论文标题
部分可观测时空混沌系统的无模型预测
The Mean Field Fokker-Planck Equation with Nonlinear No-flux Boundary Conditions
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 consider the mean field Fokker-Planck equation subject to nonlinear no-flux boundary conditions, which necessarily arise when subjecting a system of Brownian particles interacting via a pair potential in a bounded domain. With the additional presence of an external potential $V_1$, we show, by analysing the linearised Fokker-Planck operator, that the spectral properties of the equilibrium densities can differ considerably when compared with previous studies, e.g., with periodic boundary conditions. Amongst other mean field models of complex many-body particle systems, we present numerical experiments encompassing in a wide range of physical applications, including: generalised exponential models (Gaussian, Morse); a Kuramoto model, for noisy coupled oscillators; and an Onsager model for liquid crystals. We showcase our results by using the numerical methods developed in the pseudospectral collocation scheme 2DChebClass.
