# Deep Learning of Preconditioners for Conjugate Gradient Solvers in Urban Water Related Problems

arXiv: https://arxiv.org/abs/1906.06925 · HTML: https://mlsanity.com/paper/1906.06925/

## Key facts

- Cited by 20 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Jun 2019.

## Built on by (most cited first, 20 of 20)

- [Recent Advances on Machine Learning for Computational Fluid Dynamics: A Survey](https://mlsanity.com/paper/2408.12171/) (2024)
- [Neural incomplete factorization: learning preconditioners for the conjugate gradient method](https://mlsanity.com/paper/2305.16368/) (2023)
- [Learning from Linear Algebra: A Graph Neural Network Approach to Preconditioner Design for Conjugate Gradient Solvers](https://mlsanity.com/paper/2405.15557/) (2024)
- [The Sufficiency of Off-Policyness and Soft Clipping: PPO is still Insufficient according to an Off-Policy Measure](https://mlsanity.com/paper/2205.10047/) (2022)
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- [Neural Green's Operators for Parametric Partial Differential Equations](https://mlsanity.com/paper/2406.01857/) (2024)
- [Learning incomplete factorization preconditioners for GMRES](https://mlsanity.com/paper/2409.08262/) (2024)
- [Verification and Validation for Trustworthy Scientific Machine Learning](https://mlsanity.com/paper/2502.15496/) (2025)
- [PEARL: Preconditioner Enhancement through Actor-critic Reinforcement Learning](https://mlsanity.com/paper/2501.10750/) (2025)
- [Towards Neural Sparse Linear Solvers](https://mlsanity.com/paper/2203.06944/) (2022)
- [A Neural-preconditioned Poisson Solver for Mixed Dirichlet and Neumann Boundary Conditions](https://mlsanity.com/paper/2310.00177/) (2023)
- [Accelerating Legacy Numerical Solvers by Non-intrusive Gradient-based Meta-solving](https://mlsanity.com/paper/2405.02952/) (2024)
- [Generative modeling of Sparse Approximate Inverse Preconditioners](https://mlsanity.com/paper/2405.11007/) (2024)
- [Message-Passing GNNs Fail to Approximate Sparse Triangular Factorizations](https://mlsanity.com/paper/2502.01397/) (2025)
- [Deep Learning-Enhanced Preconditioning for Efficient Conjugate Gradient Solvers in Large-Scale PDE Systems](https://mlsanity.com/paper/2412.07127/) (2024)
- [Generalized Neural Operator for Parametric and Boundary-Value Problems](https://mlsanity.com/paper/2607.21932/) (2026)
- [Matrix-free Neural Preconditioner for the Dirac Operator in Lattice Gauge Theory](https://mlsanity.com/paper/2509.10378/) (2025)
- [IV-Net: A neural network for elliptic PDEs with random and highly varying coefficients](https://mlsanity.com/paper/2605.24876/) (2026)
- [Hybrid Lagrangian-Eulerian Model for Lagrangian Fluid Simulation](https://mlsanity.com/paper/2608.01164/) (2026)
- [An overview of machine learning-enhanced iterative methods for systems of linear and nonlinear equations](https://mlsanity.com/paper/2610.07211/) (2026)

## Abstract (arXiv)

Solving systems of linear equations is a problem occuring frequently in water engineering applications. Usually the size of the problem is too large to be solved via direct factorization. One can resort to iterative approaches, in particular the conjugate gradients method if the matrix is symmetric positive definite. Preconditioners further enhance the rate of convergence but hitherto only handcrafted ones requiring expert knowledge have been used. We propose an innovative approach employing Machine Learning, in particular a Convolutional Neural Network, to unassistedly design preconditioning matrices specifically for the problem at hand. Based on an in-depth case study in fluid simulation we are able to show that our learned preconditioner is able to improve the convergence rate even beyond well established methods like incomplete Cholesky factorization or Algebraic MultiGrid.