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

2025-08-27

AI Summary

This work addresses the convection–diffusion transport problem in porous media driven by a sharp Gaussian source. Methodologically, it introduces a hybrid solver coupling the finite element method (FEM) with a physics-informed deep operator network (DeepONet): an adaptive collocation-point sampling strategy resolves the source’s steep gradient; FEM accurately computes the Darcy velocity field; and a physics-constrained DeepONet learns the nonlinear mapping from source functions to concentration fields. The key contributions are a modular coupling architecture and a gradient-aware sampling mechanism—jointly ensuring physical fidelity and generalization efficiency. Numerical experiments demonstrate that the method achieves high accuracy—L² error < 1.2% relative to high-fidelity reference solutions—while accelerating inference by three to four orders of magnitude over traditional full-order simulations. This enables real-time prediction across multi-source and multi-parameter scenarios.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Computer Vision: Diffusion Models for Vision
• Machine Learning: Deep Neural Architectures and Foundation Models

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• Modeling convective transport in porous media with sharp Gaussian sources
• Coupling FEM with physics-informed DeepONet for fluid transport
• Handling steep gradients from localized sources through adaptive sampling

Innovation

• Hybrid FEM-DeepONet framework for porous media
• Adaptive sampling for handling sharp gradients
• Fast inference with physics-informed neural operator