Derivation of the probability density evolution provides invaluable insight into the behavior of many stochastic systems and their performance. However, for most real-time applica-tions, numerical determination of the probability density evolution is a formidable task. The latter is due to the required temporal and spatial discretization schemes that render most computational solutions prohibitive and impractical. In this respect, the development of an efficient computational surrogate model is of paramount importance. Recent studies on the physics-constrained networks show that a suitable surrogate can be achieved by encoding the physical insight into a deep neural network. To this aim, the present work introduces DeepPDEM which utilizes the concept of physics-informed networks to solve the evolution of the probability density via proposing a deep learning method. DeepPDEM learns the General Density Evolution Equation (GDEE) of stochastic structures. This approach paves the way for a mesh-free learning method that can solve the density evolution problem with-out prior simulation data. Moreover, it can also serve as an efficient surrogate for the solu-tion at any other spatiotemporal points within optimization schemes or real-time applica-tions. To demonstrate the potential applicability of the proposed framework, two network architectures with different activation functions as well as two optimizers are investigated. Numerical implementation on three different problems verifies the accuracy and efficacy of the proposed method.
翻译:概率密度演进的推导为许多随机系统的行为及其性能提供了非常宝贵的洞察力。然而,对于大多数实时复制者来说,对概率密度演进的数值确定是一项艰巨的任务,因为所需要的时间和空间分解计划使得大多数计算解决方案无法使用和不切实际。在这方面,开发高效的计算替代模型至关重要。最近对物理学限制的网络进行的研究表明,通过将物理洞察编入深层神经网络,可以实现适当的替代。为此,目前的工作引入了深电磁数据,利用物理知情网络的概念,通过提出深度学习方法解决概率密度的演进。深电磁磁数据小组学习了大多数计算解决方案的时空分化方案,使大多数计算解决方案变得不可及不切实际。这一方法为无网化学习方法铺平铺平了道路,该方法可以用先前模拟数据解决密度演进问题。此外,它还可以作为在任何其它具有物理知识的网络应用性网络概念概念概念的概念化概念化概念化概念化概念化概念化概念化概念化概念化概念化概念的高效的替代工具。深层平时,将两个拟议优化的模型化模型化模型化方案或再展示两种方法化框架内的拟议优化方法化方法化方法化模式化过程。