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ERROR ESTIMATES OF PHYSICS-INFORMED NEURAL NETWORKS FOR INITIAL VALUE PROBLEMS

  • JIHAHM YOO (KOREA SCIENCE ACADEMY OF KAIST) ;
  • JAYWON KIM (KOREA SCIENCE ACADEMY OF KAIST) ;
  • MINJUNG GIM (NATIONAL INSTITUTE FOR MATHEMATICAL SCIENCES) ;
  • HAESUNG LEE (DEPARTMENT OF MATHEMATICS AND BIG DATA SCIENCE, KUMOH NATIONAL INSTITUTE OF TECHNOLOGY)
  • Received : 2024.03.03
  • Accepted : 2024.03.25
  • Published : 2024.03.25

Abstract

This paper reviews basic concepts for Physics-Informed Neural Networks (PINN) applied to the initial value problems for ordinary differential equations. In particular, using only basic calculus, we derive the error estimates where the error functions (the differences between the true solution and the approximations expressed by neural networks) are dominated by training loss functions. Numerical experiments are conducted to validate our error estimates, visualizing the relationship between the error and the training loss for various first-order differential equations and a second-order linear equation.

Keywords

Acknowledgement

This research was supported by Kumoh National Institute of Technology(2023 ~ 2024)

References

  1. K. Hornik, M. Stinchcombe, H. White, Multilayer feedforward networks are universal approximators, Neural Networks, 2 (1989), 359-366.  https://doi.org/10.1016/0893-6080(89)90020-8
  2. E.K. Ryu, Infinitely Large Neural Networks, Lecture Notes in Mathematics, Research Institute of Mathematics, Number 58 (2023). 
  3. N. Yadav, A. Yadav, M. Kumar, An introduction to neural network methods for differential equations, SpringerBriefs Appl. Sci. Technol., Springer, Dordrecht, 2015. 
  4. R.T.Q. Chen, Y. Rubanova, J. Bettencourt, D.K. Duvenaud, Neural Ordinary Differential Equations, Proceedings of 32nd Conference on Neural Information Processing Systems(NeurIPS2018), Montreal, Canada 2018.
  5. M. Raissi, P. Perdikaris, G.E. Karniadakis, Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, Journal of Computational Physics, , 378 (2019), 686-707.  https://doi.org/10.1016/j.jcp.2018.10.045
  6. G.E. Karniadakis, I.G. Kevrekidis, L. Lu, P. Perdikaris, S. Wang, L. Yang, Physics-informed machine learning, Nature Reviews Physics, 3 (2021), 422-440.  https://doi.org/10.1038/s42254-021-00314-5
  7. I.E. Lagaris, A Likas, D.I. Fotiadis, Artificial Neural Networks for Solving Ordinary and Partial Differential Equations, IEEE Transactions on Neural Networks, 9 (1998), 987-1000.  https://doi.org/10.1109/72.712178
  8. A. Malek, R.S. Beidokhti, Numerical solution for high order differential equations using a hybrid neural network-optimization method, Appl. Math. Comput., 183 (2006), 260-271.  https://doi.org/10.1016/j.amc.2006.05.068
  9. H. Lee, I. Kang, Neural Algorithm for Solving Differential Equations, Journal of Computational Physics, 91 (1990), 110-131 .  https://doi.org/10.1016/0021-9991(90)90007-N
  10. M. Dissanayake, N. Phan-Thien, Neural-Network-Based Approximations for Solving Partial Differential equations, Communications in Numerical Methods in Engineering, 10 (1994), 195-201.  https://doi.org/10.1002/cnm.1640100303
  11. B. Hillebrecht, B. Unger, Certified machine learning: A posteriori error estimation for physics-informed neural networks, Proceedings of 2022 International Joint Conference on Neural Networks (IJCNN), Padua, Italy, 2022. 
  12. E. Hairer, S.P. Norsett, G. Wanner, Solving Ordinary Differential Equations, I, 3rd ed. Berlin, Heidelberg: Springer, 2008. 
  13. E.A. Coddington, An introduction to ordinary differential equations, Prentice-Hall Mathematics Series Prentice-Hall, Inc., Englewood Cliffs, NJ, 1961. 
  14. S. Mishra, R. Molinaro, Estimates on the generalization error of physics-informed neural networks for approximating a class of inverse problems for PDEs, IMA J. Numer. Anal., 42 (2022), 981-1022.  https://doi.org/10.1093/imanum/drab032
  15. S. Mishra, R. Molinaro, Estimates on the generalization error of physics-informed neural networks for approximating PDEs, IMA J. Numer. Anal., 43 (2023), 1-43. 
  16. T. De Ryck, A.D. Jagtap, S. Mishra, Error estimates for physics-informed neural networks approximating the Navier-Stokes equations, IMA J. Numer. Anal., 44 (2024), 83-119.