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

TL;DR

This work introduces Binary Structured PINNs (BsPINNs), which replace fully connected networks with Binary Structured Neural Networks to better capture localized, rapidly changing PDE solutions. By focusing on local features through a multi-channel, binary-tree connectivity, BsPINNs achieve faster convergence and higher accuracy than conventional PINNs across Burgers, Euler, Helmholtz, Möbius-knot, and high-dimensional Poisson problems, while mitigating over-smoothing and tail-region sampling issues. The approach preserves the physics-informed loss framework but attains improved stability and efficiency, suggesting significant practical impact for solving challenging PDEs with complex solution structures. The authors also discuss interpretability via channel-wise features and outline future work on theory, optimizer choices, and software integration.

Abstract

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.

Figures (20)

• Figure 2.1: The architecture of a FNN. The purple arrows depict the fully connected relationships between different components.
• Figure 3.2: Function $f$ on $\Omega$ (a) and the FNN-based approximation of function $f$ (b).
• Figure 3.3: (a) The general network structure of a BsNN. (b) The structure of a BsNN consists of $4$ hidden layers, each with $8$ neurons.
• Figure 3.4: The BsNN-based approximation of function $f$.
• Figure 3.5: Cross-sections of the function $f$ and its approximations by the FNN and BsNN on the planes $x=0$ (a) and $y=0$ (b). The blue solid lines represent the true values of $f$ , the green dashed lines represent the predicted values by the FNN, and the green dot-dashed lines represent the predicted values by the BsNN.