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 chosen as the space of DNNs, and the test space is constructed by functions compactly supported in extremely small regions whose centers are particles. To train the neural networks, an R-adaptive strategy is designed to adaptively modify the radius of regions during training. The ParticleWNN inherits the advantages of weak/variational formulation, such as requiring less regularity of the solution and a small number of quadrature points for computing the integrals. Moreover, due to the special construction of the test functions, the ParticleWNN allows local training of networks, parallel implementation, and integral calculations only in extremely small regions. The framework is particularly desirable for solving problems with high-dimensional and complex domains. The efficiency and accuracy of the ParticleWNN are demonstrated with several numerical examples. The numerical results show clear advantages of the ParticleWNN over the state-of-the-art methods.

翻译：深度神经网络（DNNs）近年来已被广泛用于求解偏微分方程（PDEs）。本文提出了一种名为粒子型弱形式神经网络（ParticleWNN）的新型深度学习框架，用于在弱形式下求解PDEs。在该框架中，试验空间选为DNNs的空间，而测试空间则由支撑在中心为粒子的极小区域内的函数构造。为训练神经网络，设计了一种R-自适应策略，在训练过程中自适应地调整区域的半径。ParticleWNN继承了弱/变分公式的优点，例如对解的正则性要求较低，且计算积分所需的求积点数量少。此外，由于测试函数的特殊构造，ParticleWNN允许网络的局部训练、并行实现以及在极小区域内进行积分计算。该框架特别适用于求解高维和复杂域问题。通过多个数值算例验证了ParticleWNN的效率与精度。数值结果表明，ParticleWNN相较于现有最优方法具有明显优势。