# RINS: Residual-Image Neural Subspace Solvers for Large Sparse Linear Systems

arXiv: https://arxiv.org/abs/2610.02217 · HTML: https://mlsanity.com/paper/2610.02217/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 20 of its 47 references that are in the map.
- First posted on arXiv 9 Sep 2026.

## Builds on (most cited first, 20 of 20)

- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (2017)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering](https://mlsanity.com/paper/1606.09375/) (2016)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Learning to Simulate Complex Physics with Graph Networks](https://mlsanity.com/paper/2002.09405/) (2020)
- [Learning Mesh-Based Simulation with Graph Networks](https://mlsanity.com/paper/2010.03409/) (2020)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [PDEBENCH: An Extensive Benchmark for Scientific Machine Learning](https://mlsanity.com/paper/2210.07182/) (2022)
- [Message Passing Neural PDE Solvers](https://mlsanity.com/paper/2202.03376/) (2022)
- [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020)
- [Model Reduction and Neural Networks for Parametric PDEs](https://mlsanity.com/paper/2005.03180/) (2020)
- [hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decomposition](https://mlsanity.com/paper/2003.05385/) (2020)
- [Multiwavelet-based Operator Learning for Differential Equations](https://mlsanity.com/paper/2109.13459/) (2021)
- [Learning to Optimize Multigrid PDE Solvers](https://mlsanity.com/paper/1902.10248/) (2019)
- [Can Physics-Informed Neural Networks beat the Finite Element Method?](https://mlsanity.com/paper/2302.04107/) (2023)
- [HAMLET: Graph Transformer Neural Operator for Partial Differential Equations](https://mlsanity.com/paper/2402.03541/) (2024)
- [Scale-PINN: Learning Efficient Physics-Informed Neural Networks Through Sequential Correction](https://mlsanity.com/paper/2602.19475/) (2026)

## Abstract (arXiv)

Large sparse linear systems from PDE discretizations require correction subspaces whose operator images explain the current residual. We study this residual-image viewpoint and propose Gate-RINS, a neural subspace solver that generates polynomial correction bases from cached residual probes and modulates them with a lightweight residual- and coordinate-dependent pointwise gate. The projected least-squares update remains unchanged, so the neural component only chooses the expansion directions while the numerical closure tests them through (operatornamerange(AQt)). We also introduce a hybrid controller schedule that composes GRANS-style graph controllers with Gate-RINS under the same projected solver. Across six PDE-derived benchmark tasks and two scales, Gate-RINS reaches fixed relative-residual thresholds faster in synchronized wall-clock time than GMRES and a recent graph-only neural baseline in most settings, and hybrid schedules further improve the residual trajectory. Difficult-mode diagnostics and trajectory visualizations support the interpretation that these gains are associated with operator-image subspaces that align more effectively with the current residual.