# NeuraLSP: A Neural Spectral Preconditioner for Accelerating PDE Solvers

arXiv: https://arxiv.org/abs/2601.20174 · HTML: https://mlsanity.com/paper/2601.20174/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 49 references that are in the map.
- First posted on arXiv 28 Jan 2026.

## Builds on (most cited first, 14 of 14)

- [Gaussian Error Linear Units (GELUs)](https://mlsanity.com/paper/1606.08415/) (2016)
- [Spectral Normalization for Generative Adversarial Networks](https://mlsanity.com/paper/1802.05957/) (2018)
- [SpectralNet: Spectral Clustering using Deep Neural Networks](https://mlsanity.com/paper/1801.01587/) (2018)
- [An Analysis of SVD for Deep Rotation Estimation](https://mlsanity.com/paper/2006.14616/) (2020)
- [Learning to Optimize Multigrid PDE Solvers](https://mlsanity.com/paper/1902.10248/) (2019)
- [Learning Algebraic Multigrid Using Graph Neural Networks](https://mlsanity.com/paper/2003.05744/) (2020)
- [EigenGame: PCA as a Nash Equilibrium](https://mlsanity.com/paper/2010.00554/) (2020)
- [Spectral Inference Networks: Unifying Deep and Spectral Learning](https://mlsanity.com/paper/1806.02215/) (2018)
- [Backpropagation-Friendly Eigendecomposition](https://mlsanity.com/paper/1906.09023/) (2019)
- [NeuralEF: Deconstructing Kernels by Deep Neural Networks](https://mlsanity.com/paper/2205.00165/) (2022)
- [Regularized linear autoencoders recover the principal components, eventually](https://mlsanity.com/paper/2007.06731/) (2020)
- [Optimization-Based Algebraic Multigrid Coarsening Using Reinforcement Learning](https://mlsanity.com/paper/2106.01854/) (2021)
- [Operator SVD with Neural Networks via Nested Low-Rank Approximation](https://mlsanity.com/paper/2402.03655/) (2024)
- [A Deep Learning algorithm to accelerate Algebraic Multigrid methods in Finite Element solvers of 3D elliptic PDEs](https://mlsanity.com/paper/2304.10832/) (2023)

## Abstract (arXiv)

Solving large-scale sparse linear systems originating from partial differential equations (PDEs) is a fundamental topic in high-performance scientific computing, where preconditioners are crucial. Multigrid methods are among the most effective preconditioners, yet their performance is dictated by the accurate construction of grid transfer operators. Current neural multigrid methods learn such operators with graph neural networks (GNNs), typically by extracting connectivity from discretized system matrices. While effective, these graph-based constructions suffer from rank inflation, resulting in unnecessarily large coarse spaces and slower convergence. To ameliorate, this paper advocates NeuraLSP, a new neural multigrid preconditioner that replaces graph aggregation with a fixed low-rank spectral representation derived from the left singular subspace of near-nullspace components. At the network design level, NeuraLSP is trained with a novel subspace loss function, which preserves the error modes most relevant to multigrid convergence while suppressing rank inflation. This paper's grand innovation hinges upon both theoretical guarantees and empirical robustness to rank inflation, affording up to a 53% speedup over SOTA neural preconditioners across a variety of PDE families.