通过机器搜索$J$-全纯曲线：初步探索
Searching for $J$-holomorphic curves via machine: first steps
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中文总结 AI 辅助
本研究组装了结合经典傅里叶展开与深度神经网络的数值算法，用于辛流形中$J$-全纯曲线的搜索，验证了算法在显式已知全纯曲线场景的有效性，并实现了近复结构形变下邻近全纯曲线的搜寻。
中文摘要 AI 辅助
我们组装了用于在辛流形中搜索$J$-全纯曲线的数值算法。每种算法都采用了若干不同的数值技术，每项技术分别解决该几何问题的不同方面。我们在算法中分别考量了经典傅里叶展开与深度神经网络，并对二者的性能进行了比较。我们的算法以给定同调类中的一条光滑曲线作为输入，搜索同一同调类中的$J$-全纯曲线。我们首先验证了算法可以生成复流形中已知的显式全纯曲线，例如环面上的Weierstrass $\boldsymbol{\text{℘}}$函数，以及配备标准复结构的$S^2\times S^2$中的曲线。随后我们搜索了配备不可积近复结构的$S^2\times S^2$中的$J$-全纯曲线：本质上，我们从可积近复结构$J_0$中的一条已知全纯曲线出发，将$J_0$形变为邻近的不可积近复结构$J_ε$，再利用我们的方法找到邻近的$J_ε$-全纯曲线。
英文摘要
We assemble numerical algorithms to search for $J$-holomorphic curves in symplectic manifolds. Each algorithm employs several different numerical techniques, each technique addressing a different aspect of the geometric problem. We separately consider both classical Fourier expansion and deep neural networks in our algorithms and compare their performance. Our algorithms take as input a smooth curve in a given homology class and search for a $J$-holomorphic curve in the same homology class. We first verify we can produce explicitly known holomorphic curves in complex manifolds, for example the Weierstrass $\wp$ function on the torus and curves in $S^2\times S^2$ with the standard complex structure. Then we search for $J$-holomorphic curves in $S^2\times S^2$ with non-integrable almost complex structures: essentially we start with a known holomorphic curve in an integrable almost complex structure $J_0$, deform $J_0$ to a nearby nonintegrable almost complex structure $J_ε$, and use our methods to find the nearby $J_ε$-holomorphic curve.
