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The Finite Element Neural Network Method: One Dimensional Study

arXiv 2501.12508 ↗First posted 21 Jan 2025cs.CE cs.AIcited by 0

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Abstract (arXiv)

The potential of neural networks (NN) in engineering is rooted in their capacity to understand intricate patterns and complex systems, leveraging their universal nonlinear approximation capabilities and high expressivity. Meanwhile, conventional numerical methods, backed by years of meticulous refinement, continue to be the standard for accuracy and dependability. Bridging these paradigms, this research introduces the finite element neural network method (FENNM) within the framework of the Petrov-Galerkin method using convolution operations to approximate the weighted residual of the differential equations. The NN generates the global trial solution, while the test functions belong to the Lagrange test function space. FENNM introduces several key advantages. Notably, the weak-form of the differential equations introduces flux terms that contribute information to the loss function compared to VPINN, hp-VPINN, and cv-PINN. This enables the integration of forcing terms and natural boundary conditions into the loss function similar to conventional finite element method (FEM) solvers, facilitating its optimization, and extending its applicability to more complex problems, which will ease industrial adoption. This study will elaborate on the derivation of FENNM, highlighting its similarities with FEM. Additionally, it will provide insights into optimal utilization strategies and user guidelines to ensure cost-efficiency. Finally, the study illustrates the robustness and accuracy of FENNM by presenting multiple numerical case studies and applying adaptive mesh refinement techniques.

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References · 18

  1. 2014Adam: A Method for Stochastic Optimizationcited by 36,960
  2. 2015Automatic differentiation in machine learning: a surveycited by 846
  3. 2020hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decompositioncited by 236
  4. 2022A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networkscited by 233
  5. 2020Self-Adaptive Physics-Informed Neural Networks using a Soft Attention Mechanismcited by 231
  6. 2017Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equationscited by 208
  7. 2019Variational Physics-Informed Neural Networks For Solving Partial Differential Equationscited by 144
  8. 2021Finite Basis Physics-Informed Neural Networks (FBPINNs): a scalable domain decomposition approach for solving differential equationscited by 142
  9. 2020Understanding and mitigating gradient pathologies in physics-informed neural networkscited by 134
  10. 2024From PINNs to PIKANs: Recent Advances in Physics-Informed Machine Learningcited by 73
  11. 2020A Dual-Dimer Method for Training Physics-Constrained Neural Networks with Minimax Architecturecited by 50
  12. 2019Nonlinear Approximation and (Deep) ReLU Networkscited by 45
6 more
  1. 2019VarNet: Variational Neural Networks for the Solution of Partial Differential Equationscited by 35
  2. 2021Variational Physics Informed Neural Networks: the role of quadratures and test functionscited by 17
  3. 2022Finite Element Method-enhanced Neural Network for Forward and Inverse Problemscited by 13
  4. 2019Function approximation by deep networkscited by 10
  5. 2024FastVPINNs: Tensor-Driven Acceleration of VPINNs for Complex Geometriescited by 5
  6. 2016Greedy Deep Dictionary Learningcited by 2

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