论文标题
部分可观测时空混沌系统的无模型预测
On the fourth order semipositone problem in $\mathbb{R}^N$
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。
For $N \geq 5$ and $a>0$, we consider the following semipositone problem \begin{align*} Δ^2 u= g(x)f_a(u) \text { in } \mathbb{R}^N, \, \text{ and } \, u \in \mathcal{D}^{2,2}(\mathbb{R}^N),\ \ \ \qquad \quad \mathrm{(SP)} \end{align*} where $g \in L^1_{loc}(\mathbb{R}^N)$ is an indefinite weight function, $f_a:\mathbb{R} \to \mathbb{R}$ is a continuous function that satisfies $f_a(t)=-a$ for $t \in \mathbb{R}^-$, and $\mathcal{D}^{2,2}(\mathbb{R}^N)$ is the completion of $\mathcal{C}_c^{\infty}(\mathbb{R}^N)$ with respect to $(\int_{\mathbb{R}^N} (Δu)^2)^{1/2}$. For $f_a$ satisfying subcritical nonlinearity and a weaker Ambrosetti-Rabinowitz type growth condition, we find the existence of $a_1>0$ such that for each $a \in (0,a_1)$, (SP) admits a mountain pass solution. Further, we show that the mountain pass solution is positive if $a$ is near zero. For the positivity, we derive uniform regularity estimates of the solutions of (SP) for certain ranges in $(0,a_1)$, relying on the Riesz potential of the biharmonic operator.
