# Banach neural operator for Navier-Stokes equations

arXiv: https://arxiv.org/abs/2512.09070 · HTML: https://mlsanity.com/paper/2512.09070/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 36 references that are in the map.
- First posted on arXiv 28 Nov 2025.

## Builds on (most cited first, 12 of 12)

- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [U-FNO -- An enhanced Fourier neural operator-based deep-learning model for multiphase flow](https://mlsanity.com/paper/2109.03697/) (2021)
- [Neural Operators for Accelerating Scientific Simulations and Design](https://mlsanity.com/paper/2309.15325/) (2023)
- [Physics-Informed Deep Neural Operator Networks](https://mlsanity.com/paper/2207.05748/) (2022)
- [Multifidelity deep neural operators for efficient learning of partial differential equations with application to fast inverse design of nanoscale heat transport](https://mlsanity.com/paper/2204.06684/) (2022)
- [Discovering and forecasting extreme events via active learning in neural operators](https://mlsanity.com/paper/2204.02488/) (2022)
- [Koopman neural operator as a mesh-free solver of non-linear partial differential equations](https://mlsanity.com/paper/2301.10022/) (2023)
- [Neural Integral Equations](https://mlsanity.com/paper/2209.15190/) (2022)
- [Multi-fidelity prediction of fluid flow and temperature field based on transfer learning using Fourier Neural Operator](https://mlsanity.com/paper/2304.06972/) (2023)

## Built on by (most cited first, 1 of 1)

- [A Physics-Informed Neural Network Framework for Elastodynamic Wave Propagation in Bimaterial Systems](https://mlsanity.com/paper/2607.06479/) (2026)

## Abstract (arXiv)

Classical neural networks are known for their ability to approximate mappings between finite-dimensional spaces, but they fall short in capturing complex operator dynamics across infinite-dimensional function spaces. Neural operators, in contrast, have emerged as powerful tools in scientific machine learning for learning such mappings. However, standard neural operators typically lack mechanisms for mixing or attending to input information across space and time. In this work, we introduce the Banach neural operator (BNO) -- a novel framework that integrates Koopman operator theory with deep neural networks to predict nonlinear, spatiotemporal dynamics from partial observations. The BNO approximates a nonlinear operator between Banach spaces by combining spectral linearization (via Koopman theory) with deep feature learning (via convolutional neural networks and nonlinear activations). This sequence-to-sequence model captures dominant dynamic modes and allows for mesh-independent prediction. Numerical experiments on the Navier-Stokes equations demonstrate the method's accuracy and generalization capabilities. In particular, BNO achieves robust zero-shot super-resolution in unsteady flow prediction and consistently outperforms conventional Koopman-based methods and deep learning models.