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An extrapolation-driven network architecture for physics-informed deep learning.

Yong Wang1, Yanzhong Yao2, Zhiming Gao2

  • 1Institute of Applied Physics and Computational Mathematics, Beijing 100088, China; Graduate School of China Academy of Engineering Physics, Beijing 100088, China; National Key Laboratory of Computational Physics, Beijing 100088, China.

Neural Networks : the Official Journal of the International Neural Network Society
|December 10, 2024
PubMed
Summary

Physics-informed neural networks (PINNs) struggle with sequential learning. This study introduces an extrapolation-driven network architecture that overcomes these limitations, enabling accurate and continuous solutions for time-dependent PDEs over large domains.

Keywords:
Deep learningEvolution equationExtrapolationPhysics-informed neural networks

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Area of Science:

  • Computational Science
  • Applied Mathematics
  • Artificial Intelligence

Background:

  • Current physics-informed neural network (PINN) methods face challenges in sequential learning for time-dependent partial differential equations (PDEs).
  • These challenges include reproducibility issues with single networks and continuity/smoothness problems with multiple networks, alongside increased computational costs.
  • Existing approaches struggle to efficiently solve evolution equations over extensive time domains.

Purpose of the Study:

  • To investigate and leverage the extrapolation capability of PINNs for time-dependent PDEs.
  • To develop a novel neural network architecture that addresses the limitations of current sequential learning strategies in PINNs.
  • To achieve accurate, continuous, and smooth solutions for PDEs across large time intervals using a single network.

Main Methods:

  • Investigated the extrapolation property of PINNs for time-dependent PDEs.
  • Developed a correction term to generalize training results from subintervals to larger intervals.
  • Introduced the extrapolation-driven network architecture, coupling network parameters with time variables via an extrapolation control function.
  • Employed a single neural network trained chronologically across multiple subintervals, respecting causality.

Main Results:

  • The extrapolation-driven network architecture successfully inherits local solutions from previous training intervals.
  • Strict continuity and smoothness are maintained at interval nodes, matching the true solution.
  • The method effectively overcomes the difficulties of conventional PINNs in solving evolution equations over large time domains.
  • Numerical experiments validated the proposed method's performance.

Conclusions:

  • The extrapolation-driven network architecture offers a robust solution for sequential learning in PINNs.
  • This approach enhances accuracy, continuity, and efficiency when solving time-dependent PDEs over large domains.
  • The method respects causality and simplifies the process of handling complex evolution equations.