# A Neural-preconditioned Poisson Solver for Mixed Dirichlet and Neumann Boundary Conditions

arXiv: https://arxiv.org/abs/2310.00177 · HTML: https://mlsanity.com/paper/2310.00177/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 38 references that are in the map.
- First posted on arXiv 29 Sep 2023.

## Builds on (most cited first, 15 of 15)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Neural Machine Translation by Jointly Learning to Align and Translate](https://mlsanity.com/paper/1409.0473/) (2014)
- [Deep multi-scale video prediction beyond mean square error](https://mlsanity.com/paper/1511.05440/) (2015)
- [Accelerating Eulerian Fluid Simulation With Convolutional Networks](https://mlsanity.com/paper/1607.03597/) (2016)
- [MgNet: A Unified Framework of Multigrid and Convolutional Neural Network](https://mlsanity.com/paper/1901.10415/) (2019)
- [Multigrid-augmented deep learning preconditioners for the Helmholtz equation](https://mlsanity.com/paper/2203.11025/) (2022)
- [Poisson CNN: Convolutional neural networks for the solution of the Poisson equation on a Cartesian mesh](https://mlsanity.com/paper/1910.08613/) (2019)
- [Deep Learning of Preconditioners for Conjugate Gradient Solvers in Urban Water Related Problems](https://mlsanity.com/paper/1906.06925/) (2019)
- [Learning optimal multigrid smoothers via neural networks](https://mlsanity.com/paper/2102.12071/) (2021)
- [A Deep Conjugate Direction Method for Iteratively Solving Linear Systems](https://mlsanity.com/paper/2205.10763/) (2022)
- [Machine-Learned Preconditioners for Linear Solvers in Geophysical Fluid Flows](https://mlsanity.com/paper/2010.02866/) (2020)
- [Using neural networks to solve the 2D Poisson equation for electric field computation in plasma fluid simulations](https://mlsanity.com/paper/2109.13076/) (2021)
- [MGIC: Multigrid-in-Channels Neural Network Architectures](https://mlsanity.com/paper/2011.09128/) (2020)

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

- [Metamizer: a versatile neural optimizer for fast and accurate physics simulations](https://mlsanity.com/paper/2410.19746/) (2024)
- [Learning Sparse Approximate Inverse Preconditioners for Conjugate Gradient Solvers on GPUs](https://mlsanity.com/paper/2510.27517/) (2025)

## Abstract (arXiv)

We introduce a neural-preconditioned iterative solver for Poisson equations with mixed boundary conditions. Typical Poisson discretizations yield large, ill-conditioned linear systems. Iterative solvers can be effective for these problems, but only when equipped with powerful preconditioners. Unfortunately, effective preconditioners like multigrid require costly setup phases that must be re-executed every time domain shapes or boundary conditions change, forming a severe bottleneck for problems with evolving boundaries. In contrast, we present a neural preconditioner trained to efficiently approximate the inverse of the discrete Laplacian in the presence of such changes. Our approach generalizes to domain shapes, boundary conditions, and grid sizes outside the training set. The key to our preconditioner's success is a novel, lightweight neural network architecture featuring spatially varying convolution kernels and supporting fast inference. We demonstrate that our solver outperforms state-of-the-art methods like algebraic multigrid as well as recently proposed neural preconditioners on challenging test cases arising from incompressible fluid simulations.