论文标题
关于超级汉密尔顿 - 雅各比方程的添加特征值的域的规律性
The regularity with respect to domains of the additive eigenvalues of superquadratic Hamilton--Jacobi equation
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 study the additive eigenvalues on changing domains, along with the associated vanishing discount problems. We consider the convergence of the vanishing discount problem on changing domains for a general scaling type $Ω_λ= (1+r(λ))Ω$ with a continuous function $r$ and a positive constant $λ$. We characterize all solutions to the ergodic problem on $Ω$ in terms of $r$. In addition, we demonstrate that the additive eigenvalue $λ\mapsto c_{Ω_λ}$ on a rescaled domain $Ω_λ= (1+λ)Ω$ possesses one-sided derivatives everywhere. Additionally, the limiting solution can be parameterized by a real function, and we establish a connection between the regularity of this real function and the regularity of $λ\mapsto c_{Ω_λ}$. We provide examples where higher regularity is achieved.
