多输出物理信息神经网络解非线性PDE正逆问题
Multi-Output Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Uncertainties
摘要：
物理信息神经网络（PINN）最近被用于解决由偏微分方程（PDE）支配的各种计算问题。在本文中，我们提出了一个多输出的物理信息神经网络（MO-PINN），它可以提供解决方案的不确定性分布的正向和逆偏微分方程问题的噪声数据。该框架首先利用自举方法将噪声数据中的不确定性转化为关于先验噪声分布的多个测量值，然后设计神经网络的输出，使其既满足测量值又满足物理规律，训练结束时得到目标参数的后验估计，进而用于不确定性量化和决策.在本文中，MO-PINNs证明了一系列的数值实验，包括线性和非线性，正问题和反问题。结果表明，MO-PINN能够在噪声数据下提供准确的预测结果，并且在相同的数据和先验知识下，MO-PINN的预测结果和后验分布与传统的有限元法和Monte Carlo方法的结果一致.最后，我们表明，额外的统计知识可以纳入训练，以提高预测，如果可用的。
英文摘要：
Physics-informed neural networks (PINNs) have recently been used to solve various computational problems which are governed by partial differential equations (PDEs). In this paper, we propose a multi-output physics-informed neural network (MO-PINN) which can provide solutions with uncertainty distributions for both forward and inverse PDE problems with noisy data. In this framework, the uncertainty arising from the noisy data is first translated into multiple measurements regarding the prior noise distribution using the bootstrap method, and then the outputs of neural networks are designed to satisfy the measurements as well as the underlying physical laws.The posterior estimation of target parameters can be obtained at the end of training, which can be further used for uncertainty quantification and decision making. In this paper, MO-PINNs are demonstrated with a series of numerical experiments including both linear and nonlinear, forward and inverse problems. The results show that MO-PINN is able to provide accurate predictions with noisy data.In addition, we also demonstrate that the prediction and posterior distributions from MO-PINNs are consistent with the solutions from traditional a finite element method (FEM) solver and Monte Carlo methods given the same data and prior knowledge. Finally, we show that additional statistical knowledge can be incorporated into the training to improve the prediction if available.
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。
