Physics-informed neural networks (PINNs), rooted in deep learning, have emerged as a promising approach for solving partial differential equations (PDEs). By embedding the physical information described by PDEs into feedforward neural networks, PINNs are trained as surrogate models to approximate solutions without the need for label data. Nevertheless, even though PINNs have shown remarkable performance, they can face difficulties, especially when dealing with equations featuring rapidly changing solutions. These difficulties encompass slow convergence, susceptibility to becoming trapped in local minima, and reduced solution accuracy. To address these issues, we propose a binary structured physics-informed neural network (BsPINN) framework, which employs binary structured neural network (BsNN) as the neural network component. By leveraging a binary structure that reduces inter-neuron connections compared to fully connected neural networks, BsPINNs excel in capturing the local features of solutions more effectively and efficiently. These features are particularly crucial for learning the rapidly changing in the nature of solutions. In a series of numerical experiments solving Burgers equation, Euler equation, Helmholtz equation, and high-dimension Poisson equation, BsPINNs exhibit superior convergence speed and heightened accuracy compared to PINNs. From these experiments, we discover that BsPINNs resolve the issues caused by increased hidden layers in PINNs resulting in over-smoothing, and prevent the decline in accuracy due to non-smoothness of PDEs solutions.

翻译：基于深度学习的物理信息神经网络（PINNs）已成为求解偏微分方程（PDEs）的一种有前景的方法。通过将PDEs描述的物理信息嵌入前馈神经网络，PINNs被训练为无需标签数据即可逼近解的代理模型。然而，尽管PINNs表现出色，但它们在处理具有快速变化解的方程时仍面临困难，包括收敛缓慢、易陷入局部极小值和解精度降低。为解决这些问题，我们提出了一种二元结构化物理信息神经网络（BsPINN）框架，采用二元结构化神经网络（BsNN）作为其神经网络组件。通过利用减少神经元间连接的二元结构（相较于全连接神经网络），BsPINNs能更高效地捕获解的局部特征。这些特征对于学习解的快速变化特性尤为关键。在求解Burgers方程、Euler方程、Helmholtz方程和高维Poisson方程的一系列数值实验中，BsPINNs相比PINNs展现出更优的收敛速度和更高的精度。通过这些实验我们发现，BsPINNs解决了PINNs因隐藏层增加导致的过平滑问题，并防止了因PDEs解非光滑性引起的精度下降。