Fourier-MIONet: Fourier-enhanced multiple-input neural operators for multiphase modeling of geological carbon sequestration

Geologic Carbon Storage (GCS) is an important technology that aims to reduce the amount of carbon dioxide in the atmosphere. Multiphase flow in porous media is essential to understand CO2 migration and pressure fields in the subsurface associated with GCS. However, numerical simulation for such problems in 4D is computationally challenging and expensive, due to the multiphysics and multiscale nature of the highly nonlinear governing partial differential equations (PDEs). It prevents us from considering multiple subsurface scenarios and conducting real-time optimization. Here, we develop a Fourier-enhanced multiple-input neural operator (Fourier-MIONet) to learn the solution operator of the problem of multiphase flow in porous media. Fourier-MIONet utilizes the recently developed framework of the multiple-input deep neural operators (MIONet) and incorporates the Fourier neural operator (FNO) in the network architecture. Once Fourier-MIONet is trained, it can predict the evolution of saturation and pressure of the multiphase flow under various reservoir conditions, such as permeability and porosity heterogeneity, anisotropy, injection configurations, and multiphase flow properties. Compared to the enhanced FNO (U-FNO), the proposed Fourier-MIONet has 90% fewer unknown parameters, and it can be trained in significantly less time (about 3.5 times faster) with much lower CPU memory (< 15%) and GPU memory (< 35%) requirements, to achieve similar prediction accuracy. In addition to the lower computational cost, Fourier-MIONet can be trained with only 6 snapshots of time to predict the PDE solutions for 30 years. The excellent generalizability of Fourier-MIONet is enabled by its adherence to the physical principle that the solution to a PDE is continuous over time.

Traversing within the Gaussian Typical Set: Differentiable Gaussianization Layers for Inverse Problems Augmented by Normalizing Flows

Generative networks such as normalizing flows can serve as a learning-based prior to augment inverse problems to achieve high-quality results. However, the latent space vector may not remain a typical sample from the desired high-dimensional standard Gaussian distribution when traversing the latent space during an inversion. As a result, it can be challenging to attain a high-fidelity solution, particularly in the presence of noise and inaccurate physics-based models. To address this issue, we propose to re-parameterize and Gaussianize the latent vector using novel differentiable data-dependent layers wherein custom operators are defined by solving optimization problems. These proposed layers enforce an inversion to find a feasible solution within a Gaussian typical set of the latent space. We tested and validated our technique on an image deblurring task and eikonal tomography -- a PDE-constrained inverse problem and achieved high-fidelity results.