  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.
