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Minglang Yin1,2, Nicolas Charon3, Ryan Brody1,4

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This study introduces Diffeomorphic Mapping Operator Learning (DIMON), an AI framework for efficiently solving partial differential equations (PDEs) across diverse geometries. DIMON significantly reduces computational costs for complex simulations, from hours to seconds.

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Area of Science:

  • Computational mathematics
  • Artificial intelligence
  • Scientific computing

Background:

  • Numerical methods for solving partial differential equations (PDEs) are essential in engineering and medicine.
  • High computational costs hinder PDE solution evaluations on multiple geometries.

Purpose of the Study:

  • Introduce Diffeomorphic Mapping Operator Learning (DIMON), a novel AI framework.
  • Enable efficient, geometry-dependent solution operator learning for various PDEs.

Main Methods:

  • Developed a generic artificial intelligence framework, DIMON.
  • Applied DIMON to learn solution operators for static and time-dependent PDEs.
  • Demonstrated framework performance on parameterized and non-parameterized domains.

Main Results:

  • DIMON effectively learns solution operators for Laplace, reaction-diffusion, and multiscale PDEs.
  • Framework shows strong performance, efficiency, and scalability across diverse geometries.
  • Reduced computational time for PDE solutions on multiple geometries from hours to seconds.

Conclusions:

  • DIMON offers a significant advancement in accelerating PDE solutions.
  • The framework provides a computationally efficient alternative for complex simulations.
  • DIMON enables rapid analysis on large-scale, personalized models like digital twins.