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DeepONet: Learning Nonlinear Operators for Identifying Differential Equations Based on the Universal Approximation Theorem of Operators

Basic Information

  • Title: DeepONet: Learning Nonlinear Operators for Identifying Differential Equations Based on the Universal Approximation Theorem of Operators
  • Authors: Lu Lu, Pengzhan Jin, George Em Karniadakis
  • Publication Date: Not specified
  • Field: Computer Graphics/Deep Learning

Introduction

  • Objective: The paper introduces Deep Operator Networks (DeepONets), aimed at learning nonlinear operators to efficiently identify dynamics systems and partial differential equations from data.
  • Background: Traditional approaches in modeling dynamic systems and PDEs often require extensive computational resources and data. DeepONets propose a novel way to tackle these challenges using deep learning.

Details

  • Techniques: DeepONets consist of two sub-networks: a branch network that encodes input functions at predetermined sensor locations and a trunk network that encodes locations for the output function. This architecture is designed to leverage the universal approximation theorem of operators, facilitating the learning process.
  • Experiments: The paper showcases experiments comparing DeepONets to fully connected networks, demonstrating DeepONets' superior ability to reduce generalization errors and achieve high-order error convergence in identifying differential equations.
  • Results: Theoretical analysis and computational results indicate that the approximation error of DeepONets depends on the number of sensors and the types of input functions. These findings are supported by empirical evidence showing significant performance improvements over traditional methods.

Conclusion

  • Contributions: The paper successfully demonstrates the potential of DeepONets in revolutionizing the way dynamic systems and PDEs are modeled and solved, offering a significant reduction in computational cost and an increase in efficiency.
  • Insights: The proposed framework highlights the importance of learning nonlinear operators in the context of differential equations and sets a new standard for research in this area. The findings suggest a promising direction for future research, especially in terms of refining the network architecture and exploring its applications in more complex systems.