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Machine Learning and Modern Numerical Methods in Partial Differential Equations

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A special issue of Entropy (ISSN 1099-4300). This special issue belongs to the section "Complexity".

Deadline for manuscript submissions: closed (31 March 2023) | Viewed by 25126

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Special Issue Editors

Dr. Xinlong Feng

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Guest Editor

College of Mathematics and Systems Science, Xinjiang University, Urumqi 830049, China
Interests: numerical methods for PDEs; computational fluid dynamics; computational mathematics; uncertainty quantification; machine learning
Special Issues, Collections and Topics in MDPI journals

Dr. Hui Xu

E-Mail Website
Guest Editor

School of Aeronautics & Astronautics, Shanghai Jiao Tong University, Shanghai 200240, China
Interests: complex flows and applied mathematics; high-order numerical methods; laminar-turbulent transition; machine learning

Special Issue Information

Dear Colleagues,

Machine learning and its related numerical modelling are emerging tools in solving partial differential equations which underpin science and engineering. It needs to mention that in practical applications, machine learning has recently attracted great attentions in modelling and solving complex systems. This special issue will bring together experts in applied mathematics and computational mathematics to discuss fundamental problems and practical applications of machine learning and its related methodologies in solving partial differential equations and modelling complex systems. This special issue also includes papers around modern numerical techniques based on machine learning in interdisciplinary fields.

Topics of interest include, but not limited to:

Machine learning-based numerical algorithm for solving high dimensional PDEs.
Deep reinforcement learning in control problems
Machine learning approach in uncertainty quantification
Reduced order modelling in complex systems
Mathematical methods in kernel learning

Machine learning and artificial intelligence in fluid dynamics, which includes recent significant developments on modelling and computations.

Dr. Xinlong Feng
Dr. Hui Xu
Guest Editors

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Keywords

partial differential equations
computational fluid dynamics
modern numerical methods
machine learning
model reduction

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Published Papers (11 papers)

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Research

14 pages, 1381 KiB

Open AccessArticle

An Optimized Schwarz Method for the Optical Response Model Discretized by HDG Method

by Jia-Fen Chen, Xian-Ming Gu, Liang Li and Ping Zhou

Entropy 2023, 25(4), 693; https://doi.org/10.3390/e25040693 - 19 Apr 2023

Viewed by 1364

Abstract

An optimized Schwarz domain decomposition method (DDM) for solving the local optical response model (LORM) is proposed in this paper. We introduce a hybridizable discontinuous Galerkin (HDG) scheme for the discretization of such a model problem based on a triangular mesh of the computational domain. The discretized linear system of the HDG method on each subdomain is solved by a sparse direct solver. The solution of the interface linear system in the domain decomposition framework is accelerated by a Krylov subspace method. We study the spectral radius of the iteration matrix of the Schwarz method for the LORM problems, and thus propose an optimized parameter for the transmission condition, which is different from that for the classical electromagnetic problems. The numerical results show that the proposed method is effective. Full article

(This article belongs to the Special Issue Machine Learning and Modern Numerical Methods in Partial Differential Equations)

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21 pages, 3482 KiB

Open AccessArticle

Improved Physics-Informed Neural Networks Combined with Small Sample Learning to Solve Two-Dimensional Stefan Problem

by Jiawei Li, Wei Wu and Xinlong Feng

Entropy 2023, 25(4), 675; https://doi.org/10.3390/e25040675 - 18 Apr 2023

Cited by 1 | Viewed by 2386

Abstract

With the remarkable development of deep learning in the field of science, deep neural networks provide a new way to solve the Stefan problem. In this paper, deep neural networks combined with small sample learning and a general deep learning framework are proposed to solve the two-dimensional Stefan problem. In the case of adding less sample data, the model can be modified and the prediction accuracy can be improved. In addition, by solving the forward and inverse problems of the two-dimensional single-phase Stefan problem, it is verified that the improved method can accurately predict the solutions of the partial differential equations of the moving boundary and the dynamic interface. Full article

(This article belongs to the Special Issue Machine Learning and Modern Numerical Methods in Partial Differential Equations)

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22 pages, 2700 KiB

Open AccessArticle

A Second-Order Network Structure Based on Gradient-Enhanced Physics-Informed Neural Networks for Solving Parabolic Partial Differential Equations

by Kuo Sun and Xinlong Feng

Entropy 2023, 25(4), 674; https://doi.org/10.3390/e25040674 - 18 Apr 2023

Cited by 5 | Viewed by 2970

Abstract

Physics-informed neural networks (PINNs) are effective for solving partial differential equations (PDEs). This method of embedding partial differential equations and their initial boundary conditions into the loss functions of neural networks has successfully solved forward and inverse PDE problems. In this study, we considered a parametric light wave equation, discretized it using the central difference, and, through this difference scheme, constructed a new neural network structure named the second-order neural network structure. Additionally, we used the adaptive activation function strategy and gradient-enhanced strategy to improve the performance of the neural network and used the deep mixed residual method (MIM) to reduce the high computational cost caused by the enhanced gradient. At the end of this paper, we give some numerical examples of nonlinear parabolic partial differential equations to verify the effectiveness of the method. Full article

(This article belongs to the Special Issue Machine Learning and Modern Numerical Methods in Partial Differential Equations)

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18 pages, 4416 KiB

Open AccessArticle

Large-Scale Simulation of Full Three-Dimensional Flow and Combustion of an Aero-Turbofan Engine on Sunway TaihuLight Supercomputer

by Quanyong Xu, Hu Ren, Hanfeng Gu, Jie Wu, Jingyuan Wang, Zhifeng Xie and Guangwen Yang

Entropy 2023, 25(3), 436; https://doi.org/10.3390/e25030436 - 1 Mar 2023

Cited by 3 | Viewed by 2236

Abstract

Computational fluid dynamics- (CFD-) based component-level numerical simulation technology has been widely used in the design of aeroengines. However, due to the strong coupling effects between components, the numerical simulation of the whole engine considering the full three-dimensional flow and multi-component chemical reaction is still very difficult at present. Aimed at this problem, an efficient implicit solver, ‘sprayDyMFoam’ for an unstructured mesh, is developed in this paper based on the Sunway TaihuLight supercomputer. This sprayDyMFoam solver improves the PIMPLE algorithm in the solution of aerodynamic force and adjusts the existing droplet atomization model in the solution of the combustion process so as to meet the matching situation between components and the combustion chamber in the solution process. Meanwhile, the parallel communication mechanism of AMI boundary processing is optimized based on the hardware environment of the Sunway supercomputer. The sprayDyMFoam solver is used to simulate a typical double-rotor turbofan engine: the calculation capacity and efficiency meet the use requirements, and the obtained compressor performance can form a good match with the test. The research proposed in this paper has strong application value in high-confidence computing, complex phenomenon capturing, and time and cost reduction for aeroengine development. Full article

(This article belongs to the Special Issue Machine Learning and Modern Numerical Methods in Partial Differential Equations)

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18 pages, 3923 KiB

Open AccessArticle

Local Inverse Mapping Implicit Hole-Cutting Method for Structured Cartesian Overset Grid Assembly

by Jingyuan Wang, Feng Wu, Quanyong Xu and Lei Tan

Entropy 2023, 25(3), 432; https://doi.org/10.3390/e25030432 - 28 Feb 2023

Cited by 1 | Viewed by 1823

Abstract

An automatic hole-cutting method is proposed to search donor cells between a structured Cartesian mesh and an overlapping body-fitted mesh. The main flow is simulated on the structured Cartesian mesh and the viscous flow near the solid boundary is simulated on the body-fitted mesh. Through the spatial interpolation of flux, the surface boundary information on the body-fitted mesh is transferred to the Cartesian mesh nodes near the surface. Cartesian mesh box near a body-fitted mesh cell is selected as a local inverse map. The Cartesian nodes located inside the donor cells are marked by the relative coordinate transformation, so that all Cartesian nodes can be classified and the hole boundaries are implicitly cut. This hole-cutting process for overset grid assembly is called Local Inverse Mapping (LIM) method. In the LIM method, spatial interpolation of flux is carried out synchronously with the marking of Cartesian nodes. The LIM method is combined with the in-house finite-difference solver to simulate the unsteady flow field of moving bodies. The numerical results show that the LIM method can accurately mark the Cartesian hole boundary nodes, the search efficiency of donor cells is high, and the result of spatial interpolation is accurate. The calculation time of overset grid assembly (OGA) can be less than 3% of the total simulation time. Full article

(This article belongs to the Special Issue Machine Learning and Modern Numerical Methods in Partial Differential Equations)

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19 pages, 6025 KiB

Open AccessArticle

Sand Discharge Simulation and Flow Path Optimization of a Particle Separator

by Zhou Du, Yulin Ma, Quanyong Xu and Feng Wu

Entropy 2023, 25(1), 147; https://doi.org/10.3390/e25010147 - 11 Jan 2023

Cited by 2 | Viewed by 1655

Abstract

A numerical simulation method is used to optimize the removal of sand from a helicopter engine particle separator. First, the classic configuration of a particle separator based on the literature is simulated using two boundary conditions. The results show that the boundary conditions for the total pressure inlet and mass flow outlet are much more closely aligned with the experimental environment. By modifying the material at the front of the shroud, the separation efficiencies of coarse Arizona road dust (AC-Coarse) and MIL-E-5007C (C-Spec) can be improved to 93.3% and 97.6%, respectively. Configuration modifications of the particle separator with dual protection can increase the separation efficiencies of AC-Coarse and C-Spec to 91.7% and 97.7%. Full article

(This article belongs to the Special Issue Machine Learning and Modern Numerical Methods in Partial Differential Equations)

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18 pages, 4834 KiB

Open AccessArticle

A Finite Element Approximation for Nematic Liquid Crystal Flow with Stretching Effect Based on Nonincremental Pressure-Correction Method

by Zhaoxia Meng, Meng Liu and Hongen Jia

Entropy 2022, 24(12), 1844; https://doi.org/10.3390/e24121844 - 18 Dec 2022

Viewed by 1433

Abstract

In this paper, a new decoupling method is proposed to solve a nematic liquid crystal flow with stretching effect. In the finite element discrete framework, the director vector is calculated by introducing a new auxiliary variable w, and the velocity vector and scalar pressure are decoupled by a nonincremental pressure-correction projection method. Then, the energy dissipation law and unconditional energy stability of the resulting system are given. Finally, some numerical examples are given to verify the effects of various parameters on the singularity annihilation, stability and accuracy in space and time. Full article

(This article belongs to the Special Issue Machine Learning and Modern Numerical Methods in Partial Differential Equations)

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18 pages, 966 KiB

Open AccessArticle

Two-Level Finite Element Iterative Algorithm Based on Stabilized Method for the Stationary Incompressible Magnetohydrodynamics

by Qili Tang, Min Hou, Yajie Xiao and Lina Yin

Entropy 2022, 24(10), 1426; https://doi.org/10.3390/e24101426 - 7 Oct 2022

Cited by 1 | Viewed by 1427

Abstract

In this paper, based on the stabilization technique, the Oseen iterative method and the two-level finite element algorithm are combined to numerically solve the stationary incompressible magnetohydrodynamic (MHD) equations. For the low regularity of the magnetic field, when dealing with the magnetic field sub-problem, the Lagrange multiplier technique is used. The stabilized method is applied to approximate the flow field sub-problem to circumvent the inf-sup condition restrictions. One- and two-level stabilized finite element algorithms are presented, and their stability and convergence analysis is given. The two-level method uses the Oseen iteration to solve the nonlinear MHD equations on a coarse grid of size H, and then employs the linearized correction on a fine grid with grid size h. The error analysis shows that when the grid sizes satisfy

h = O (H^{2})

, the two-level stabilization method has the same convergence order as the one-level one. However, the former saves more computational cost than the latter one. Finally, through some numerical experiments, it has been verified that our proposed method is effective. The two-level stabilized method takes less than half the time of the one-level one when using the second class Nédélec element to approximate magnetic field, and even takes almost a third of the computing time of the one-level one when adopting the first class Nédélec element. Full article

(This article belongs to the Special Issue Machine Learning and Modern Numerical Methods in Partial Differential Equations)

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15 pages, 2662 KiB

Open AccessArticle

AP Shadow Net: A Remote Sensing Shadow Removal Network Based on Atmospheric Transport and Poisson’s Equation

by Fan Li, Zhiyi Wang and Guoliang He

Entropy 2022, 24(9), 1301; https://doi.org/10.3390/e24091301 - 14 Sep 2022

Cited by 1 | Viewed by 2154

Abstract

Shadow is one of the fundamental indicators of remote sensing image which could cause loss or interference of the target data. As a result, the detection and removal of shadow has already been the hotspot of current study because of the complicated background information. In the following passage, a model combining the Atmospheric Transport Model (hereinafter abbreviated as ATM) with the Poisson Equation, AP ShadowNet, is proposed for the shadow detection and removal of remote sensing images by unsupervised learning. This network based on a preprocessing network based on ATM, A Net, and a network based on the Poisson Equation, P Net. Firstly, corresponding mapping between shadow and unshaded area is generated by the ATM. The brightened image will then enter the Confrontation identification in the P Net. Lastly, the reconstructed image is optimized on color consistency and edge transition by Poisson Equation. At present, most shadow removal models based on neural networks are significantly data-driven. Fortunately, by the model in this passage, the unsupervised shadow detection and removal could be released from the data source restrictions from the remote sensing images themselves. By verifying the shadow removal on our model, the result shows a satisfying effect from a both qualitative and quantitative angle. From a qualitative point of view, our results have a prominent effect on tone consistency and removal of detailed shadows. From the quantitative point of view, we adopt the non-reference evaluation indicators: gradient structure similarity (NRSS) and Natural Image Quality Evaluator (NIQE). Combining various evaluation factors such as reasoning speed and memory occupation, it shows that it is outstanding among other current algorithms. Full article

(This article belongs to the Special Issue Machine Learning and Modern Numerical Methods in Partial Differential Equations)

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16 pages, 4613 KiB

Open AccessArticle

Physics-Informed Neural Networks for Solving Coupled Stokes–Darcy Equation

by Ruilong Pu and Xinlong Feng

Entropy 2022, 24(8), 1106; https://doi.org/10.3390/e24081106 - 11 Aug 2022

Cited by 5 | Viewed by 3913

Abstract

In this paper, a grid-free deep learning method based on a physics-informed neural network is proposed for solving coupled Stokes–Darcy equations with Bever–Joseph–Saffman interface conditions. This method has the advantage of avoiding grid generation and can greatly reduce the amount of computation when solving complex problems. Although original physical neural network algorithms have been used to solve many differential equations, we find that the direct use of physical neural networks to solve coupled Stokes–Darcy equations does not provide accurate solutions in some cases, such as rigid terms due to small parameters and interface discontinuity problems. In order to improve the approximation ability of a physics-informed neural network, we propose a loss-function-weighted function strategy, a parallel network structure strategy, and a local adaptive activation function strategy. In addition, the physical information neural network with an added strategy provides inspiration for solving other more complicated problems of multi-physical field coupling. Finally, the effectiveness of the proposed strategy is verified by numerical experiments. Full article

(This article belongs to the Special Issue Machine Learning and Modern Numerical Methods in Partial Differential Equations)

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16 pages, 779 KiB

Open AccessArticle

A Time Two-Mesh Compact Difference Method for the One-Dimensional Nonlinear Schrödinger Equation

by Siriguleng He, Yang Liu and Hong Li

Entropy 2022, 24(6), 806; https://doi.org/10.3390/e24060806 - 9 Jun 2022

Cited by 5 | Viewed by 1954

Abstract

The nonlinear Schrödinger equation is an important model equation in the study of quantum states of physical systems. To improve the computing efficiency, a fast algorithm based on the time two-mesh high-order compact difference scheme for solving the nonlinear Schrödinger equation is studied. The fourth-order compact difference scheme is used to approximate the spatial derivatives and the time two-mesh method is designed for efficiently solving the resulting nonlinear system. Comparing to the existing time two-mesh algorithm, the novelty of the new algorithm is that the fine mesh solution, which becomes available, is also used as the initial guess of the linear system, which can improve the calculation accuracy of fine mesh solutions. Compared to the two-grid finite element methods (or finite difference methods) for nonlinear Schrödinger equations, the numerical calculation of this method is relatively simple, and its two-mesh algorithm is implemented in the temporal direction. Taking advantage of the discrete energy, the result with

O (τ_{C}^{4} + τ_{F}^{2} + h^{4})

in the discrete

L_{2}

-norm is obtained. Here,

τ_{C}

and

τ_{F}

are the temporal parameters on the coarse and fine mesh, respectively, and h is the space step size. Finally, some numerical experiments are conducted to demonstrate its efficiency and accuracy. The numerical results show that the new algorithm gives highly accurate results and preserves conservation laws of charge and energy. Furthermore, by comparing with the standard nonlinear implicit compact difference scheme, it can reduce the CPU time without loss of accuracy. Full article

(This article belongs to the Special Issue Machine Learning and Modern Numerical Methods in Partial Differential Equations)

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