Neural operator architecture that combines Dynamic Mode Decomposition (DMD) with deep learning for solving partial differential equations (PDEs).
DMD-Neural-Operator is a novel neural operator architecture that synergistically combines Dynamic Mode Decomposition (DMD) with deep learning to efficiently solve partial differential equations (PDEs). By leveraging DMD for dimensionality reduction and feature extraction, the architecture identifies key modes and system dynamics within PDE solutions. These extracted features are then integrated with neural networks to facilitate operator learning, providing an efficient means of approximating PDE solutions in various parameterized settings. This hybrid approach significantly reduces computational costs compared to traditional methods like FEM, FDM, and FVM, while maintaining high solution reconstruction accuracy, as demonstrated on benchmark problems such as the heat equation, Laplace's equation, and Burgers' equation.
Sakovich, N., Aksenov, D., Pleshakova, E., & Gataullin, S. (2025). A Neural Operator based on Dynamic Mode Decomposition. arXiv preprint arXiv:2507.01117. https://doi.org/10.48550/arXiv.2507.01117
A neural operator using dynamic mode decomposition analysis to approximate the partial differential equations
Abstract
Solving partial differential equations (PDEs) for various initial and boundary conditions requires significant computational resources. We propose a neural operator $G_\theta: \mathcal{A} \to \mathcal{U}$, mapping functional spaces, which combines dynamic mode decomposition (DMD) and deep learning for efficient modeling of spatiotemporal processes. The method automatically extracts key modes and system dynamics and uses them to construct predictions, reducing computational costs compared to traditional methods (FEM, FDM, FVM). The approach is demonstrated on the heat equation, Laplace's equation, and Burgers' equation, where it achieves high solution reconstruction accuracy.
Uses DMD for dimensionality reduction and feature extraction from PDE solutions
Combines DMD modes and dynamics with neural networks for operator learning
Provides an efficient way to approximate PDE solutions in parameterized settings
Technology Stack
Core: Python 3.8+
Deep Learning: PyTorch 2.6+
DMD: PyDMD 2025.4+
Numerical Computing: NumPy, SciPy
Visualization: Matplotlib
Development: tqdm, torchviz
Features
Dimensionality reduction using DMD analysis
Neural operator architecture for function space mapping
Efficient processing of spatiotemporal data
Customizable network architecture with multiple branches
Algorithm
Article
@article{sakovich2025neural,
title={A Neural Operator based on Dynamic Mode Decomposition},
author={Sakovich, Nikita and Aksenov, Dmitry and Pleshakova, Ekaterina and Gataullin, Sergey},
journal={arXiv preprint arXiv:2507.01117},
year={2025}
}
@article{sakovich2025neural,
title={A neural operator using dynamic mode decomposition analysis to approximate partial differential equations},
author={Sakovich, Nikita and Aksenov, Dmitry and Pleshakova, Ekaterina and Gataullin, Sergey},
journal={AIMS Mathematics},
volume={10},
number={9},
pages={22432--22444},
year={2025}
}
