# Latent Mamba Operator for Partial Differential Equations

arXiv: https://arxiv.org/abs/2505.19105 · HTML: https://mlsanity.com/paper/2505.19105/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Placed in Time series forecasting with SSMs, one of 10 directions in State-space sequence models (S4, Mamba).
- Builds on 46 of its 61 references that are in the map.
- First posted on arXiv 25 May 2025.

## Placed in

- State-space sequence models (S4, Mamba) / [Time series forecasting with SSMs](https://mlsanity.com/ssm/time-series-forecasting-with-ssms/)

## Builds on (most cited first, 30 of 46)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Swin Transformer: Hierarchical Vision Transformer using Shifted Windows](https://mlsanity.com/paper/2103.14030/) (2021)
- [Layer Normalization](https://mlsanity.com/paper/1607.06450/) (2016)
- [Mamba: Linear-Time Sequence Modeling with Selective State Spaces](https://mlsanity.com/paper/2312.00752/) (2023)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Efficiently Modeling Long Sequences with Structured State Spaces](https://mlsanity.com/paper/2111.00396/) (2021)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Decoupled Weight Decay Regularization](https://mlsanity.com/paper/1711.05101/) (2017)
- [Transformers are SSMs: Generalized Models and Efficient Algorithms Through Structured State Space Duality](https://mlsanity.com/paper/2405.21060/) (2024)
- [Perceiver: General Perception with Iterative Attention](https://mlsanity.com/paper/2103.03206/) (2021)
- [Vision Mamba: Efficient Visual Representation Learning with Bidirectional State Space Model](https://mlsanity.com/paper/2401.09417/) (2024)
- [Sigmoid-Weighted Linear Units for Neural Network Function Approximation in Reinforcement Learning](https://mlsanity.com/paper/1702.03118/) (2017)
- [Combining Recurrent, Convolutional, and Continuous-time Models with Linear State-Space Layers](https://mlsanity.com/paper/2110.13985/) (2021)
- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [Choose a Transformer: Fourier or Galerkin](https://mlsanity.com/paper/2105.14995/) (2021)
- [GNOT: A General Neural Operator Transformer for Operator Learning](https://mlsanity.com/paper/2302.14376/) (2023)
- [Transolver: A Fast Transformer Solver for PDEs on General Geometries](https://mlsanity.com/paper/2402.02366/) (2024)
- [U-FNO -- An enhanced Fourier neural operator-based deep-learning model for multiphase flow](https://mlsanity.com/paper/2109.03697/) (2021)
- [Transformer for Partial Differential Equations' Operator Learning](https://mlsanity.com/paper/2205.13671/) (2022)
- [Factorized Fourier Neural Operators](https://mlsanity.com/paper/2111.13802/) (2021)
- [Adaptive Fourier Neural Operators: Efficient Token Mixers for Transformers](https://mlsanity.com/paper/2111.13587/) (2021)
- [Multiwavelet-based Operator Learning for Differential Equations](https://mlsanity.com/paper/2109.13459/) (2021)
- [U-NO: U-shaped Neural Operators](https://mlsanity.com/paper/2204.11127/) (2022)
- [FourCastNet: Accelerating Global High-Resolution Weather Forecasting using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2208.05419/) (2022)
- [Fast Sampling of Diffusion Models via Operator Learning](https://mlsanity.com/paper/2211.13449/) (2022)
- [DPOT: Auto-Regressive Denoising Operator Transformer for Large-Scale PDE Pre-Training](https://mlsanity.com/paper/2403.03542/) (2024)

## Built on by (most cited first, 10 of 10)

- [Structure-Preserving Learning Improves Geometry Generalization in Neural PDEs](https://mlsanity.com/paper/2602.02788/) (2026)
- [Stable Long-Horizon PDE Forecasting via Latent Structured Spectral Propagators](https://mlsanity.com/paper/2605.10154/) (2026)
- [Agentic Neural Architecture Search](https://mlsanity.com/paper/2607.07984/) (2026)
- [Adaptive Mamba Neural Operators](https://mlsanity.com/paper/2607.18043/) (2026)
- [SPAMoE: Spectrum-Aware Hybrid Operator Framework for Full-Waveform Inversion](https://mlsanity.com/paper/2604.07421/) (2026)
- [Graph Mamba Operator: A Latent Simulator for Interacting Particle Systems](https://mlsanity.com/paper/2606.09432/) (2026)
- [Higher-Order Fourier Neural Operator: Explicit Mode Mixer for Nonlinear PDEs](https://mlsanity.com/paper/2606.28122/) (2026)
- [Beyond Pairwise Graphs in Science: Hypergraph Adaptive Wavelet Operators for Parametric PDEs](https://mlsanity.com/paper/2608.27883/) (2026)
- [Advectra: Asymmetric Latent Transport for Non-Stationary Physics](https://mlsanity.com/paper/2610.05098/) (2026)
- [METRO: Metric-Enhanced Token Routing Operator](https://mlsanity.com/paper/2610.05100/) (2026)

## Abstract (arXiv)

Neural operators have emerged as powerful data-driven frameworks for solving Partial Differential Equations (PDEs), offering significant speedups over numerical methods. However, existing neural operators struggle with scalability in high-dimensional spaces, incur high computational costs, and face challenges in capturing continuous and long-range dependencies in PDE dynamics. To address these limitations, we introduce the Latent Mamba Operator (LaMO), which integrates the efficiency of state-space models (SSMs) in latent space with the expressive power of kernel integral formulations in neural operators. We also establish a theoretical connection between state-space models (SSMs) and the kernel integral of neural operators. Extensive experiments across diverse PDE benchmarks on regular grids, structured meshes, and point clouds covering solid and fluid physics datasets, LaMOs achieve consistent state-of-the-art (SOTA) performance, with a 32.3% improvement over existing baselines in solution operator approximation, highlighting its efficacy in modeling complex PDE solutions.