A framework for data-driven solution and parameter estimation of PDEs using conditional generative adversarial...

Area of Science:

• Computational Science
• Geophysics
• Machine Learning

Background:

• Solving partial differential equations (PDEs) for coupled hydromechanical processes in heterogeneous porous media is computationally intensive.
• Traditional reduced-order modeling techniques struggle with parameterizing spatially heterogeneous coefficients.

Purpose of the Study:

• To adapt image-to-image translation using conditional generative adversarial networks (cGANs) for learning forward and inverse solution operators of PDEs.
• To efficiently model steady-state solutions of coupled hydromechanical processes in heterogeneous porous media.

Main Methods:

• Employing conditional generative adversarial networks (cGANs) for image-to-image translation to learn PDE solution operators.
• Developing a framework for parameterizing spatially heterogeneous coefficients in porous media.

Main Results:

• Achieved a speed-up of at least 2,000 times compared to finite-element solvers for forward modeling.
• Obtained a relative root-mean-square error (r.m.s.e.) of less than 2% for forward modeling.
• Estimated heterogeneous coefficients with a relative r.m.s.e. of less than 7% for inverse modeling, even with noisy and incomplete data.
• Demonstrated a 120,000 times speed-up for inverse modeling compared to Gaussian prior-based methods, with improved accuracy.

Conclusions:

• The cGAN-based framework offers a highly efficient and accurate approach for solving forward and inverse problems in heterogeneous porous media.
• This method overcomes limitations of traditional techniques in parameterizing complex geological properties.
• The framework shows significant potential for accelerating scientific discovery and engineering applications involving PDEs.