PAGP：一个物理辅助的偏微分方程正逆问题主动学习高斯过程框架
PAGP: A physics-assisted Gaussian process framework with active learning for forward and inverse problems of partial differential equations
摘要：
在这项工作中，一个高斯过程回归（GPR）模型结合了偏微分方程（PDE）中给定的物理信息：物理辅助高斯过程（PAGP）。该模型的目标可以分为两类：求解或发现具有初始和边界条件的偏微分方程的未知系数。我们介绍了三种不同的模型：连续时间，离散时间和混合模型。通过我们设计的GP损失函数将给定的物理信息整合到高斯过程模型中。本文根据训练标准GP模型的两种不同方法，提出了三种损失函数。本文第一部分介绍了将时间域和空间域同等对待的连续时间模型。通过最小化设计的损失函数，可以与GP超参数联合学习给定偏微分方程中的未知系数。在离散时间模型中，我们首先选择一个时间离散化方案来离散时间域。然后在每个时间步应用PAGP模型和该方案来逼近最终时刻给定测试点处的偏微分方程解。为了发现这种情况下的未知系数，需要在两个特定时间的观测值，并构造混合均方误差函数来获得最佳系数。最后提出了一种新的连续时间模型和离散时间模型相结合的混合模型。它融合了连续时间模型的灵活性和离散时间模型的准确性。本文还讨论了在不同GP损失函数下选择不同模型的性能。在我们的数值部分中说明了所提出的PAGP方法的有效性。
英文摘要：
In this work, a Gaussian process regression(GPR) model incorporated with given physical information in partial differential equations(PDEs) is developed: physics-assisted Gaussian processes(PAGP). The targets of this model can be divided into two types of problem: finding solutions or discovering unknown coefficients of given PDEs with initial and boundary conditions. We introduce three different models: continuous time, discrete time and hybrid models. The given physical information is integrated into Gaussian process model through our designed GP loss functions. Three types of loss function are provided in this paper based on two different approaches to train the standard GP model. The first part of the paper introduces the continuous time model which treats temporal domain the same as spatial domain. The unknown coefficients in given PDEs can be jointly learned with GP hyper-parameters by minimizing the designed loss function. In the discrete time models, we first choose a time discretization scheme to discretize the temporal domain. Then the PAGP model is applied at each time step together with the scheme to approximate PDE solutions at given test points of final time. To discover unknown coefficients in this setting, observations at two specific time are needed and a mixed mean square error function is constructed to obtain the optimal coefficients. In the last part, a novel hybrid model combining the continuous and discrete time models is presented. It merges the flexibility of continuous time model and the accuracy of the discrete time model. The performance of choosing different models with different GP loss functions is also discussed. The effectiveness of the proposed PAGP methods is illustrated in our numerical section.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
