，做可积性研究的朋友可以看看。
arXiv 论文《From Flow-Box Invariants to Global Conservation Laws》针对数据驱动方法只在训练采样区域内学习守恒律、区域外行为未知的问题，提出局部到全局延拓问题的形式化。方法采用扩张域神经消解（expanding-domain neural deflation）框架，在嵌套域上追踪独立守恒律的数量。该方法能恢复流盒定理预期的 d-1 个局部不变量，并随域扩张检测其逐级失效，收敛到预期的全局守恒律极限。研究还发现额外守恒律存续的域尺度可反映系统接近 Liouville 可积性的程度，逐点损失可定位孤立奇点。
From Flow-Box Invariants to Global Conservation Laws: Machine Learning-Assisted Detection of Singularities and Near-Integrability
Conservation 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. Data-driven methods typically learn conservation laws on a prescribed region of phase space from which training samples are drawn, leaving their behavior beyond that region unresolved. We formulate this local-to-global continuation problem and develop an {expanding-domain neural deflation} framework that tracks the number of independent conservation laws over nested domains. The method recovers the expected $d-1$ local invariants and detects their successive breakdown as the domain expands, relaxing to the expected global conservation law limit. It distinguishes the local invariants (associated with Frobenius integrability) from persistent global structure and shows that the domain scale over which additional laws survive reflects proximity to Liouville integrability. Pointwise losses can localize isolated obstructions, whereas in more complicated dynamics the breakdown of our loss functions is gradual and spatially diffuse. The framework thus provides a data-driven approach for studying the persistence and breakdown of (so-far-learned) conservation laws across phase space.