Physics-Informed Neural Networks (PINNs) have emerged as a powerful tool for solving partial differential equations (PDEs) in various scientific and engineering domains. However, traditional PINN architectures typically rely on fully connected multilayer perceptrons (MLPs), lacking the sparsity and modularity inherent in many traditional numerical solvers. This study investigates a novel approach by merging established PINN methodologies with brain-inspired neural network techniques to address this architectural limitation. We leverage Brain-Inspired Modular Training (BIMT), leveraging concepts such as locality, sparsity, and modularity inspired by the organization of the brain. Through BIMT, we demonstrate the evolution of PINN architectures from fully connected structures to highly sparse and modular forms, resulting in reduced computational complexity and memory requirements. We showcase the efficacy of this approach by solving differential equations with varying spectral components, revealing insights into the spectral bias phenomenon and its impact on neural network architecture. Moreover, we derive basic PINN building blocks through BIMT training on simple problems akin to convolutional and attention modules in deep neural networks, enabling the construction of modular PINN architectures. Our experiments show that these modular architectures offer improved accuracy compared to traditional fully connected MLP PINNs, showcasing their potential for enhancing PINN performance while reducing computational overhead. Overall, this study contributes to advancing the understanding and development of efficient and effective neural network architectures for solving PDEs, bridging the gap between PINNs and traditional numerical methods.

翻译：物理信息神经网络（PINNs）已成为解决科学和工程领域中偏微分方程（PDEs）的强大工具。然而，传统的PINN架构通常依赖于全连接多层感知机（MLPs），缺乏许多传统数值求解器固有的稀疏性和模块化特性。本研究通过将成熟的PINN方法与脑启发式神经网络技术相结合，探索一种新方法来解决这一架构局限性。我们利用脑启发式模块化训练（BIMT），借鉴大脑组织中的局域性、稀疏性和模块化等概念。通过BIMT，我们展示了PINN架构从全连接结构向高度稀疏和模块化形式的演化，从而降低了计算复杂度和内存需求。通过求解具有不同频谱成分的微分方程，我们展示了该方法的有效性，揭示了对频谱偏差现象及其对神经网络架构影响的见解。此外，通过在简单问题上进行BIMT训练，我们推导出基本的PINN构建模块——类似于深度神经网络中的卷积和注意力模块，从而能够构建模块化PINN架构。我们的实验表明，与传统全连接MLP PINN相比，这些模块化架构在降低计算开销的同时提供了更高的精度，展示了其提升PINN性能的潜力。总体而言，本研究有助于推进高效能神经网络架构在解决PDEs方面的理解与发展，弥合PINN与传统数值方法之间的差距。