We propose a novel method for fast and accurate training of physics-informed neural networks (PINNs) to find solutions to boundary value problems (BVPs) and initial boundary value problems (IBVPs). By combining the methods of training deep neural networks (DNNs) and Extreme Learning Machines (ELMs), we develop a model which has the expressivity of DNNs with the fine-tuning ability of ELMs. We showcase the superiority of our proposed method by solving several BVPs and IBVPs which include linear and non-linear ordinary differential equations (ODEs), partial differential equations (PDEs) and coupled PDEs. The examples we consider include a stiff coupled ODE system where traditional numerical methods fail, a 3+1D non-linear PDE, Kovasznay flow and Taylor-Green vortex solutions to incompressible Navier-Stokes equations and pure advection solution of 1+1 D compressible Euler equation. The Theory of Functional Connections (TFC) is used to exactly impose initial and boundary conditions (IBCs) of (I)BVPs on PINNs. We propose a modification to the TFC framework named Reduced TFC and show a significant improvement in the training and inference time of PINNs compared to IBCs imposed using TFC. Furthermore, Reduced TFC is shown to be able to generalize to more complex boundary geometries which is not possible with TFC. We also introduce a method of applying boundary conditions at infinity for BVPs and numerically solve the pure advection in 1+1 D Euler equations using these boundary conditions.

我们提出了一种快速准确训练物理信息神经网络（PINNs）解决边界值问题（BVPs）和初始边界值问题（IBVPs）的新方法。该方法结合了深度神经网络（DNNs）和极限学习机（ELMs）的训练方法，将DNN的表达能力与ELMs的微调能力相结合，通过解决线性和非线性常微分方程（ODEs）、偏微分方程（PDEs）以及耦合的PDEs等多个问题来展示该方法的优越性。使用功能连接理论（TFC）来确切地强加初值条件和边界条件（IBCs）到PINNs上，并提出了缩减TFC框架来改进PINNs的训练和推测时间。此外，通过引入无穷远处边界条件的方法，我们成功地数值求解了一维可压缩Euler方程的纯对流问题。

通过极小化优化来微调用于求解边界值问题的物理 信息神经网络

通过极小化优化来微调用于求解边界值问题的物理信息神经网络