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.
翻译:物理知情神经网络(PINNs)最近被用来解决由部分差异方程式(PDEs)管理的各种计算问题。在本文件中,我们提议建立一个多输出物理知情神经网络(MO-PINN),这个网络可以提供未来和反PDE问题的不确定性分布的解决方案;在这个框架内,噪音数据产生的不确定性首先转化为对使用靴子捕捉方法先前的噪音分布的多种测量,然后神经网络的产出旨在满足测量和基本物理法律的要求。 目标参数的事后估计可以在培训结束时获得,可以进一步用于不确定性的量化和决策。在本文件中,MO-PINNs通过一系列数字实验(包括线性和非线性、前向和反向问题)得到证明。结果显示,MO-PINNs能够用噪音数据提供准确的预测。此外,我们还表明,MO-PINNs的预测和后方程式分布与传统定要素要素方法(FEM)的解决方案一致,可以进一步用于不确定性的量化和决策。在本文中,通过一系列数字实验,包括线性和非线性、前期、前期和反向后期的预测,如果我们所提供的统计数据能显示更多的数据,则反映先前的预测方法。