Improving Pseudo-Time Stepping Convergence for CFD Simulations With Neural Networks

• URL: http://arxiv.org/abs/2310.06717v1
• Date: Tue, 10 Oct 2023 15:45:19 GMT
• Title: Improving Pseudo-Time Stepping Convergence for CFD Simulations With Neural Networks
• Authors: Anouk Zandbergen, Tycho van Noorden, Alexander Heinlein
• Abstract summary: Navier-Stokes equations may exhibit a highly nonlinear behavior. The system of nonlinear equations resulting from the discretization of the Navier-Stokes equations can be solved using nonlinear iteration methods, such as Newton's method. In this paper, pseudo-transient continuation is employed in order to improve nonlinear convergence.
• Score: 44.99833362998488
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Computational fluid dynamics (CFD) simulations of viscous fluids described by the Navier-Stokes equations are considered. Depending on the Reynolds number of the flow, the Navier-Stokes equations may exhibit a highly nonlinear behavior. The system of nonlinear equations resulting from the discretization of the Navier-Stokes equations can be solved using nonlinear iteration methods, such as Newton's method. However, fast quadratic convergence is typically only obtained in a local neighborhood of the solution, and for many configurations, the classical Newton iteration does not converge at all. In such cases, so-called globalization techniques may help to improve convergence. In this paper, pseudo-transient continuation is employed in order to improve nonlinear convergence. The classical algorithm is enhanced by a neural network model that is trained to predict a local pseudo-time step. Generalization of the novel approach is facilitated by predicting the local pseudo-time step separately on each element using only local information on a patch of adjacent elements as input. Numerical results for standard benchmark problems, including flow through a backward facing step geometry and Couette flow, show the performance of the machine learning-enhanced globalization approach; as the software for the simulations, the CFD module of COMSOL Multiphysics is employed.

Related papers

• Quantum algorithm for the advection-diffusion equation and the Koopman-von Neumann approach to nonlinear dynamical systems [0.0]
  We propose an explicit algorithm to simulate both the advection-diffusion equation and a nonunitary discretized version of the Koopman-von Neumann formulation of nonlinear dynamics. The proposed algorithm is universal and can be used for modeling a broad class of linear and nonlinear differential equations.
  arXiv  Detail & Related papers  (2024-10-04T23:58:12Z)
• FEM-based Neural Networks for Solving Incompressible Fluid Flows and Related Inverse Problems [41.94295877935867]
  numerical simulation and optimization of technical systems described by partial differential equations is expensive. A comparatively new approach in this context is to combine the good approximation properties of neural networks with the classical finite element method. In this paper, we extend this approach to saddle-point and non-linear fluid dynamics problems, respectively.
  arXiv  Detail & Related papers  (2024-09-06T07:17:01Z)
• Enhancing Data-Assimilation in CFD using Graph Neural Networks [0.0]
  We present a novel machine learning approach for data assimilation applied in fluid mechanics, based on adjoint-optimization augmented by Graph Neural Networks (GNNs) models. We obtain our results using direct numerical simulations based on a Finite Element Method (FEM) solver; a two-fold interface between the GNN model and the solver allows the GNN's predictions to be incorporated into post-processing steps of the FEM analysis.
  arXiv  Detail & Related papers  (2023-11-29T19:11:40Z)
• Tensor network reduced order models for wall-bounded flows [0.0]
  We introduce a widely applicable tensor network-based framework for developing reduced order models. We consider the incompressible Navier-Stokes equations and the lid-driven cavity in two spatial dimensions.
  arXiv  Detail & Related papers  (2023-03-06T10:33:00Z)
• Neural Basis Functions for Accelerating Solutions to High Mach Euler Equations [63.8376359764052]
  We propose an approach to solving partial differential equations (PDEs) using a set of neural networks. We regress a set of neural networks onto a reduced order Proper Orthogonal Decomposition (POD) basis. These networks are then used in combination with a branch network that ingests the parameters of the prescribed PDE to compute a reduced order approximation to the PDE.
  arXiv  Detail & Related papers  (2022-08-02T18:27:13Z)
• Deep Random Vortex Method for Simulation and Inference of Navier-Stokes Equations [69.5454078868963]
  Navier-Stokes equations are significant partial differential equations that describe the motion of fluids such as liquids and air. With the development of AI techniques, several approaches have been designed to integrate deep neural networks in simulating and inferring the fluid dynamics governed by incompressible Navier-Stokes equations. We propose the emphDeep Random Vortex Method (DRVM), which combines the neural network with a random vortex dynamics system equivalent to the Navier-Stokes equation.
  arXiv  Detail & Related papers  (2022-06-20T04:58:09Z)
• Deep Equilibrium Optical Flow Estimation [80.80992684796566]
  Recent state-of-the-art (SOTA) optical flow models use finite-step recurrent update operations to emulate traditional algorithms. These RNNs impose large computation and memory overheads, and are not directly trained to model such stable estimation. We propose deep equilibrium (DEQ) flow estimators, an approach that directly solves for the flow as the infinite-level fixed point of an implicit layer.
  arXiv  Detail & Related papers  (2022-04-18T17:53:44Z)
• Provably Efficient Neural Estimation of Structural Equation Model: An Adversarial Approach [144.21892195917758]
  We study estimation in a class of generalized Structural equation models (SEMs) We formulate the linear operator equation as a min-max game, where both players are parameterized by neural networks (NNs), and learn the parameters of these neural networks using a gradient descent. For the first time we provide a tractable estimation procedure for SEMs based on NNs with provable convergence and without the need for sample splitting.
  arXiv  Detail & Related papers  (2020-07-02T17:55:47Z)
• Multipole Graph Neural Operator for Parametric Partial Differential Equations [57.90284928158383]
  One of the main challenges in using deep learning-based methods for simulating physical systems is formulating physics-based data. We propose a novel multi-level graph neural network framework that captures interaction at all ranges with only linear complexity. Experiments confirm our multi-graph network learns discretization-invariant solution operators to PDEs and can be evaluated in linear time.
  arXiv  Detail & Related papers  (2020-06-16T21:56:22Z)
• Convergence and sample complexity of gradient methods for the model-free linear quadratic regulator problem [27.09339991866556]
  We show that ODE searches for optimal control for an unknown computation system by directly searching over the corresponding space of controllers. We take a step towards demystifying the performance and efficiency of such methods by focusing on the gradient-flow dynamics set of stabilizing feedback gains and a similar result holds for the forward disctization of the ODE.
  arXiv  Detail & Related papers  (2019-12-26T16:56:59Z)

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.