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.

通过在前馈神经网络中嵌入偏微分方程所描述的物理信息，物理信息导向的神经网络（PINNs）作为深度学习的一种有前途的方法，用于解决偏微分方程（PDEs）。然而，尽管 PINNs 表现出卓越的性能，但它们在处理解快速变化的方程时可能面临困难。为了解决这些问题，我们提出了一种二进制结构的物理信息导向神经网络（BsPINN）框架，其中采用二进制结构神经网络（BsNN）作为神经网络组件。通过利用相对于完全连接的神经网络减少神经元连接的二进制结构，BsPINNs 在更有效和高效地捕捉解的局部特征方面表现出色。在一系列解决 Burgers 方程、Euler 方程、Helmholtz 方程和高维 Poisson 方程的数值实验中，BsPINNs 展现出优异的收敛速度和更高的准确性，相较于 PINNs。从这些实验中，我们发现 BsPINNs 解决了 PINNs 中增加的隐藏层导致过度平滑的问题，并防止了 PDEs 解不光滑导致准确性下降的问题。