# Generative modeling of Sparse Approximate Inverse Preconditioners

arXiv: https://arxiv.org/abs/2405.11007 · HTML: https://mlsanity.com/paper/2405.11007/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 35 references that are in the map.
- First posted on arXiv 17 May 2024.

## Builds on (most cited first, 11 of 11)

- [Conditional Generative Adversarial Nets](https://mlsanity.com/paper/1411.1784/) (2014)
- [Variational Graph Auto-Encoders](https://mlsanity.com/paper/1611.07308/) (2016)
- [How Attentive are Graph Attention Networks?](https://mlsanity.com/paper/2105.14491/) (2021)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)
- [Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10566/) (2017)
- [Multigrid-augmented deep learning preconditioners for the Helmholtz equation](https://mlsanity.com/paper/2203.11025/) (2022)
- [Deep Learning of Preconditioners for Conjugate Gradient Solvers in Urban Water Related Problems](https://mlsanity.com/paper/1906.06925/) (2019)
- [Neural incomplete factorization: learning preconditioners for the conjugate gradient method](https://mlsanity.com/paper/2305.16368/) (2023)
- [Diversity-Promoting Human Motion Interpolation via Conditional Variational Auto-Encoder](https://mlsanity.com/paper/2111.06762/) (2021)

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

- [Learning incomplete factorization preconditioners for GMRES](https://mlsanity.com/paper/2409.08262/) (2024)
- [Message-Passing GNNs Fail to Approximate Sparse Triangular Factorizations](https://mlsanity.com/paper/2502.01397/) (2025)

## Abstract (arXiv)

We present a new deep learning paradigm for the generation of sparse approximate inverse (SPAI) preconditioners for matrix systems arising from the mesh-based discretization of elliptic differential operators. Our approach is based upon the observation that matrices generated in this manner are not arbitrary, but inherit properties from differential operators that they discretize. Consequently, we seek to represent a learnable distribution of high-performance preconditioners from a low-dimensional subspace through a carefully-designed autoencoder, which is able to generate SPAI preconditioners for these systems. The concept has been implemented on a variety of finite element discretizations of second- and fourth-order elliptic partial differential equations with highly promising results.