# Physics-Informed DeepONet Coupled with FEM for Convective Transport in Porous Media with Sharp Gaussian Sources

## TL;DR

This paper introduces a hybrid FEM and physics-informed DeepONet framework for modeling convective transport in porous media with sharp Gaussian sources, achieving accurate and fast predictions.

## Contribution

It combines FEM with DeepONet to efficiently model fluid transport, incorporating adaptive sampling for steep gradients, and demonstrates significant speedups over traditional methods.

## Key findings

- Good agreement with reference solutions
- Orders of magnitude speedup
- Effective handling of sharp source gradients

## Abstract

We present a hybrid framework that couples finite element methods (FEM) with physics-informed DeepONet to model fluid transport in porous media from sharp, localized Gaussian sources. The governing system consists of a steady-state Darcy flow equation and a time-dependent convection-diffusion equation. Our approach solves the Darcy system using FEM and transfers the resulting velocity field to a physics-informed DeepONet, which learns the mapping from source functions to solute concentration profiles. This modular strategy preserves FEM-level accuracy in the flow field while enabling fast inference for transport dynamics. To handle steep gradients induced by sharp sources, we introduce an adaptive sampling strategy for trunk collocation points. Numerical experiments demonstrate that our method is in good agreement with the reference solutions while offering orders of magnitude speedups over traditional solvers, making it suitable for practical applications in relevant scenarios. Implementation of our proposed method is available at https://github.com/erkara/fem-pi-deeponet.