论文标题
扩散过程的概率密度函数的神经网络表示
Neural network representation of the probability density function of diffusion processes
论文作者
论文摘要
物理信息的神经网络的开发是为了表征随机环境中动力学系统的状态。神经网络近似于满足Fokker-Planck方程或在高斯和/或Poisson White噪声下满足Fokker-Planck方程的这些系统状态的概率密度函数（PDF）或特征函数（CHF）。我们在分析和数值上检查解决每种类型的微分方程以表征状态的优势和缺点。还证明了如何利用动态系统的先前信息来设计和简化神经网络体系结构。数值示例表明：1）神经网络解决方案即使对于描述PDF/CHF的时间演化的部分局部差异方程和PDE系统的局部解决方案也可以近似于目标解决方案；随机强制的类型。
Physics-informed neural networks are developed to characterize the state of dynamical systems in a random environment. The neural network approximates the probability density function (pdf) or the characteristic function (chf) of the state of these systems which satisfy the Fokker-Planck equation or an integro-differential equation under Gaussian and/or Poisson white noises. We examine analytically and numerically the advantages and disadvantages of solving each type of differential equation to characterize the state. It is also demonstrated how prior information of the dynamical system can be exploited to design and simplify the neural network architecture. Numerical examples show that: 1) the neural network solution can approximate the target solution even for partial integro-differential equations and system of PDEs describing the time evolution of the pdf/chf, 2) solving either the Fokker-Planck equation or the chf differential equation using neural networks yields similar pdfs of the state, and 3) the solution to these differential equations can be used to study the behavior of the state for different types of random forcings.
