# Prediction of Multiscale Features Using Deep Learning-based Preconditioner-Solver Architecture for Darcy Equation in High-Contrast Media

arXiv: https://arxiv.org/abs/2411.02431 · HTML: https://mlsanity.com/paper/2411.02431/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 33 references that are in the map.
- First posted on arXiv 1 Nov 2024.

## Builds on (most cited first, 7 of 7)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Very Deep Convolutional Networks for Large-Scale Image Recognition](https://mlsanity.com/paper/1409.1556/) (2014)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [Mitigating spectral bias for the multiscale operator learning](https://mlsanity.com/paper/2210.10890/) (2022)
- [Generalized Multiscale Finite Element Methods (GMsFEM)](https://mlsanity.com/paper/1301.2866/) (2013)
- [Lipschitz Continuity Retained Binary Neural Network](https://mlsanity.com/paper/2207.06540/) (2022)

## Built on by (most cited first, 1 of 1)

- [Applying Two-Grid Preconditioner for Subsurface Flow Simulation using Attention-enhanced Hybrid Network to Accelerate Multiscale Discretization in High-contrast Media](https://mlsanity.com/paper/2606.02582/) (2026)

## Abstract (arXiv)

Modeling subsurface fluid flow in porous media is crucial for applications such as oil and gas exploration. However, the inherent heterogeneity and multi-scale characteristics of these systems pose significant challenges in accurately reconstructing fluid flow behaviors. To address this issue, we proposed Fourier Preconditioner-based Hierarchical Multiscale Net (FP-HMsNet), an efficient hierarchical preconditioner-learner architecture that combines Fourier Neural Operators (FNO) with multi-scale neural networks to reconstruct multi-scale basis functions of high-dimensional subsurface fluid flow. Using a dataset comprising 102,757 training samples, 34,252 validation samples, and 34,254 test samples, we ensured the reliability and generalization capability of the model. Experimental results showed that FP-HMsNet achieved an MSE of 0.0036, an MAE of 0.0375, and an R2 of 0.9716 on the testing set, significantly outperforming existing models and demonstrating exceptional accuracy and generalization ability. Additionally, robustness tests revealed that the model maintained stability under various levels of noise interference. Ablation studies confirmed the critical contribution of the preconditioner and multi-scale pathways to the model's performance. Compared to current models, FP-HMsNet not only achieved lower errors and higher accuracy but also demonstrated faster convergence and improved computational efficiency, establishing itself as the state-of-the-art (SOTA) approach. This model offers a novel method for efficient and accurate subsurface fluid flow modeling, with promising potential for more complex real-world applications.