论文标题
hamilton-jacobi方程的粘度解决方案$ rcd（k，\ infty）$ spaces and应用程序的应用
Viscosity solutions of Hamilton-Jacobi equation in $RCD(K,\infty)$ spaces and applications to large deviations
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。
The aim of this paper is twofold. - In the setting of RCD(K,$\infty$) metric measure spaces, we derive uniform gradient and Laplacian contraction estimates along solutions of the viscous approximation of the Hamilton--Jacobi equation. We use these estimates to prove that, as the viscosity tends to zero, solutions of this equation converge to the evolution driven by the Hopf--Lax formula, in accordance with the smooth case. - We then use such convergence to study the small-time Large Deviation Principle for both the heat kernel and the Brownian motion: we obtain the expected behavior under the additional assumption that the space is proper. As an application of the latter point, we also discuss the $Γ$-convergence of the Schrödinger problem to the quadratic optimal transport problem in proper RCD(K,$\infty$) spaces.
