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Physics-informed neural networks (PINNs), rooted in deep learning, haveemerged as a promising approach for solving partial differential equations(PDEs). By embedding the physical information described by PDEs intofeedforward neural networks, PINNs are trained as surrogate models toapproximate solutions without the need for label data. Nevertheless, eventhough 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 becomingtrapped in local minima, and reduced solution accuracy. To address theseissues, we propose a binary structured physics-informed neural network (BsPINN)framework, which employs binary structured neural network (BsNN) as the neuralnetwork component. By leveraging a binary structure that reduces inter-neuronconnections compared to fully connected neural networks, BsPINNs excel incapturing the local features of solutions more effectively and efficiently.These features are particularly crucial for learning the rapidly changing inthe nature of solutions. In a series of numerical experiments solving Burgersequation, Euler equation, Helmholtz equation, and high-dimension Poissonequation, BsPINNs exhibit superior convergence speed and heightened accuracycompared to PINNs. From these experiments, we discover that BsPINNs resolve theissues caused by increased hidden layers in PINNs resulting in over-smoothing,and prevent the decline in accuracy due to non-smoothness of PDEs solutions.

Binary structured physics-informed neural networks for solving equations with rapidly changing solutions