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Error Analysis and Numerical Algorithm for PDE Approximation with Hidden-Layer Concatenated Physics Informed Neural Networks

arXiv 2406.06350 ↗First posted 10 Jun 2024math.NA cs.LG cs.NAcited by 0

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Abstract (arXiv)

We present the hidden-layer concatenated physics informed neural network (HLConcPINN) method, which combines hidden-layer concatenated feed-forward neural networks, a modified block time marching strategy, and a physics informed approach for approximating partial differential equations (PDEs). We analyze the convergence properties and establish the error bounds of this method for two types of PDEs: parabolic (exemplified by the heat and Burgers' equations) and hyperbolic (exemplified by the wave and nonlinear Klein-Gordon equations). We show that its approximation error of the solution can be effectively controlled by the training loss for dynamic simulations with long time horizons. The HLConcPINN method in principle allows an arbitrary number of hidden layers not smaller than two and any of the commonly-used smooth activation functions for the hidden layers beyond the first two, with theoretical guarantees. This generalizes several recent neural-network techniques, which have theoretical guarantees but are confined to two hidden layers in the network architecture and the tanh activation function. Our theoretical analyses subsequently inform the formulation of appropriate training loss functions for these PDEs, leading to physics informed neural network (PINN) type computational algorithms that differ from the standard PINN formulation. Ample numerical experiments are presented based on the proposed algorithm to validate the effectiveness of this method and confirm aspects of the theoretical analyses.

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References · 29

  1. 2017DGM: A deep learning algorithm for solving partial differential equationscited by 678
  2. 2021Characterizing possible failure modes in physics-informed neural networkscited by 643
  3. 2020When and why PINNs fail to train: A neural tangent kernel perspectivecited by 634
  4. 2019DeepXDE: A deep learning library for solving differential equationscited by 619
  5. 2022Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's nextcited by 608
  6. 2017The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problemscited by 583
  7. 2020Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating a class of inverse problems for PDEscited by 102
  8. 2020A Method for Representing Periodic Functions and Enforcing Exactly Periodic Boundary Conditions with Deep Neural Networkscited by 87
  9. 2020Local Extreme Learning Machines and Domain Decomposition for Solving Linear and Nonlinear Partial Differential Equationscited by 81
  10. 2021On the approximation of functions by tanh neural networkscited by 81
  11. 2017Machine learning approximation algorithms for high-dimensional fully nonlinear partial differential equations and second-order backward stochastic differential equationscited by 65
  12. 2022Error estimates for physics informed neural networks approximating the Navier-Stokes equationscited by 63
17 more
  1. 2021When Do Extended Physics-Informed Neural Networks (XPINNs) Improve Generalization?cited by 58
  2. 2023A unified scalable framework for causal sweeping strategies for Physics-Informed Neural Networks (PINNs) and their temporal decompositionscited by 48
  3. 2018Analysis of the Generalization Error: Empirical Risk Minimization over Deep Artificial Neural Networks Overcomes the Curse of Dimensionality in the Numerical Approximation of Black-Scholes Partial Differential Equationscited by 45
  4. 2019Robust Training and Initialization of Deep Neural Networks: An Adaptive Basis Viewpointcited by 42
  5. 2019MgNet: A Unified Framework of Multigrid and Convolutional Neural Networkcited by 41
  6. 2020Physics Informed Neural Networks for Simulating Radiative Transfercited by 35
  7. 2019Deep least-squares methods: an unsupervised learning-based numerical method for solving elliptic PDEscited by 28
  8. 2021On Computing the Hyperparameter of Extreme Learning Machines: Algorithm and Application to Computational PDEs, and Comparison with Classical and High-Order Finite Elementscited by 26
  9. 2023An Extreme Learning Machine-Based Method for Computational PDEs in Higher Dimensionscited by 21
  10. 2022Parsimonious Physics-Informed Random Projection Neural Networks for Initial-Value Problems of ODEs and index-1 DAEscited by 19
  11. 2021A Modified Batch Intrinsic Plasticity Method for Pre-training the Random Coefficients of Extreme Learning Machinescited by 14
  12. 2022A Method for Computing Inverse Parametric PDE Problems with Random-Weight Neural Networkscited by 14
  13. 2020VAE-KRnet and its applications to variational Bayescited by 11
  14. 2022Numerical Approximation of Partial Differential Equations by a Variable Projection Method with Artificial Neural Networkscited by 9
  15. 2022Numerical Computation of Partial Differential Equations by Hidden-Layer Concatenated Extreme Learning Machinecited by 9
  16. 2023Error convergence and engineering-guided hyperparameter search of PINNs: towards optimized I-FENN performancecited by 5
  17. 2023PINNs error estimates for nonlinear equations in ℝ-smooth Banach spacescited by 2

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