论文标题
傅立叶变换的不规则自动d型，无态函数无穷大的奇异性和不规则的特征周期
Fourier transforms of irregular holonomic D-modules, singularities at infinity of meromorphic functions and irregular characteristic cycles
论文作者
论文摘要
储层计算是预测湍流的有力工具，其简单的架构具有处理大型系统的计算效率。然而，其实现通常需要完整的状态向量测量和系统非线性知识。我们使用非线性投影函数将系统测量扩展到高维空间，然后将其输入到储层中以获得预测。我们展示了这种储层计算网络在时空混沌系统上的应用，该系统模拟了湍流的若干特征。我们表明，使用径向基函数作为非线性投影器，即使只有部分观测并且不知道控制方程，也能稳健地捕捉复杂的系统非线性。最后，我们表明，当测量稀疏、不完整且带有噪声，甚至控制方程变得不准确时，我们的网络仍然可以产生相当准确的预测，从而为实际湍流系统的无模型预测铺平了道路。
Based on the recent developments in the irregular Riemann-Hilbert correspondence for holonomic D-modules and the Fourier-Sato transforms for enhanced ind-sheaves, we study the Fourier transforms of some irregular holonomic D-modules. For this purpose, the singularities of rational and meromorphic functions on complex affine varieties will be studied precisely, with the help of some new methods and tools such as meromorphic vanishing cycle functors. As a consequence, we show that the exponential factors and the irregularities of the Fourier transform of a holonomic D-module are described geometrically by the stationary phase method, as in the classical case of dimension one. A new feature in the higher-dimensional case is that we have some extra rank jump of the Fourier transform produced by the singularities of the linear perturbations of the exponential factors at their points of indeterminacy. In the course of our study, not necessarily homogeneous Lagrangian cycles that we call irregular characteristic cycles will play a crucial role.
