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.

文章提出了一种基于深度神经网络的新型弱形式粒子网格神经网络 (ParticleWNN) 来解决偏微分方程 (PDEs)，借助试探空间作为 DNNs，测试空间由中心为粒子的紧支撑函数构成。通过 R - 自适应策略调整区域的半径来训练神经网络，具有较高效性和准确性，特别适用于高维和复杂的问题。