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

arXiv: https://arxiv.org/abs/2508.19847 · HTML: https://mlsanity.com/paper/2508.19847/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 64 references that are in the map.
- First posted on arXiv 27 Aug 2025.

## Builds on (most cited first, 24 of 24)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [MIONet: Learning multiple-input operators via tensor product](https://mlsanity.com/paper/2202.06137/) (2022)
- [Improved architectures and training algorithms for deep operator networks](https://mlsanity.com/paper/2110.01654/) (2021)
- [When Physics Meets Machine Learning: A Survey of Physics-Informed Machine Learning](https://mlsanity.com/paper/2203.16797/) (2022)
- [Multifidelity Deep Operator Networks For Data-Driven and Physics-Informed Problems](https://mlsanity.com/paper/2204.09157/) (2022)
- [JAX-FEM: A differentiable GPU-accelerated 3D finite element solver for automatic inverse design and mechanistic data science](https://mlsanity.com/paper/2212.00964/) (2022)
- [JAX-FLUIDS: A fully-differentiable high-order computational fluid dynamics solver for compressible two-phase flows](https://mlsanity.com/paper/2203.13760/) (2022)
- [Applications of physics informed neural operators](https://mlsanity.com/paper/2203.12634/) (2022)
- [Neural Chaos: A Spectral Stochastic Neural Operator](https://mlsanity.com/paper/2502.11835/) (2025)
- [Physics-informed Neural Networks for Solving Nonlinear Diffusivity and Biot's equations](https://mlsanity.com/paper/2002.08235/) (2020)
- [Synergistic Learning with Multi-Task DeepONet for Efficient PDE Problem Solving](https://mlsanity.com/paper/2408.02198/) (2024)
- [Basis-to-Basis Operator Learning Using Function Encoders](https://mlsanity.com/paper/2410.00171/) (2024)
- [Separable DeepONet: Breaking the Curse of Dimensionality in Physics-Informed Machine Learning](https://mlsanity.com/paper/2407.15887/) (2024)
- [A multifidelity approach to continual learning for physical systems](https://mlsanity.com/paper/2304.03894/) (2023)
- [A Gradient-based Deep Neural Network Model for Simulating Multiphase Flow in Porous Media](https://mlsanity.com/paper/2105.02652/) (2021)
- [Inverse modeling of nonisothermal multiphase poromechanics using physics-informed neural networks](https://mlsanity.com/paper/2209.03276/) (2022)
- [iPINNs: Incremental learning for Physics-informed neural networks](https://mlsanity.com/paper/2304.04854/) (2023)
- [Neural Operator-Based Proxy for Reservoir Simulations Considering Varying Well Settings, Locations, and Permeability Fields](https://mlsanity.com/paper/2407.09728/) (2024)

## Built on by (most cited first, 3 of 3)

- [ANCHOR: Error-Controlled Adaptive Numerical Correction for Neural Operator Time Marching](https://mlsanity.com/paper/2512.19643/) (2025)
- [A hybrid analytical-PINN model for subsurface simulation of geothermal heat exchangers in heterogeneous underground](https://mlsanity.com/paper/2607.12271/) (2026)
- [Learning Unsteady Aneurysm Hemodynamics with Physics-Informed DeepONets](https://mlsanity.com/paper/2608.13629/) (2026)

## Abstract (arXiv)

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.