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

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.