A Neural Operator based on Dynamic Mode Decomposition

• URL: http://arxiv.org/abs/2507.01117v1
• Date: Tue, 01 Jul 2025 18:23:28 GMT
• Title: A Neural Operator based on Dynamic Mode Decomposition
• Authors: Nikita Sakovich, Dmitry Aksenov, Ekaterina Pleshakova, Sergey Gataullin,
• Abstract summary: The study presents a neural based operator on the dynamic mode decomposition algorithm (DMD), mapping functional spaces.<n>The method suggested automatically extracts key modes and system dynamics using them to construct predictions, reducing computational costs compared to traditional numerical methods.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The scientific computation methods development in conjunction with artificial intelligence technologies remains a hot research topic. Finding a balance between lightweight and accurate computations is a solid foundation for this direction. The study presents a neural operator based on the dynamic mode decomposition algorithm (DMD), mapping functional spaces, which combines DMD and deep learning (DL) for spatiotemporal processes efficient modeling. Solving PDEs for various initial and boundary conditions requires significant computational resources. The method suggested automatically extracts key modes and system dynamics using them to construct predictions, reducing computational costs compared to traditional numerical methods. The approach has demonstrated its efficiency through comparative analysis of performance with closest analogues DeepONet and FNO in the heat equation, Laplaces equation, and Burgers equation solutions approximation, where it achieves high reconstruction accuracy.

Related papers

• Self-Supervised Coarsening of Unstructured Grid with Automatic Differentiation [55.88862563823878]
  In this work, we present an original algorithm to coarsen an unstructured grid based on the concepts of differentiable physics.<n>We demonstrate performance of the algorithm on two PDEs: a linear equation which governs slightly compressible fluid flow in porous media and the wave equation.<n>Our results show that in the considered scenarios, we reduced the number of grid points up to 10 times while preserving the modeled variable dynamics in the points of interest.
  arXiv  Detail & Related papers  (2025-07-24T11:02:13Z)
• Equation discovery framework EPDE: Towards a better equation discovery [50.79602839359522]
  We enhance the EPDE algorithm -- an evolutionary optimization-based discovery framework.<n>Our approach generates terms using fundamental building blocks such as elementary functions and individual differentials.<n>We validate our algorithm's noise resilience and overall performance by comparing its results with those from the state-of-the-art equation discovery framework SINDy.
  arXiv  Detail & Related papers  (2024-12-28T15:58:44Z)
• LE-PDE++: Mamba for accelerating PDEs Simulations [4.7505178698234625]
  The Latent Evolution of PDEs method is designed to address the computational intensity of classical and deep learning-based PDE solvers. Our method doubles the inference speed compared to the LE-PDE while retaining the same level of parameter efficiency.
  arXiv  Detail & Related papers  (2024-11-04T09:04:11Z)
• Scaling up Probabilistic PDE Simulators with Structured Volumetric Information [23.654711580674885]
  We propose a framework combining a discretization scheme based on the popular Finite Volume Method with complementary numerical linear algebra techniques. Experiments, including atemporal tsunami simulation, demonstrate substantially improved scaling behavior of this approach over previous collocation-based techniques.
  arXiv  Detail & Related papers  (2024-06-07T15:38:27Z)
• Monte Carlo Neural PDE Solver for Learning PDEs via Probabilistic Representation [59.45669299295436]
  We propose a Monte Carlo PDE solver for training unsupervised neural solvers.<n>We use the PDEs' probabilistic representation, which regards macroscopic phenomena as ensembles of random particles.<n>Our experiments on convection-diffusion, Allen-Cahn, and Navier-Stokes equations demonstrate significant improvements in accuracy and efficiency.
  arXiv  Detail & Related papers  (2023-02-10T08:05:19Z)
• Solving High-Dimensional PDEs with Latent Spectral Models [74.1011309005488]
  We present Latent Spectral Models (LSM) toward an efficient and precise solver for high-dimensional PDEs. Inspired by classical spectral methods in numerical analysis, we design a neural spectral block to solve PDEs in the latent space. LSM achieves consistent state-of-the-art and yields a relative gain of 11.5% averaged on seven benchmarks.
  arXiv  Detail & Related papers  (2023-01-30T04:58:40Z)
• Long-time integration of parametric evolution equations with physics-informed DeepONets [0.0]
  We introduce an effective framework for learning infinite-dimensional operators that map random initial conditions to associated PDE solutions within a short time interval. Global long-time predictions across a range of initial conditions can be then obtained by iteratively evaluating the trained model. This introduces a new approach to temporal domain decomposition that is shown to be effective in performing accurate long-time simulations.
  arXiv  Detail & Related papers  (2021-06-09T20:46:17Z)
• Stochastic analysis of heterogeneous porous material with modified neural architecture search (NAS) based physics-informed neural networks using transfer learning [0.0]
  modified neural architecture search method (NAS) based physics-informed deep learning model is presented. A three dimensional flow model is built to provide a benchmark to the simulation of groundwater flow in highly heterogeneous aquifers.
  arXiv  Detail & Related papers  (2020-10-03T19:57:54Z)
• Large-scale Neural Solvers for Partial Differential Equations [48.7576911714538]
  Solving partial differential equations (PDE) is an indispensable part of many branches of science as many processes can be modelled in terms of PDEs. Recent numerical solvers require manual discretization of the underlying equation as well as sophisticated, tailored code for distributed computing. We examine the applicability of continuous, mesh-free neural solvers for partial differential equations, physics-informed neural networks (PINNs) We discuss the accuracy of GatedPINN with respect to analytical solutions -- as well as state-of-the-art numerical solvers, such as spectral solvers.
  arXiv  Detail & Related papers  (2020-09-08T13:26:51Z)
• Interpolation Technique to Speed Up Gradients Propagation in Neural ODEs [71.26657499537366]
  We propose a simple literature-based method for the efficient approximation of gradients in neural ODE models. We compare it with the reverse dynamic method to train neural ODEs on classification, density estimation, and inference approximation tasks.
  arXiv  Detail & Related papers  (2020-03-11T13:15:57Z)
• Non-linear reduced modeling of dynamical systems using kernel methods and low-rank approximation [5.935306543481018]
  We propose a new efficient algorithm for data-driven reduced modeling of non-linear dynamics based on linear approximations in a kernel Hilbert space.<n>This algorithm takes advantage of the closed-form solution of a low-rank constraint optimization problem while exploiting advantageously kernel-based computations.
  arXiv  Detail & Related papers  (2017-10-30T13:06:19Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.