论文标题
使用PIE框架对PDE的计算稳定性分析
Computational stability analysis of PDEs with integral terms using the PIE framework
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 Partial Integral Equation (PIE) framework was developed to computationally analyze linear Partial Differential Equations (PDEs) where the PDE is first converted to a PIE and then the analysis problem is solved by solving operator-valued optimization problems. Previous works on the PIE framework focused on the analysis of PDEs with spatial derivatives up to $2^{nd}$-order. In this paper, we extend the class of PDEs by including integral terms and performing stability analysis using the PIE framework. More specifically, we show that PDEs with the integral terms where the integration is with respect to the spatial variable and the kernel of the integral operator is matrix-valued polynomials can be converted to PIEs if the boundary conditions satisfy certain criteria. The conversion is performed by using a change of variable where every PDE state is substituted in terms of its highest derivative and boundary values to obtain a new equation (a PIE) in a variable that does not have any continuity requirements. Later, we show that this change of variable can be represented using explicit maps from the parameters of the PDE to the parameters of the PIE and the stability test can be posed as an optimization problem involving these parameters. Lastly, we present numerical examples to demonstrate the simplicity and application of this method.
