mlsanityData to 7 Oct 2026

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

arXiv 2305.12433 ↗First posted 21 May 2023cs.LG cs.NA math.NAcited by 4

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Abstract (arXiv)

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.

Key facts. Cited by 4 papers in the map (data through 7 Oct 2026). Builds on 23 of its 44 references that are in the map. First posted on arXiv 21 May 2023.

Builds on

23 of 44 references.

References · 23

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11 more
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Citing papers per year: 2024: 1; 2025: 3.

4 papers.

  1. 2025DGenNO: A Novel Physics-aware Neural Operator for Solving Forward and Inverse PDE Problems based on Deep, Generative Probabilistic Modelingcited by 5
  2. 2024WANCO: Weak Adversarial Networks for Constrained Optimization problemscited by 3
  3. 2025DGNN: A Neural PDE Solver Induced by Discontinuous Galerkin Methodscited by 3
  4. 2025A unified physics-informed generative operator framework for general inverse problemscited by 1