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Physics-informed radial basis network (PIRBN): A local approximating neural network for solving nonlinear PDEs

arXiv 2304.06234 ↗First posted 13 Apr 2023cs.LGcited by 0

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Abstract (arXiv)

Our recent intensive study has found that physics-informed neural networks (PINN) tend to be local approximators after training. This observation leads to this novel physics-informed radial basis network (PIRBN), which can maintain the local property throughout the entire training process. Compared to deep neural networks, a PIRBN comprises of only one hidden layer and a radial basis "activation" function. Under appropriate conditions, we demonstrated that the training of PIRBNs using gradient descendent methods can converge to Gaussian processes. Besides, we studied the training dynamics of PIRBN via the neural tangent kernel (NTK) theory. In addition, comprehensive investigations regarding the initialisation strategies of PIRBN were conducted. Based on numerical examples, PIRBN has been demonstrated to be more effective and efficient than PINN in solving PDEs with high-frequency features and ill-posed computational domains. Moreover, the existing PINN numerical techniques, such as adaptive learning, decomposition and different types of loss functions, are applicable to PIRBN. The programs that can regenerate all numerical results can be found at https://github.com/JinshuaiBai/PIRBN.

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References · 30

  1. 2018Neural Tangent Kernel: Convergence and Generalization in Neural Networkscited by 2,597
  2. 2018Gradient Descent Provably Optimizes Over-parameterized Neural Networkscited by 655
  3. 2020When and why PINNs fail to train: A neural tangent kernel perspectivecited by 629
  4. 2018Gradient Descent Finds Global Minima of Deep Neural Networkscited by 572
  5. 2020On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networkscited by 279
  6. 2020hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decompositioncited by 236
  7. 2021Gradient-enhanced physics-informed neural networks for forward and inverse PDE problemscited by 220
  8. 2017Deep Potential Molecular Dynamics: a scalable model with the accuracy of quantum mechanicscited by 213
  9. 2020Spectrum Dependent Learning Curves in Kernel Regression and Wide Neural Networkscited by 200
  10. 2020Understanding and mitigating gradient pathologies in physics-informed neural networkscited by 134
  11. 2021Solving and Learning Nonlinear PDEs with Gaussian Processescited by 128
  12. 2019An Energy Approach to the Solution of Partial Differential Equations in Computational Mechanics via Machine Learning: Concepts, Implementation and Applicationscited by 118
18 more
  1. 2021Parallel Physics-Informed Neural Networks via Domain Decompositioncited by 112
  2. 2021Long-time integration of parametric evolution equations with physics-informed DeepONetscited by 87
  3. 2017DeePMD-kit: A deep learning package for many-body potential energy representation and molecular dynamicscited by 79
  4. 2020Local Extreme Learning Machines and Domain Decomposition for Solving Linear and Nonlinear Partial Differential Equationscited by 78
  5. 2020An overview on deep learning-based approximation methods for partial differential equationscited by 65
  6. 2022Bayesian Physics-Informed Neural Networks for real-world nonlinear dynamical systemscited by 54
  7. 2021Physics informed neural networks for continuum micromechanicscited by 46
  8. 2022Transfer learning based physics-informed neural networks for solving inverse problems in engineering structures under different loading scenarioscited by 44
  9. 2020A Physics-Guided Neural Network Framework for Elastic Plates: Comparison of Governing Equations-Based and Energy-Based Approachescited by 34
  10. 2022A mixed formulation for physics-informed neural networks as a potential solver for engineering problems in heterogeneous domains: comparison with finite element methodcited by 33
  11. 2021The mixed deep energy method for resolving concentration features in finite strain hyperelasticitycited by 28
  12. 2020A nonlocal physics-informed deep learning framework using the peridynamic differential operatorcited by 21
  13. 2020Why Spectral Normalization Stabilizes GANs: Analysis and Improvementscited by 21
  14. 2022EINNs: Epidemiologically-informed Neural Networkscited by 21
  15. 2020Improved unsupervised physics-informed deep learning for intravoxel incoherent motion modeling and evaluation in pancreatic cancer patientscited by 16
  16. 2020Deep Autoencoder based Energy Method for the Bending, Vibration, and Buckling Analysis of Kirchhoff Platescited by 12
  17. 2022Utilising physics-guided deep learning to overcome data scarcitycited by 1
  18. 2023Physics-informed neural network for friction-involved nonsmooth dynamics problemscited by 1

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