用Fokker-Planck方程和物理信息神经网络求解离散粒子观测的逆随机问题
Solving Inverse Stochastic Problems from Discrete Particle Observations Using the Fokker-Planck Equation and Physics-informed Neural Networks
摘要：
控制概率密度函数（PDF）演化的福克-普朗克（FP）方程适用于许多学科，但它需要为每种情况指定系数，这些系数可以是时空的函数而不仅仅是常数，因此需要开发数据驱动的建模方法。当可用的数据直接在PDF上时，则存在用于反问题的方法，其可以用于推断系数，从而确定FP方程并随后获得其解。在本文中，我们解决了一个更现实的场景，其中仅在几个时刻给出关于粒子位置的稀疏数据，即使在那些时间从现有方法也不足以准确地直接构建PDF，例如，核估计算法为此，我们开发了一个基于物理信息神经网络（PINNs）的通用框架，该框架引入了一个新的损失函数，使用Kullback-Leibler散度将随机样本与FP方程连接起来，同时学习方程并随时推断多维PDF。特别是，我们考虑两种类型的反问题，I型FP方程是已知的，但初始PDF是未知的，和II型，其中，除了未知的初始PDF，漂移和扩散项也是未知的。在这两种情况下，我们调查的问题，无论是布朗噪声或列维噪声或两者的组合。我们详细展示了新的PINN框架在一维情况下（1D），但我们也提供了高达5D的结果，证明我们可以同时推断FP方程和动力学在任何时候都具有高精度，只使用非常少的离散观测粒子。
英文摘要：
The Fokker-Planck (FP) equation governing the evolution of the probability density function (PDF) is applicable to many disciplines but it requires specification of the coefficients for each case, which can be functions of space-time and not just constants, hence requiring the development of a data-driven modeling approach. When the data available is directly on the PDF, then there exist methods for inverse problems that can be employed to infer the coefficients and thus determine the FP equation and subsequently obtain its solution. Herein, we address a more realistic scenario, where only sparse data are given on the particles' positions at a few time instants, which are not sufficient to accurately construct directly the PDF even at those times from existing methods, e.g., kernel estimation algorithms. To this end, we develop a general framework based on physics-informed neural networks (PINNs) that introduces a new loss function using the Kullback-Leibler divergence to connect the stochastic samples with the FP equation, to simultaneously learn the equation and infer the multi-dimensional PDF at all times. In particular, we consider two types of inverse problems, type I where the FP equation is known but the initial PDF is unknown, and type II in which, in addition to unknown initial PDF, the drift and diffusion terms are also unknown. In both cases, we investigate problems with either Brownian or Levy noise or a combination of both. We demonstrate the new PINN framework in detail in the one-dimensional case (1D) but we also provide results for up to 5D demonstrating that we can infer both the FP equation and} dynamics simultaneously at all times with high accuracy using only very few discrete observations of the particles.
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