Time-dependent Nonlinear Partial Differential Equations Research Articles

TCAS-PINN: Physics-informed neural networks with a novel temporal causality-based adaptive sampling method

Physics-informed neural networks (PINNs) have become an attractive machine learning framework for obtaining solutions to partial differential equations (PDEs). PINNs embed initial, boundary, and PDE constraints into the loss function. The performance of PINNs is generally affected by both training and sampling. Specifically, training methods focus on how to overcome the training difficulties caused by the special PDE residual loss of PINNs, and sampling methods are concerned with the location and distribution of the sampling points upon which evaluations of PDE residual loss are accomplished. However, a common problem among these original PINNs is that they omit special temporal information utilization during the training or sampling stages when dealing with an important PDE category, namely, time-dependent PDEs, where temporal information plays a key role in the algorithms used. There is one method, called Causal PINN, that considers temporal causality at the training level but not special temporal utilization at the sampling level. Incorporating temporal knowledge into sampling remains to be studied. To fill this gap, we propose a novel temporal causality-based adaptive sampling method that dynamically determines the sampling ratio according to both PDE residual and temporal causality. By designing a sampling ratio determined by both residual loss and temporal causality to control the number and location of sampled points in each temporal sub-domain, we provide a practical solution by incorporating temporal information into sampling. Numerical experiments of several nonlinear time-dependent PDEs, including the Cahn–Hilliard, Korteweg–de Vries, Allen–Cahn and wave equations, show that our proposed sampling method can improve the performance. We demonstrate that using such a relatively simple sampling method can improve prediction performance by up to two orders of magnitude compared with the results from other methods, especially when points are limited.

Magneto-thermal convection and entropy production of hybrid nanofluid in an inclined chamber having a solid block

PurposeThis paper aims to examine the thermal transmission and entropy generation of hybrid nanofluid filled containers with solid body inside. The solid body is seen as being both isothermal and capable of producing heat. A time-dependent non-linear partial differential equation is used to represent the transfer of heat through a solid body. The current study’s objective is to investigate the key properties of nanoparticles, external forces and particular attention paid to the impact of hybrid nanoparticles on entropy formation. This investigation is useful for researchers studying in the area of cavity flows to know features of the flow structures and nature of hybrid nanofluid characteristics. In addition, a detailed entropy generation analysis has been performed to highlight possible regimes with minimal entropy generation rates. Hybrid nanofluid has been proven to have useful qualities, making it an attractive coolant for an electrical device. The findings would help scientists and engineers better understand how to analyse convective heat transmission and how to forecast better heat transfer rates in cutting-edge technological systems used in industries such as heat transportation, power generation, chemical production and passive cooling systems for electronic devices.Design/methodology/approachThermal transmission and entropy generation of hybrid nanofluid are analysed within the enclosure. The domain of interest is a square chamber of size L, including a square solid block. The solid body is considered to be isothermal and generating heat. The flow driven by temperature gradient in the cavity is two-dimensional. The governing equations, formulated in dimensionless primitive variables with corresponding initial and boundary conditions, are worked out by using the finite volume technique with the SIMPLE algorithm on a uniformly staggered mesh. QUICK and central difference schemes were used to handle convective and diffusive elements. In-house code is developed using FORTRAN programming to visualize the isotherms, streamlines, heatlines and entropy contours, which are handled by Tecplot software. The influence of nanoparticles volume fraction, heat generation factor, external magnetic forces and an irreversibility ratio on energy transport and flow patterns is examined.FindingsThe results show that the hybrid nanoparticles concentration augments the thermal transmission and the entropy production increases also while the augmentation of temperature difference results in a diminution of entropy production. Finally, magnetic force has the significant impact on heat transfer, isotherms, streamlines and entropy. It has been observed that the external magnetic force plays a good role in thermal regulations.Research limitations/implicationsHybrid nanofluid is a desirable coolant for an electrical device. Various nanoparticles and their combinations can be analysed. Ferro-copper hybrid nanofluid considered with the help of prevailing literature review. The research would benefit scientists and engineers by improving their comprehension of how to analyses convective heat transmission and forecast more accurate heat transfer rates in various fields.Practical implicationsDue to its helpful characteristics, ferrous-copper hybrid nanofluid is a desirable coolant for an electrical device. The research would benefit scientists and engineers by improving their comprehension of how to analyse convective heat transmission and forecast more accurate heat transfer rates in cutting-edge technological systems used in sectors like thermal transportation, cooling systems for electronic devices, etc.Social implicationsEntropy generation is used for an evaluation of the system’s performance, which is an indicator of optimal design. Hence, in recent times, it does a good engineering sense to draw attention to irreversibility under magnetic force, and it has an indispensable impact on investigation of electronic devices.Originality/valueAn efficient numerical technique has been developed to solve this problem. The originality of this work is to analyse convective energy transport and entropy generation in a chamber with internal block, which is capable of maintaining heat and producing heat. Effects of irreversibility ratio are scrutinized for the first time. Analysis of convective heat transfer and entropy production in an enclosure with internal isothermal/heat generating blocks gives the way to predict enhanced heat transfer rate and avoid the failure of advanced technical systems in industrial sectors.

Approximations of Time-Dependent Nonlinear Partial Differential Equations using Galerkin Optimal Auxiliary Function Method

The purpose of this research work is to employ the Optimal Auxiliary Function Method (OAFM) for obtaining numerical approximations of time-dependent nonlinear partial differential equations (PDEs) that arise in many disciplines of science and engineering. The initial and first approximations of parabolic nonlinear PDEs associated with initial conditions have been generated by utilizing this method. Then the Galerkin method is applied to estimate the coefficients that remain unknown. Finally, the values of the coefficients generated by the Galerkin method have been inserted into the first approximation. In each example, all numerical computations and corresponding absolute errors are provided in schematic representations. The rate of convergence attained by the proposed method is depicted in tabular form.
 GANIT J. Bangladesh Math. Soc. 43.1 (2023) 001- 016

Impact of magnetic dipole on unsteady ferromagnetic Casson nanofluid flow over a static/moving wedge in the presence of viscous dissipation

This article mainly focused on magnetic dipole effects on unsteady two-dimensional ferromagnetic Casson nanofluid Falkner–Skan flow along a static/moving wedge under the effects of viscous dissipation. A nanofluid features i.e. Brownian motion and thermophoresis are also observed for static (λ = 0) and moving wedge ( and ). A class of appropriate similarity functions is utilized to convert time-dependent nonlinear partial differential equations (PDEs) into a set of ordinary differential equations (ODEs) and then computed numerically by applying two distinct techniques Runge–Kutta–Fehlberg (RKF) in Maple as well as bvp4c in Matlab. For limited cases, we also provided a comparison with already published work and found in good agreement. Impact of various controlling parameters active in the velocity, temperature, and concentration are observed and presented extensively. In addition, for engineering aspects the numerical computations and graphical presentations for Nusselt and Sherwood numbers (the wall heat and mass fluxes) are also performed for various wedge movements. All the three profiles ( and ) vary for a static (λ = 0) and moving wedge ( and ). Velocity and thermal profiles decelerates for rising ferromagnetic interaction and unsteady parameter. Furthermore, it was noticed that fluid velocity gets smaller for a static wedge.

Finite difference schemes for MHD mixed convective Darcy–forchheimer flow of Non-Newtonian fluid over oscillatory sheet: A computational study

This contribution proposes two third-order numerical schemes for solving time-dependent linear and non-linear partial differential equations (PDEs). For spatial discretization, a compact fourth-order scheme is deliberated. The stability of the proposed scheme is set for scalar partial differential equation, whereas its convergence is specified for a system of parabolic equations. The scheme is applied to linear scalar partial differential equation and non-linear systems of time-dependent partial differential equations. The non-linear system comprises a set of governing equations for the heat and mass transfer of magnetohydrodynamics (MHD) mixed convective Casson nanofluid flow across the oscillatory sheet with the Darcy–Forchheimer model, joule heating, viscous dissipation, and chemical reaction. It is noted that the concentration profile is escalated by mounting the thermophoresis parameter. Also, the proposed scheme converges faster than the existing Crank-Nicolson scheme. The findings that were provided in this study have the potential to serve as a helpful guide for investigations into fluid flow in closed-off industrial settings in the future.

Machine-learning-based spectral methods for partial differential equations

Spectral methods are an important part of scientific computing’s arsenal for solving partial differential equations (PDEs). However, their applicability and effectiveness depend crucially on the choice of basis functions used to expand the solution of a PDE. The last decade has seen the emergence of deep learning as a strong contender in providing efficient representations of complex functions. In the current work, we present an approach for combining deep neural networks with spectral methods to solve PDEs. In particular, we use a deep learning technique known as the Deep Operator Network (DeepONet) to identify candidate functions on which to expand the solution of PDEs. We have devised an approach that uses the candidate functions provided by the DeepONet as a starting point to construct a set of functions that have the following properties: (1) they constitute a basis, (2) they are orthonormal, and (3) they are hierarchical, i.e., akin to Fourier series or orthogonal polynomials. We have exploited the favorable properties of our custom-made basis functions to both study their approximation capability and use them to expand the solution of linear and nonlinear time-dependent PDEs. The proposed approach advances the state of the art and versatility of spectral methods and, more generally, promotes the synergy between traditional scientific computing and machine learning.

Legendre-Petrov-Galerkin Chebyshev spectral collocation method for second-order nonlinear differential equations

<p style='text-indent:20px;'>We propose a Legendre-Petrov-Galerkin Chebyshev spectral collocation method for initial value problems (IVPs) of second-order nonlinear ordinary differential equations (ODEs). The Legendre-Petrov-Galerkin method is applied to time discretization and the nonlinear term is dealt with Chebyshev spectral collocation method. The scheme results in a simple algebraic system by choosing appropriate basis functions. Optimal error estimates in <inline-formula><tex-math id="M1">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula>-norm for the single and multiple interval methods are given. As an application of the method, we construct the space-time spectral schemes for solving some nonlinear time-dependent partial differential equations (PDEs). Numerical experiments suggest the efficiency of the methods.</p>

Inspection of thermal distribution through a porous fin of triangular profile with internal heat generation and electromagnetic field

The transient thermal analysis of a triangular profiled longitudinal porous fin with the combined effect of magnetic and electric fields is explained in this article. The heat transmission through the porous fin is modeled by a time-dependent nonlinear partial differential equation (PDE). The PDE and boundary conditions are transformed into a nonlinear dimensionless PDE with the aid of dimensionless terms. The numerical solution is obtained by using the technique of FDM (finite difference method). The consequences of non-dimensional thermoelectrical and thermomagnetic parameters, and other thermo-physical constraints including generation number, ambient temperature, radiation, and the convective parameter are also examined with the graphical explanations. The outcomes reveal that the emerging radiation, convective parameter, and dimensionless time parameter, faster the fin dissipate heat to the surrounding temperature. The presence of an electromagnetic field assists the triangular porous fin to distribute transient temperature effectively.

Semi-Lagrangian forward methods for some time-dependent nonlinear partial differential equations

In this article, we study one-step Semi-Lagrangian forward method for computing the numerical solutions of time-dependent nonlinear partial differential equations with initial and boundary conditions in one space dimension. Comparing with classic Semi-Lagrangian method, this method is more straight forward to analyze and implement. This method is based on Lagrangian trajectory from the departure points (regular nodes) to the arrival points by Runge-Kutta methods. The arrival points are traced forward from the departure points along the trajectory of the path. Most likely the arrival points are not on the regular grid nodes. However, it is convenient to approximate the high order derivative terms in spatial dimension on regular nodes. The convergence and stability are studied for the explicit methods. The numerical examples show that those methods work very efficient for the time-dependent nonlinear partial differential equations. For more information see https://ejde.math.txstate.edu/conf-proc/26/g1/abstr.html

Reduced Operator Inference for Nonlinear Partial Differential Equations

We present a new scientific machine learning method that learns from data a computationally inexpensive surrogate model for predicting the evolution of a system governed by a time-dependent nonlinear partial differential equation (PDE), an enabling technology for many computational algorithms used in engineering settings. Our formulation generalizes to the function space PDE setting the Operator Inference method previously developed in [B. Peherstorfer and K. Willcox, Data-driven operator inference for non-intrusive projection-based model reduction, Computer Methods in Applied Mechanics and Engineering, 306 (2016)] for systems governed by ordinary differential equations. The method brings together two main elements. First, ideas from projection-based model reduction are used to explicitly parametrize the learned model by low-dimensional polynomial operators which reflect the known form of the governing PDE. Second, supervised machine learning tools are used to infer from data the reduced operators of this physics-informed parametrization. For systems whose governing PDEs contain more general (non-polynomial) nonlinearities, the learned model performance can be improved through the use of lifting variable transformations, which expose polynomial structure in the PDE. The proposed method is demonstrated on two examples: a heat equation model problem that demonstrates the benefits of the function space formulation in terms of consistency with the underlying continuous truth, and a three-dimensional combustion simulation with over 18 million degrees of freedom, for which the learned reduced models achieve accurate predictions with a dimension reduction of five orders of magnitude and model runtime reduction of up to nine orders of magnitude.

A space-time generalized finite difference method for solving unsteady double-diffusive natural convection in fluid-saturated porous media

In this paper, the space-time generalized finite difference scheme is proposed to effectively solve the unsteady double-diffusive natural convection problem in the fluid-saturated porous media. In such a case, it is mathematically described by nonlinear time-dependent partial differential equations based on Darcy's law. In this work, the space-time approach is applied using a combination of the generalized finite difference, Newton-Raphson, and time-marching methods. In the space-time generalized finite difference scheme, the spatial and temporal derivatives can be performed using the technique for spatial discretization. Thus, the stability of the proposed numerical scheme is determined by the generalized finite difference method. Due to the property of this numerical method, which is based on the Taylor series expansion and the moving-least square method, the resultant matrix system is a sparse matrix. Then, the Newton-Raphson method is used to solve the nonlinear system efficiently. Furthermore, the time-marching method is utilized to proceed along the time axis after a numerical process in one space-time domain. By using this method, the proposed numerical scheme can efficiently simulate the problems which have an unpredictable end time. In this study, three benchmark examples are tested to verify the capability of the proposed meshless scheme.

The space–time generalized finite difference scheme for solving the nonlinear equal-width equation in the long-time simulation

In this research, the space–time (ST) coupled generalized finite difference method (GFDM) was extended to cooperate with the Newton–Raphson and time-marching methods for stably solving the nonlinear equal-width equation in long-time simulation. The equal-width equation is a time-dependent nonlinear partial differential equation and has a third-order derivation that is mixed space and time. In such a case, the action of fluid flow is over a long period, and the numerical scheme needs to rely on the state-steady condition or set up a specific time to stop the simulation. In the ST-GFDM, the temporal and spatial discretizations can be performed using the GFDM in an ST-domain. That makes the proposed scheme can easily deal with the mixed derivative. By applying the GFDM, which is advanced from the Taylor series expansion and the moving-least square method, the process of the numerical discretizations is only related to nearby nodes on the central node. Thus, the Jacobian matrix is sparse and can be solved by the Newton–Raphson method efficiently. Furthermore, the time-marching method is used to proceed with an ST-domain along the time axis. In this paper, two numerical examples are solved to verify the capability of the proposed ST-GFDM.

A linearly implicit scheme and fast multigrid solver for 3D Fitzhugh-Nagumo equation

In this work, a second-order finite difference (FD) scheme for three-dimensional (3D) nonlinear Fitzhugh-Nagumo (FN) equation with the nonlinear term treated with semi-implicitly technique is proposed. The existence and uniqueness of the difference scheme is proved, and the stability and convergence of numerical solution in L∞-norm are also shown. Then, we employ an efficient extrapolation cascadic multigrid (EXCMG) method to solve the large linear system arising from the proposed second-order FD discretization for the 3D FN equation. Numerical results are presented to verify our theoretical findings of the difference scheme and the efficiency of the EXCMG method. The EXCMG method can also be extended to solve other kinds of time-dependent nonlinear partial differential equations.

Amperometric biosensors and coupled enzyme nonlinear reactions processes: A complete theoretical and numerical approach

The transient response of amperometric enzyme-based biosensors working in trigger mode is discussed. Nonlinear time-dependent partial differential equations for Michaelis–Menten reaction kinetics are solved analytically using a new approach of homotopy perturbation technique. The simple and closed-form analytical expression for concentration profiles are provided. Subsequently, the biosensor's current, sensitivity, resistance, and amplification are derived from the concentration profiles. The current response is predicted under steady-state conditions when T→∞, proving the validity of the mathematical analyses. The limiting situations of catalytic sites (unsaturation and saturation) are considered. The compatibility of analytical results with simulation and limiting case results can be observed from the graphs and tables presented. The existence of the moving boundary between the two categories of catalytic sites is also discussed.

POD-DL-ROM: Enhancing deep learning-based reduced order models for nonlinear parametrized PDEs by proper orthogonal decomposition

Deep learning-based reduced order models (DL-ROMs) have been recently proposed to overcome common limitations shared by conventional reduced order models (ROMs) – built, e.g., through proper orthogonal decomposition (POD) – when applied to nonlinear time-dependent parametrized partial differential equations (PDEs). These might be related to (i) the need to deal with projections onto high dimensional linear approximating trial manifolds, (ii) expensive hyper-reduction strategies, or (iii) the intrinsic difficulty to handle physical complexity with a linear superimposition of modes. All these aspects are avoided when employing DL-ROMs, which learn in a non-intrusive way both the nonlinear trial manifold and the reduced dynamics, by relying on deep (e.g., feedforward, convolutional, autoencoder) neural networks. Although extremely efficient at testing time, when evaluating the PDE solution for any new testing-parameter instance, DL-ROMs require an expensive training stage, because of the extremely large number of network parameters to be estimated. In this paper we propose a possible way to avoid an expensive training stage of DL-ROMs, by (i) performing a prior dimensionality reduction through POD, and (ii) relying on a multi-fidelity pretraining stage, where different physical models can be efficiently combined. The proposed POD-DL-ROM is tested on several (both scalar and vector, linear and nonlinear) time-dependent parametrized PDEs (such as, e.g., linear advection–diffusion–reaction, nonlinear diffusion–reaction, nonlinear elastodynamics, and Navier–Stokes equations) to show the generality of this approach and its remarkable computational savings.

Goal-oriented model reduction for parametrized time-dependent nonlinear partial differential equations

We present a projection-based model reduction formulation for parametrized time-dependent nonlinear partial differential equations (PDEs). Our approach builds on the following ingredients: reduced bases (RB), which provide rapidly convergent approximations of the parameter-temporal solution manifold; reduced quadrature (RQ) rules, which provide hyperreduction of the nonlinear residual; and the dual-weighted residual (DWR) method, which provides an error representation formula for the quantity of interest. To find the RQ rules, we develop an empirical quadrature procedure (EQP) for time-dependent problems; we analyze the output error due to hyperreduction using a space–time DWR framework and identify appropriate constraints so that the output error due to hyperreduction is controlled. We in addition equip our reduced model with an online-efficient DWR a posteriori error estimate for the output; we again analyze the error in the hyperreduced dual problem and DWR expression to find appropriate constraints for the EQP so that the error in the error estimate is controlled. In the offline stage, the RBs and RQs, as well as the finite element mesh, are simultaneously constructed using a POD-greedy algorithm that leverages the online-efficient output error estimate. We demonstrate the framework for parametrized unsteady flows in a lid-driven cavity and over a NACA0012 airfoil. Reduced models achieve over two orders of magnitude reduction in the number of degrees of freedom, number of quadrature points, and wall-clock computational time, while achieving less than 0.5% output error and providing efficient error estimates in predictive settings.

DPM: A Novel Training Method for Physics-Informed Neural Networks in Extrapolation

We present a method for learning dynamics of complex physical processes described by time-dependent nonlinear partial differential equations (PDEs). Our particular interest lies in extrapolating solutions in time beyond the range of temporal domain used in training. Our choice for a baseline method is physics-informed neural network (PINN) because the method parameterizes not only the solutions, but also the equations that describe the dynamics of physical processes. We demonstrate that PINN performs poorly on extrapolation tasks in many benchmark problems. To address this, we propose a novel method for better training PINN and demonstrate that our newly enhanced PINNs can accurately extrapolate solutions in time. Our method shows up to 72% smaller errors than state-of-the-art methods in terms of the standard L2-norm metric.

Dynamic tensor approximation of high-dimensional nonlinear PDEs

We present a new method based on functional tensor decomposition and dynamic tensor approximation to compute the solution of a high-dimensional time-dependent nonlinear partial differential equation (PDE). The idea of dynamic approximation is to project the time derivative of the PDE solution onto the tangent space of a low-rank functional tensor manifold at each time. Such a projection can be computed by minimizing a convex energy functional over the tangent space. This minimization problem yields the unique optimal velocity vector that allows us to integrate the PDE forward in time on a tensor manifold of constant rank. In the case of initial/boundary value problems defined in real separable Hilbert spaces, this procedure yields evolution equations for the tensor modes in the form of a coupled system of one-dimensional time-dependent PDEs. We apply the dynamic tensor approximation to a four-dimensional Fokker–Planck equation with non-constant drift and diffusion coefficients, and demonstrate its accuracy in predicting relaxation to statistical equilibrium.

Motion Prediction of Underwater Sensors

In this work, we will simulate the motion of a single underwater sensor knowing the current velocity to predict its location and velocity during certain time frame using a numerical approach of non-linear time-dependent partial differential equations and develop numerical computer programming code to solve the equations.
 The underwater sensor are used to collect data for many scientific and practical reasons all the sensor collected data without specifying the sensor location and time will be missing lowers valuable information and by simulating the sensor motion numerically will have many values and impact on the underwater sensor industries as this will lead to less power consumption sensors with smaller size and less network coverage required.
 This paper will study the kinetics of the underwater sensor which will resulted to a set of non-linear time-dependent partial differential equations that can be solved analytically and computer programming simulation is developed to solve the equations and predict the motion of underwater sensor.
 Different scenarios considered in the work such as simulating the result for different sensor’s density and the effect on its final position. Also, the result will include the sensor velocity simulation and comparison with the sea current velocity.
 This work is limited to the motion prediction of single underwater sensor and the result is only for mechanical aspect of the problem, the networks connectivity or coverage is out-of-scope.

Motion Prediction of Underwater Sensors

In this work, we will simulate the motion of a single underwater sensor knowing the current velocity to predict its location and velocity during certain time frame using a numerical approach of non-linear time-dependent partial differential equations and develop numerical computer programming code to solve the equations.
 The underwater sensor are used to collect data for many scientific and practical reasons all the sensor collected data without specifying the sensor location and time will be missing lowers valuable information and by simulating the sensor motion numerically will have many values and impact on the underwater sensor industries as this will lead to less power consumption sensors with smaller size and less network coverage required.
 This paper will study the kinetics of the underwater sensor which will resulted to a set of non-linear time-dependent partial differential equations that can be solved analytically and computer programming simulation is developed to solve the equations and predict the motion of underwater sensor.
 Different scenarios considered in the work such as simulating the result for different sensor’s density and the effect on its final position. Also, the result will include the sensor velocity simulation and comparison with the sea current velocity.
 This work is limited to the motion prediction of single underwater sensor and the result is only for mechanical aspect of the problem, the networks connectivity or coverage is out-of-scope.