Deep neural networks (DNNs) have been widely used to solve partial differential equations (PDEs) in recent years. In this work, a novel deep learning-based framework named Particle Weak-form based Neural Networks (ParticleWNN) is developed for solving PDEs in the weak form. In this framework, the trial space is defined as the space of DNNs, while the test space consists of functions compactly supported in extremely small regions, centered around particles. To facilitate the training of neural networks, an R-adaptive strategy is designed to adaptively modify the radius of regions during training. The ParticleWNN inherits the benefits of weak/variational formulation, requiring less regularity of the solution and a small number of quadrature points for computing integrals. Additionally, due to the special construction of the test functions, ParticleWNN enables parallel implementation and integral calculations only in extremely small regions. This framework is particularly desirable for solving problems with high-dimensional and complex domains. The efficiency and accuracy of ParticleWNN are demonstrated through several numerical examples, showcasing its superiority over state-of-the-art methods. The source code for the numerical examples presented in this paper is available at https://github.com/yaohua32/ParticleWNN.

翻译：近年来，深度神经网络已被广泛应用于求解偏微分方程。本文提出了一种名为"基于粒子弱形式的神经网络"的新型深度学习框架，用于在弱形式下求解偏微分方程。在该框架中，试函数空间定义为深度神经网络空间，而测试函数空间则由支撑域集中在粒子周围、范围极小的函数构成。为促进神经网络训练，我们设计了一种R自适应策略，可在训练过程中动态调整支撑域半径。ParticleWNN继承了弱形式/变分公式的优势，对解的正则性要求较低，且计算积分时仅需少量求积点。此外，由于测试函数的特殊构造，ParticleWNN可实现并行计算，积分运算也仅在极小的局部区域内进行。该框架特别适用于求解高维复杂区域中的问题。通过多个数值算例验证了ParticleWNN的效率和精度，证明其优于当前最先进的方法。本文数值算例的源代码已公开于https://github.com/yaohua32/ParticleWNN。