论文标题
部分可观测时空混沌系统的无模型预测
Stability conditions in geometric invariant theory
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。
We explain how structures analogous to those appearing in the theory of stability conditions on abelian and triangulated categories arise in geometric invariant theory. This leads to an axiomatic notion of a central charge on a scheme with a group action, and ultimately to a notion of a stability condition on a stack analogous to that on an abelian category. We use these ideas to introduce an axiomatic notion of a stability condition for polarised schemes, defined in such a way that K-stability is a special case. In the setting of axiomatic geometric invariant theory on a smooth projective variety, we produce an analytic counterpart to stability and explain the role of the Kempf-Ness theorem. This clarifies many of the structures involved in the study of deformed Hermitian Yang-Mills connections, Z-critical connections and Z-critical Kähler metrics.
