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Banach neural operator for Navier-Stokes equations

Bo Zhang

TL;DR

The paper introduces the Banach neural operator (BNO), a mesh-independent operator-learning framework that fuses Koopman spectral theory with CNN-based nonlinear corrections to model nonlinear Navier–Stokes dynamics from partial observations. By embedding a Koopman discrete operator within a sequence-to-sequence Banach-layer architecture and leveraging DMD for spectral propagation, BNO achieves interpretable dynamic reconstruction and robust zero-shot super-resolution across discretizations. Empirical results on a 2D compressible Navier–Stokes LES dataset show superior long-term forecasting, stability, and grid-transferability compared with DMD and competitive performance versus CNNs, with notable advantages in extrapolation and wake/ shear-region dynamics. Deeper Banach configurations introduce numerical instability, highlighting a practical preference for single-layer BNO in current implementations and outlining avenues for stability-enhancing regularization and alternative decompositions.

Abstract

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.

Banach neural operator for Navier-Stokes equations

TL;DR

The paper introduces the Banach neural operator (BNO), a mesh-independent operator-learning framework that fuses Koopman spectral theory with CNN-based nonlinear corrections to model nonlinear Navier–Stokes dynamics from partial observations. By embedding a Koopman discrete operator within a sequence-to-sequence Banach-layer architecture and leveraging DMD for spectral propagation, BNO achieves interpretable dynamic reconstruction and robust zero-shot super-resolution across discretizations. Empirical results on a 2D compressible Navier–Stokes LES dataset show superior long-term forecasting, stability, and grid-transferability compared with DMD and competitive performance versus CNNs, with notable advantages in extrapolation and wake/ shear-region dynamics. Deeper Banach configurations introduce numerical instability, highlighting a practical preference for single-layer BNO in current implementations and outlining avenues for stability-enhancing regularization and alternative decompositions.

Abstract

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.

Paper Structure

This paper contains 29 sections, 23 equations, 18 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Methods
  3. Problem setup: operator learning in Banach spaces
  4. BNO architecture and recursive update rule
  5. Banach neural operator
  6. Koopman discrete operator via DMD
  7. DMD computation steps
  8. Runtime and complexity analysis of BNO components
  9. High-fidelity numerical experiments
  10. Simulation setup and computational framework
  11. Training dataset construction
  12. Benchmarking: one-layer BNO versus DMD and CNN
  13. Training setup
  14. Convergence behavior and overfitting analysis
  15. One-step forecasting accuracy
  16. ...and 14 more sections

Figures (18)

  • Figure 1: Illustration of the BNO architecture. (a) A schematic showcasing the overall architecture of the BNO. We start with an input function $b(x,t)$. 1. Maintain the original dimensionality for the input layer; 2. Apply several layers of Banach neural operators, each composed of Koopman discrete operators, stacked convolutional operators, nonlinear activation functions and local transformations; 3. Finally output $w(x,t)$. (b) Banach layers: start from input $u(x,t)$. Top pannel: Perform DMD analysis to compute the truncated DMD modes and eigenvalues, then reconstruct the future state. Bottom panel: Apply stacked convolutional operators to the input sequence.
  • Figure 2: Flowchart of the proposed approach for learning the BNO to model the Navier-Stokes equations in an end-to-end, seq2seq manner.
  • Figure 3: Training and validation loss histories of BNO and CNN models for the transverse velocity field evaluated at a resolution of $256 \times 128$.
  • Figure 4: Spatiotemporal evolution of the transverse velocity field evaluated at a resolution of $256 \times 128$. The first and fourth columns show the ground truth and DMD prediction, respectively; the second and third columns present the learned interpolations by the BNO and CNN models. In this setting, the BNO is constructed with a single Banach layer. Furthermore, one-step-ahead prediction on the training dataset is applied to capture the spatiotemporal dynamics for both the BNO and CNN models.
  • Figure 5: Comparison of the temporal evolutions of the transverse velocity profiles predicted by the BNO, DMD and CNN models against the ground truth at $x/c=1.8$ and distinct time instants (a) $t=0.43$s, (b) $0.44$s, (c) $0.45$s, and (d) $0.46$s. A one-step-ahead prediction strategy is applied to capture the spatiotemporal dynamics. The results are obtained using the transverse velocity field evaluated at a resolution of $256 \times 128$.
  • ...and 13 more figures

Theorems & Definitions (2)

  • Definition 1: Recursive representation update
  • Definition 2: Koopman discrete operator $\mathcal{K}$