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Scalable physics-informed deep generative model for solving forward and inverse stochastic differential equations.

Shaoqian Zhou1, Wen You1, Ling Guo2

  • 1Institute of Interdisciplinary Research for Mathematics and Applied Science, School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, 430074, China.

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Summary
This summary is machine-generated.

This study introduces a scalable physics-informed deep generative model (sPI-GeM) to solve complex stochastic differential equation (SDE) problems in high-dimensional spaces. The novel model accurately handles both stochastic and spatial dimensions, overcoming limitations of existing deep learning methods.

Keywords:
Basis functionPhysics-informed deep generative modelSDEs With high-dimensional stochastic and spatial spaceScalability

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Area of Science:

  • Computational Science
  • Applied Mathematics
  • Machine Learning

Background:

  • Physics-informed deep learning effectively solves high-dimensional stochastic differential equation (SDE) problems.
  • Existing models struggle with SDEs featuring high-dimensional spatial components.

Purpose of the Study:

  • To develop a scalable physics-informed deep generative model (sPI-GeM) for SDEs with high-dimensional stochastic and spatial spaces.
  • To address the limitations of current deep learning models in handling spatial dimensionality.

Main Methods:

  • Introduced a two-component model: physics-informed basis networks (PI-BasisNet) and a physics-informed deep generative model (PI-GeM).
  • PI-BasisNet learns basis functions and coefficients; PI-GeM learns coefficient distributions.
  • Scalability in spatial dimensions is achieved similarly to principal component analysis (PCA).

Main Results:

  • The sPI-GeM accurately approximates Gaussian and non-Gaussian stochastic processes.
  • Demonstrated effectiveness in solving forward and inverse SDE problems.
  • Validated scalability for SDEs with high-dimensional stochastic and spatial characteristics.

Conclusions:

  • The proposed sPI-GeM offers a scalable solution for SDEs in high-dimensional stochastic and spatial domains.
  • Represents a significant advancement in applying physics-informed deep learning to complex SDE problems.