ESR-PINNs:Physics-informed neural networks with expansion-shrinkage resampling selection strategies

Neural network methods have been widely used in many fields of scientific research with the rapid increase of computing power.The physics-informed neural networks(PINNs)have received much attention as a major breakthrough in solving partial differential equations using neural networks.In this paper,a resampling technique based on the expansion-shrinkage point(ESP)selection strategy is developed to dynamically modify the distribution of training points in accordance with the performance of the neural networks.In this new approach both training sites with slight changes in residual values and training points with large residuals are taken into account.In order to make the distribution of training points more uniform,the concept of continuity is further introduced and incorporated.This method successfully addresses the issue that the neural network becomes ill or even crashes due to the extensive alteration of training point distribution.The effectiveness of the improved physics-informed neural networks with expansion-shrinkage resampling is demonstrated through a series of numerical experiments.

Discovering Phase Field Models from Image Data with the Pseudo-Spectral Physics Informed Neural Networks

In this paper,we introduce a new deep learning framework for discovering the phase-field models from existing image data.The new framework embraces the approximation power of physics informed neural networks(PINNs)and the computational efficiency of the pseudo-spectral methods,which we named pseudo-spectral PINN or SPINN.Unlike the baseline PINN,the pseudo-spectral PINN has several advantages.First of all,it requires less training data.A minimum of two temporal snapshots with uniform spatial resolution would be adequate.Secondly,it is computationally efficient,as the pseudo-spectral method is used for spatial discretization.Thirdly,it requires less trainable parameters compared with the baseline PINN,which significantly simplifies the training process and potentially assures fewer local minima or saddle points.We illustrate the effectiveness of pseudo-spectral PINN through several numerical examples.The newly proposed pseudo-spectral PINN is rather general,and it can be readily applied to discover other FDE-based models from image data.

A new method to solve the Reynolds equation including mass-conserving cavitation by physics informed neural networks(PINNs)with both soft and hard constraints

In this work,a new method to solve the Reynolds equation including mass-conserving cavitation by using the physics informed neural networks(PINNs)is proposed.The complementarity relationship between the pressure and the void fraction is used.There are several difficulties in problem solving,and the solutions are provided.Firstly,the difficulty for considering the pressure inequality constraint by PINNs is solved by transferring it into one equality constraint without introducing error.While the void fraction inequality constraint is considered by using the hard constraint with the max-min function.Secondly,to avoid the fluctuation of the boundary value problems,the hard constraint method is also utilized to apply the boundary pressure values and the corresponding functions are provided.Lastly,for avoiding the trivial solution the limitation for the mean value of the void fraction is applied.The results are validated against existing data,and both the incompressible and compressible lubricant are considered.Good agreement can be found for both the domain and domain boundaries.

Physical informed memory networks for solving PDEs:implementation and applications

With the advent of physics informed neural networks(PINNs),deep learning has gained interest for solving nonlinear partial differential equations(PDEs)in recent years.In this paper,physics informed memory networks(PIMNs)are proposed as a new approach to solving PDEs by using physical laws and dynamic behavior of PDEs.Unlike the fully connected structure of the PINNs,the PIMNs construct the long-term dependence of the dynamics behavior with the help of the long short-term memory network.Meanwhile,the PDEs residuals are approximated using difference schemes in the form of convolution filter,which avoids information loss at the neighborhood of the sampling points.Finally,the performance of the PIMNs is assessed by solving the Kd V equation and the nonlinear Schr?dinger equation,and the effects of difference schemes,boundary conditions,network structure and mesh size on the solutions are discussed.Experiments show that the PIMNs are insensitive to boundary conditions and have excellent solution accuracy even with only the initial conditions.

Transient Thermal Distribution in a Wavy Fin Using Finite Difference Approximation Based Physics Informed Neural Network

Heat transport has been significantly enhanced by the widespread usage of extended surfaces in various engi-neering domains.Gas turbine blade cooling,refrigeration,and electronic equipment cooling are a few prevalent applications.Thus,the thermal analysis of extended surfaces has been the subject of a significant assessment by researchers.Motivated by this,the present study describes the unsteady thermal dispersal phenomena in a wavy fin with the presence of convection heat transmission.This analysis also emphasizes a novel mathematical model in accordance with transient thermal change in a wavy profiled fin resulting from convection using the finite difference method(FDM)and physics informed neural network(PINN).The time and space-dependent governing partial differential equation(PDE)for the suggested heat problem has been translated into a dimensionless form using the relevant dimensionless terms.The graph depicts the effect of thermal parameters on the fin’s thermal profile.The temperature dispersion in the fin decreases as the dimensionless convection-conduction variable rises.The heat dispersion in the fin is decreased by increasing the aspect ratio,whereas the reverse behavior is seen with the time change.Furthermore,FDM-PINN results are validated against the outcomes of the FDM.

An Adaptive Physics-Informed Neural Network with Two-Stage Learning Strategy to Solve Partial Differential Equations

Physics-Informed Neural Network(PINN)represents a new approach to solve Partial Differential Equations(PDEs).PINNs aim to solve PDEs by integrating governing equations and the initial/boundary conditions(I/BCs)into a loss function.However,the imbalance of the loss function caused by parameter settings usually makes it difficult for PINNs to converge,e.g.because they fall into local optima.In other words,the presence of balanced PDE loss,initial loss and boundary loss may be critical for the convergence.In addition,existing PINNs are not able to reveal the hidden errors caused by non-convergent boundaries and conduction errors caused by the PDE near the boundaries.Overall,these problems have made PINN-based methods of limited use on practical situations.In this paper,we propose a novel physics-informed neural network,i.e.an adaptive physics-informed neural network with a two-stage training process.Our algorithm adds spatio-temporal coefficient and PDE balance parameter to the loss function,and solve PDEs using a two-stage training process:pre-training and formal training.The pre-training step ensures the convergence of boundary loss,whereas the formal training process completes the solution of PDE by balancing various loss functions.In order to verify the performance of our method,we consider the imbalanced heat conduction and Helmholtz equations often appearing in practical situations.The Klein-Gordon equation,which is widely used to compare performance,reveals that our method is able to reduce the hidden errors.Experimental results confirm that our algorithm can effectively and accurately solve models with unbalanced loss function,hidden errors and conduction errors.The codes developed in this manuscript are publicy available at https://github.com/callmedrcom/ATPINN.

PHYSICS INFORMED NEURAL NETWORKS (PINNs) FOR APPROXIMATING NONLINEAR DISPERSIVE PDEs

We propose a novel algorithm,based on physics-informed neural networks(PINNs)to efficiently approximate solutions of nonlinear dispersive PDEs such as the KdV-Kawahara,Camassa-Holm and Benjamin-Ono equations.The stability of solutions of these dispersive PDEs is leveraged to prove rigorous bounds on the resulting error.We present several numerical experiments to demonstrate that PINNs can approximate solutions of these dispersive PDEs very accurately.

Applying physics informed neural network for flow data assimilation

Data assimilation(DA)refers to methodologies which combine data and underlying governing equations to provide an estimation of a complex system.Physics informed neural network(PINN)provides an innovative machine learning technique for solving and discovering the physics in nature.By encoding general nonlinear partial differential equations,which govern different physical systems such as fluid flows,to the deep neural network,PINN can be used as a tool for DA.Due to its nature that neither numerical differential operation nor temporal and spatial discretization is needed,PINN is straightforward for implementation and getting more and more attention in the academia.In this paper,we apply the PINN to several flow problems and explore its potential in fluid physics.Both the mesoscopic Boltzmann equation and the macroscopic Navier-Stokes are considered as physics constraints.We first introduce a discrete Boltzmann equation informed neural network and evaluate it with a one-dimensional propagating wave and two-dimensional lid-driven cavity flow.Such laminar cavity flow is also considered as an example in an incompressible Navier-Stokes equation informed neural network.With parameterized Navier-Stokes,two turbulent flows,one within a C-shape duct and one passing a bump,are studied and accompanying pressure field is obtained.Those examples end with a flow passing through a porous media.Applications in this paper show that PINN provides a new way for intelligent flow inference and identification,ranging from mesoscopic scale to macroscopic scale,and from laminar flow to turbulent flow.

Physics Informed Neural Network-based High-frequency Modeling of Induction Motors

The high-frequency(HF)modeling of induction motors plays a key role in predicting the motor terminal overvoltage and conducted emissions in a motor drive system.In this study,a physics informed neural network-based HF modeling method,which has the merits of high accuracy,good versatility,and simple parameterization,is proposed.The proposed model of the induction motor consists of a three-phase equivalent circuit with eighteen circuit elements per phase to ensure model accuracy.The per phase circuit structure is symmetric concerning its phase-start and phase-end points.This symmetry enables the proposed model to be applicable for both star-and delta-connected induction motors without having to recalculate the circuit element values when changing the motor connection from star to delta and vice versa.Motor physics knowledge,namely per-phase impedances,are used in the artificial neural network to obtain the values of the circuit elements.The parameterization can be easily implemented within a few minutes using a common personal computer(PC).Case studies verify the effectiveness of the proposed HF modeling method.

The prediction of external flow field and hydrodynamic force with limited data using deep neural network

Predicting the external flow field with limited data or limited measurements has attracted long-time interests of researchers in many industrial applications.Physics informed neural network(PINN)provides a seamless framework for combining the measured data with the deep neural network,making the neural network capable of executing certain physical constraints.Unlike the data-driven model to learn the end-to-end mapping between the sensor data and high-dimensional flow field,PINN need no prior high-dimensional field as the training dataset and can construct the mapping from sensor data to high dimensional flow field directly.However,the extrapolation of the flow field in the temporal direction is limited due to the lack of training data.Therefore,we apply the long short-term memory(LSTM)network and physics-informed neural network(PINN)to predict the flow field and hydrodynamic force in the future temporal domain with limited data measured in the spatial domain.The physical constraints(conservation laws of fluid flow,e.g.,Navier-Stokes equations)are embedded into the loss function to enforce the trained neural network to capture some latent physical relation between the output fluid parameters and input tempo-spatial parameters.The sparsely measured points in this work are obtained from computational fluid dynamics(CFD)solver based on the local radial basis function(RBF)method.Different numbers of spatial measured points(4–35)downstream the cylinder are trained with/without the prior knowledge of Reynolds number to validate the availability and accuracy of the proposed approach.More practical applications of flow field prediction can compute the drag and lift force along with the cylinder,while different geometry shapes are taken into account.By comparing the flow field reconstruction and force prediction with CFD results,the proposed approach produces a comparable level of accuracy while significantly fewer data in the spatial domain is needed.The numerical results demonstrate that the proposed approach with a specific deep neural network configuration is of great potential for emerging cases where the measured data are often limited.

通过融合物理神经网络重构稀疏或不完整数据流场的实用方法

高分辨率流场重构被普遍认为是实验流体力学领域的一项艰巨任务,因为测量数据在时间和空间上通常是稀疏或不完整的.具体而言,由于实验设备或测量技术的限制,某些关键区域的数据无法测量.本文提出了一种基于融合物理神经网络(PINN)的不完美数据重建流场的实用方法,该网络将已知数据与物理原理相结合.通过圆柱体的尾流作为测试算例.研究了两种不完美数据训练集,一种是不同稀疏度的速度数据,另一种是不同区域缺失的速度数据.为了加速训练收敛,本文采用了余弦退火算法以提高PINN的计算效率.计算结果表明,该方法不仅可以高精度地重建真实的速度场,而且即使在数据稀疏度达到1%或核心流动区域数据被截断的情况下,也可以精确地预测压力场.这项研究提供了令人鼓舞的结论,即PINN可以作为实验流体力学的有潜力的数据同化方法.

kεNet湍流模型研究及其在低雷诺数槽道流中的应用

我们提出了一种基于物理信息的深度学习网络(kεNet),可用于RANS方程中发现封闭的湍流模型.kεNet由一个传统的典型神经网络结构和若干个基于物理信息的方程组成,如雷诺应力方程、k方程和ε方程.以低雷诺数下的槽道流动的湍流模型的修正为例,通过训练基于物理信息的神经网络,模型参数得到了修正.修正后的湍流模型参数应用于OpenFOAM软件进行计算,能够非常好地预测Re_(τ)=5200和2000下的槽道流动.