ParticleWNN: a Novel Neural Networks Framework for Solving Partial Differential Equations

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.