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General Explicit Network (GEN): A novel deep learning architecture for solving partial differential equations

arXiv 2604.03321 ↗First posted 2 Apr 2026cs.LG cs.AI math.APcited by 0

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Abstract (arXiv)

Machine learning, especially physics-informed neural networks (PINNs) and their neural network variants, has been widely used to solve problems involving partial differential equations (PDEs). The successful deployment of such methods beyond academic research remains limited. For example, PINN methods primarily consider discrete point-to-point fitting and fail to account for the potential properties of real solutions. The adoption of continuous activation functions in these approaches leads to local characteristics that align with the equation solutions while resulting in poor extensibility and robustness. A general explicit network (GEN) that implements point-to-function PDE solving is proposed in this paper. The "function" component can be constructed based on our prior knowledge of the original PDEs through corresponding basis functions for fitting. The experimental results demonstrate that this approach enables solutions with high robustness and strong extensibility to be obtained.

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References · 16

  1. 2019PyTorch: An Imperative Style, High-Performance Deep Learning Librarycited by 20,305
  2. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,908
  3. 2018Neural Tangent Kernel: Convergence and Generalization in Neural Networkscited by 2,621
  4. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 2,018
  5. 2020Neural Operator: Graph Kernel Network for Partial Differential Equationscited by 734
  6. 2021Characterizing possible failure modes in physics-informed neural networkscited by 643
  7. 2020When and why PINNs fail to train: A neural tangent kernel perspectivecited by 634
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  9. 2020On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networkscited by 284
  10. 2021Enhancing Computational Fluid Dynamics with Machine Learningcited by 138
  11. 2020Solving Allen-Cahn and Cahn-Hilliard Equations using the Adaptive Physics Informed Neural Networkscited by 123
  12. 2021Physics-based Deep Learningcited by 68
4 more
  1. 2022Applications of physics informed neural operatorscited by 33
  2. 2023Phase-Field DeepONet: Physics-informed deep operator neural network for fast simulations of pattern formation governed by gradient flows of free-energy functionalscited by 26
  3. 2022Experience report of physics-informed neural networks in fluid simulations: pitfalls and frustrationcited by 21
  4. 2023Predictive Limitations of Physics-Informed Neural Networks in Vortex Sheddingcited by 12

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