# Understanding the Efficacy of U-Net & Vision Transformer for Groundwater Numerical Modelling

arXiv: https://arxiv.org/abs/2307.04010 · HTML: https://mlsanity.com/paper/2307.04010/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 21 references that are in the map.
- First posted on arXiv 8 Jul 2023.

## Builds on (most cited first, 14 of 14)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [TransUNet: Transformers Make Strong Encoders for Medical Image Segmentation](https://mlsanity.com/paper/2102.04306/) (2021)
- [Axial Attention in Multidimensional Transformers](https://mlsanity.com/paper/1912.12180/) (2019)
- [Choose a Transformer: Fourier or Galerkin](https://mlsanity.com/paper/2105.14995/) (2021)
- [U-FNO -- An enhanced Fourier neural operator-based deep-learning model for multiphase flow](https://mlsanity.com/paper/2109.03697/) (2021)
- [Transformer for Partial Differential Equations' Operator Learning](https://mlsanity.com/paper/2205.13671/) (2022)
- [Towards Multi-spatiotemporal-scale Generalized PDE Modeling](https://mlsanity.com/paper/2209.15616/) (2022)
- [NOMAD: Nonlinear Manifold Decoders for Operator Learning](https://mlsanity.com/paper/2206.03551/) (2022)
- [Fourier-MIONet: Fourier-enhanced multiple-input neural operators for multiphase modeling of geological carbon sequestration](https://mlsanity.com/paper/2303.04778/) (2023)
- [ViTO: Vision Transformer-Operator](https://mlsanity.com/paper/2303.08891/) (2023)

## Built on by (most cited first, 2 of 2)

- [PIC2O-Sim: A Physics-Inspired Causality-Aware Dynamic Convolutional Neural Operator for Ultra-Fast Photonic Device FDTD Simulation](https://mlsanity.com/paper/2406.17810/) (2024)
- [LLT: Local Linear Transformer for PDE Operator Learning](https://mlsanity.com/paper/2607.07718/) (2026)

## Abstract (arXiv)

This paper presents a comprehensive comparison of various machine learning models, namely U-Net, U-Net integrated with Vision Transformers (ViT), and Fourier Neural Operator (FNO), for time-dependent forward modelling in groundwater systems. Through testing on synthetic datasets, it is demonstrated that U-Net and U-Net + ViT models outperform FNO in accuracy and efficiency, especially in sparse data scenarios. These findings underscore the potential of U-Net-based models for groundwater modelling in real-world applications where data scarcity is prevalent.