# AI-enhanced iterative solvers for accelerating the solution of large scale parametrized systems

arXiv: https://arxiv.org/abs/2207.02543 · HTML: https://mlsanity.com/paper/2207.02543/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 92 references that are in the map.
- First posted on arXiv 6 Jul 2022.

## Builds on (most cited first, 12 of 12)

- [A comprehensive deep learning-based approach to reduced order modeling of nonlinear time-dependent parametrized PDEs](https://mlsanity.com/paper/2001.04001/) (2020)
- [POD-DL-ROM: enhancing deep learning-based reduced order models for nonlinear parametrized PDEs by proper orthogonal decomposition](https://mlsanity.com/paper/2101.11845/) (2021)
- [Deep convolutional encoder-decoder networks for uncertainty quantification of dynamic multiphase flow in heterogeneous media](https://mlsanity.com/paper/1807.00882/) (2018)
- [Deep convolutional recurrent autoencoders for learning low-dimensional feature dynamics of fluid systems](https://mlsanity.com/paper/1808.01346/) (2018)
- [Learning Algebraic Multigrid Using Graph Neural Networks](https://mlsanity.com/paper/2003.05744/) (2020)
- [Multi-level Convolutional Autoencoder Networks for Parametric Prediction of Spatio-temporal Dynamics](https://mlsanity.com/paper/1912.11114/) (2019)
- [Non-intrusive reduced order modeling of natural convection in porous media using convolutional autoencoders: comparison with linear subspace techniques](https://mlsanity.com/paper/2107.11460/) (2021)
- [Data driven approximation of parametrized PDEs by Reduced Basis and Neural Networks](https://mlsanity.com/paper/1904.01514/) (2019)
- [A Comparison of Neural Network Architectures for Data-Driven Reduced-Order Modeling](https://mlsanity.com/paper/2110.03442/) (2021)
- [A local basis approximation approach for nonlinear parametric model order reduction](https://mlsanity.com/paper/2003.07716/) (2020)
- [Grassmannian diffusion maps based dimension reduction and classification for high-dimensional data](https://mlsanity.com/paper/2009.07547/) (2020)
- [An AI-based Domain-Decomposition Non-Intrusive Reduced-Order Model for Extended Domains applied to Multiphase Flow in Pipes](https://mlsanity.com/paper/2202.06170/) (2022)

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

- [Neural Green's Operators for Parametric Partial Differential Equations](https://mlsanity.com/paper/2406.01857/) (2024)
- [NOWS: Neural Operator Warm Starts for Accelerating Iterative Solvers](https://mlsanity.com/paper/2511.02481/) (2025)
- [Integrating Mechanistic and Data-Driven Models for Neurological Disorders through Differentiable Programming](https://mlsanity.com/paper/2606.06094/) (2026)

## Abstract (arXiv)

Recent advances in the field of machine learning open a new era in high performance computing. Applications of machine learning algorithms for the development of accurate and cost-efficient surrogates of complex problems have already attracted major attention from scientists. Despite their powerful approximation capabilities, however, surrogates cannot produce the `exact' solution to the problem. To address this issue, this paper exploits up-to-date ML tools and delivers customized iterative solvers of linear equation systems, capable of solving large-scale parametrized problems at any desired level of accuracy. Specifically, the proposed approach consists of the following two steps. At first, a reduced set of model evaluations is performed and the corresponding solutions are used to establish an approximate mapping from the problem's parametric space to its solution space using deep feedforward neural networks and convolutional autoencoders. This mapping serves a means to obtain very accurate initial predictions of the system's response to new query points at negligible computational cost. Subsequently, an iterative solver inspired by the Algebraic Multigrid method in combination with Proper Orthogonal Decomposition, termed POD-2G, is developed that successively refines the initial predictions towards the exact system solutions. The application of POD-2G as a standalone solver or as preconditioner in the context of preconditioned conjugate gradient methods is demonstrated on several numerical examples of large scale systems, with the results indicating its superiority over conventional iterative solution schemes.