Partial Differential Equations Research Articles

Extracting fiber paths from the optimized lamination parameters of variable-stiffness laminated shells based on physic-informed neural network

This paper presents a novel approach for extracting fiber paths from the optimized lamination parameters (LPs) of variable stiffness laminated shells, utilizing the framework of physics-informed neural network (PINN). In this methodology, each fiber layer is associated with a specific stream function, which is approximated by an independent neural network. The stream function is governed by a partial differential equation (PDE) derived from the fiber orientation field in the parameter space. Moreover, the isocontours of the stream function are transformed into the actual fiber paths in the physical space. To account for manufacturing constraints, Riemannian geometry serves as a computational tool to determine the intrinsic distance between adjacent fiber paths and the geodesic curvature of the isocontours. By incorporating regularization terms into the loss function based on the physical relationships, the constrained optimization problem is converted into an unconstrained one, making it more suitable for neural network training. Meanwhile, a fiber path extraction (FPE) algorithm is used to minimize the loss function at randomly sampled points through gradient descent. The numerical results suggest that the extraction of fiber paths using PINN can achieve satisfactory levels of accuracy while effectively satisfying the imposed constraints.

Prediction of ILI following the COVID-19 pandemic in China by using a partial differential equation

Abstract The COVID-19 outbreak has significantly disrupted the lives of individuals worldwide. Following the lifting of COVID-19 interventions, there is a heightened risk of future outbreaks from other circulating respiratory infections, such as influenza-like illness (ILI). Accurate prediction models for ILI cases are crucial in enabling governments to implement necessary measures and persuade individuals to adopt personal precautions against the disease. This paper aims to provide a forecasting model for ILI cases with actual cases. We propose a specific model utilizing the partial differential equation (PDE) that will be developed and validated using real-world data obtained from the Chinese National Influenza Center. Our model combines the effects of transboundary spread among regions in China mainland and human activities’ impact on ILI transmission dynamics. The simulated results demonstrate that our model achieves excellent predictive performance. Additionally, relevant factors influencing the dissemination are further examined in our analysis. Furthermore, we investigate the effectiveness of travel restrictions on ILI cases. Results can be used to utilize to mitigate the spread of disease.

Optimization of heat transfer in a channel with stretching walls using a magnetized tetra-hybrid nanofluid

Nanoparticles have numerous applications and are used frequently in different cooling, heating, treatment of cancer cells and manufacturing processes. The current investigation covers the utilization of tetra hybrid nanofluid (aluminium oxide, iron dioxide, titanium dioxide and copper) for Crossflow model over a vertical disk by considering the shape effects (bricks, cylindrical and platelet) of nanoparticles, electro-magneto-hydrodynamic effect and quadratic thermal radiation. The present study is devoted to present the mathematical formulation of the tetra-hybrid nanofluid flow in a porous channel with stretching/shrinking walls. The present inspection model is derived from given partial differential equations (PDEs) and then transformed into a system of ordinary differential equations (ODEs) by incorporating similarity variables. The transformed ODEs are solved using the bvp4c methodology, which yields numerical results. From the obtained results it is observed that the maximum amount of [Formula: see text] happens when there is slightly increase in [Formula: see text] with stretched wall that is [Formula: see text]. A high radiation value indicates a considerable contribution from radiative heat transfer, whereas stretching and shrinking increase and reduce the size of the boundary. The combined impact caused an increase in the Nusselt number. The graphical and tabular representations are used to explain the physical behaviour of various model parameters. Previous outcomes are also contrasted with the current outcomes.

Investigating neural networks with groundwater flow equation loss

Physics-Informed Neural Networks (PINNs) are considered a powerful tool for solving partial differential equations (PDEs), particularly for the groundwater flow (GF) problem. In this paper, we investigate how the deep learning (DL) architecture, within the PINN framework, is connected to the ability to compute a more or less accurate numerical GF solution, so the link ‘PINN architecture - numerical performance’ is explored. Specifically, this paper explores the effect of various DL components, such as different activation functions and neural network structures, on the computational framework. Through numerical results and on the basis of some theoretical foundations of PINNs, this research aims to improve the explicability of PINNs to resolve, in this case, the one-dimensional GF equation. Moreover, our problem involves source terms described by a Dirac delta function, providing insights into the role of DL architecture in solving complex PDEs.

General-Kindred physics-informed neural network to the solutions of singularly perturbed differential equations

Physics-informed neural networks (PINNs) have become a promising research direction in the field of solving partial differential equations (PDEs). Dealing with singular perturbation problems continues to be a difficult challenge in the field of PINN. The solution of singular perturbation problems often exhibits sharp boundary layers and steep gradients, and traditional PINN cannot achieve approximation of boundary layers. In this manuscript, we propose the General-Kindred physics-informed neural network (GKPINN) for solving singular perturbation differential equations (SPDEs). This approach utilizes asymptotic analysis to acquire prior knowledge of the boundary layer from the equation and establishes a novel network to assist PINN in approximating the boundary layer. It is compared with traditional PINN by solving examples of one-dimensional, two-dimensional, and time-varying SPDE equations. The research findings underscore the exceptional performance of our novel approach, GKPINN, which delivers a remarkable enhancement in reducing the L2 error by two to four orders of magnitude compared to the established PINN methodology. This significant improvement is accompanied by a substantial acceleration in convergence rates, without compromising the high precision that is critical for our applications. Furthermore, GKPINN still performs well in extreme cases with perturbation parameters of 1×10−38, demonstrating its excellent generalization ability.

Numerical Methods and Computation in Financial Mathematics: Solving PDEs, Monte Carlo Simulations, and Machine Learning Applications

Abstract. This paper delves into three fundamental numerical methods and computational techniques in financial mathematics: Finite Difference Methods (FDM), Monte Carlo Simulations (MCS), and Machine Learning (ML) applications. Finite Difference Methods are widely utilized for solving partial differential equations (PDEs) in option pricing, with various schemes offering different stability and convergence properties. Monte Carlo Simulations provide a powerful approach for pricing complex derivatives and risk management, addressing the challenges of high-dimensionality and computational complexity. Machine Learning has revolutionized predictive modeling in finance, enabling sophisticated analysis of large datasets to uncover hidden patterns and enhance trading strategies. Through a detailed examination of these methods, including specific examples and data, this paper highlights their theoretical foundations, practical implementations, and the advancements they bring to computational finance. By bridging theoretical approaches with practical applications, we aim to offer insights into the future directions and challenges in financial mathematics.

Solutions to elliptic and parabolic problems via finite difference based unsupervised small linear convolutional neural networks

In recent years, there has been a growing interest in leveraging deep learning and neural networks to address scientific problems, particularly in solving partial differential equations (PDEs). However, many neural network-based methods like PINNs rely on auto differentiation and sampling collocation points, leading to a lack of interpretability and lower accuracy than traditional numerical methods. As a result, we propose a fully unsupervised approach, requiring no training data, to estimate finite difference solutions for PDEs directly via small linear convolutional neural networks. Our proposed approach uses substantially fewer parameters than similar finite difference-based approaches while also demonstrating comparable accuracy to the true solution for several selected elliptic and parabolic problems compared to the finite difference method.

A micropolar phase-field model for size-dependent electro-mechanical fracture

This article proposes a micropolar phase-field model for size-dependent brittle fracture in solids under electro-mechanical loading conditions. Considering displacement, micro-rotation, electric potential, and phase-field variable as the kinematic descriptors and employing the virtual power principle, we derive a set of coupled governing partial differential equations (PDEs) for size-dependent solids. Invoking the first and second laws of thermodynamics, we determine the constitutive relations for the thermodynamic fluxes. Carrying out the finite element implementation of the derived governing PDEs using the open-source Gridap package in Julia, we demonstrate the efficacy of the proposed phase-field model through a few representative numerical examples. Especially the importance of the proposed model in incorporating the effect of relative rotation, i.e., the difference between macro- and micro-rotation, on the response of solids under electro-mechanical loading is shown that may not be possible with the existing non-local models such as strain-gradient or couple-stress approaches. To capture the experimentally observed size effects in solids under electro-mechanical loading, the proposed model does not demand higher-order continuity of finite element shape functions, unlike a typically used strain gradient model. To demonstrate the efficacy of the proposed model, we have compared our results against demanding experimental and numerical benchmark results available in the literature. We provide a parametric study to unravel the effect of different micropolar material parameters on the electro-mechanical response of a brittle solid. Interestingly, the proposed micropolar model is less sensitive to the phase-field length scale than the conventional non-polar phase-field models.

Exploring the Thermal Behavior of Cu-water and CuO-water Power-law Nanofluids on a Rotating Circular Disc: A Computational Analysis

The current theoretical investigation focuses on the thermal behavior of a power-law nanofluid flow on a rotating circular disc. In which nanofluid is made by taking colloidal suspension of copper (Cu) and copper oxide (CuO) nanoparticles in a base fluid like water. A theoretical investigation is conducted by formulating a mathematical model of a power-law fluid (which exhibits shear thickening and thinning effects) along with the Tiwari and Das model. A similarity transformation is employed to transform the nonlinear partial differential equations (PDEs) into ordinary differential equations (ODEs) and these ODEs are then numerically solved using the shooting technique. The study's findings are analyzed graphically by plotting radial, tangential, and axial velocity profiles, temperature profiles, Nusselt number, and skin friction coefficients to show how various factors affect the fluid's flow and heat transfer. It is revealed that drag force reduces in Cu-water nanofluid as compared to CuO-water nanofluid for all types of fluids (Newtonian, pseudoplastic, and dilatant fluids) and Nusselt number becomes high in Cu-water nanofluid for both Newtonian and dilatant fluids as compared to CuO-water nanofluid but in case of pseudoplastic reverse behavior is observed. Heat transfer rate in Cu-water dilatant and pseudoplastic nanofluids increases up to 20.4% and 12.5% with respect to base fluid and in CuO-water dilatant and pseudoplastic water nanofluids, it increases up to 18.6% and 15.2%.

Locally linearized physics-informed neural networks for Riemann problems of hyperbolic conservation laws

In this work, we demonstrate that physics-informed neural networks (PINNs) tend to propagate predicted shock wave information bidirectionally in time, which does not align with the actual evolution direction of solutions to hyperbolic conservation laws. This mismatch results in instability and hinders the reduction of the loss of governing equations, as well as the initial condition loss by the deep neural network. In order to tackle this problem, we simplify the complexity of the problem by constructing equivalent linear transport equations in the region of shock wave generation. The speeds of these linearized waves are governed by the Rankine–Hugoniot relations of conservation laws. This approach is termed the Locally Linearized PINNs method. Specifically, an appropriate shock wave detector is initially designed to identify domains where shock waves occur. Near shock waves, the original nonlinear equations are transformed into their linearized forms, thereby modifying the residual terms of the partial differential equations. Additionally, an equilibrium factor is introduced in fluid compression regions to reduce prediction errors and stabilize the training of deep neural networks. Numerical examples illustrate that Locally Linearized PINNs effectively address the challenge of predicting global solutions in PINNs and significantly improve shock-capturing performance for hyperbolic conservation laws.

Well-balanced High-order Finite Difference Weighted Essentially Nonoscillatory Schemes for a First-order Z4 Formulation of the Einstein Field Equations

We develop a new class of high-order accurate well-balanced finite difference (FD) weighted essentially nonoscillatory (WENO) methods for numerical general relativity (GR), which can be applied to any first-order reduction of the Einstein field equations, even if nonconservative terms are present. We choose the first-order nonconservative Z4 formulation of the Einstein equations, which has a built-in cleaning procedure that accounts for the Einstein constraints and that has already shown its ability in keeping stationary solutions stable over long timescales. By introducing auxiliary variables, the vacuum Einstein equations in first-order form constitute a partial differential equation system of 54 equations that is naturally nonconservative. We show how to design FD-WENO schemes that can handle nonconservative products. Different variants of FD WENO are discussed, with an eye to their suitability for higher-order accurate formulations for numerical GR. We successfully solve a set of fundamental tests of numerical GR with up to ninth-order spatial accuracy. Due to their intrinsic robustness, flexibility, and ease of implementation, FD-WENO schemes can effectively replace traditional central finite differencing in any first-order formulation of the Einstein field equations, without any artificial viscosity. When used in combination with well-balancing, the new numerical schemes preserve stationary equilibrium solutions of the Einstein equations exactly. This is particularly relevant in view of the numerical study of the quasi-normal modes of oscillations of relevant astrophysical sources. In conclusion, general relativistic high-energy astrophysics could benefit from this new class of numerical schemes and the ecosystem of desirable capabilities built around them.

Interactions of localized wave and dynamics analysis in the new generalized stochastic fractional potential-KdV equation.

In this paper, we investigate the new generalized stochastic fractional potential-Korteweg-de Vries equation, which describes nonlinear optical solitons and photon propagation in circuits and multicomponent plasmas. Inspired by Kolmogorov-Arnold network and our earlier work, we enhance the improved bilinear neural network method by using a large number of activation functions instead of neurons. This method incorporates the concept of simulating more complicated activation functions with fewer parameters, with more diverse activation functions to generate more complex and rare analytical solutions. On this basis, constraints are introduced into the method, reducing a significant amount of computational workload. We also construct neural network architectures, such as "2-3-1," "2-2-3-1," "2-3-3-1," and "2-3-2-1" using this method. Maple software is employed to obtain many exact analytical solutions by selecting appropriate parameters, such as the superposition of double-period lump solutions, lump-rogue wave solutions, and three interaction solutions. The results show that these solutions exhibit more complex waveforms than those obtained by conventional methods, which is of great significance for the electrical systems and multicomponent fluids to which the equation is applied. This novel method shows significant advantages when applied to fractional-order equations and is expected to be increasingly widely used in the study of nonlinear partial differential equations.

Enhancing high-resolution reconstruction of flow fields using physics-informed diffusion model with probability flow sampling

The application of artificial intelligence (AI) technology in fluid dynamics is becoming increasingly prevalent, particularly in accelerating the solution of partial differential equations and predicting complex flow fields. Researchers have extensively explored deep learning algorithms for flow field super-resolution reconstruction. However, purely data-driven deep learning models in this domain face numerous challenges. These include susceptibility to variations in data distribution during model training and a lack of physical and mathematical interpretability in the predictions. These issues significantly impact the effectiveness of the models in practical applications, especially when input data exhibit irregular distributions and noise. In recent years, the rapid development of generative artificial intelligence and physics-informed deep learning algorithms has created significant opportunities for complex physical simulations. This paper proposes a novel approach that combines diffusion models with physical constraint information. By integrating physical equation constraints into the training process of diffusion models, this method achieves high-fidelity flow field reconstruction from low-resolution inputs. Thus, it not only leverages the advantages of diffusion models but also enhances the interpretability of the models. Experimental results demonstrate that, compared to traditional methods, our approach excels in generating high-resolution flow fields with enhanced detail and physical consistency. This advancement provides new insights into developing more accurate and generalized flow field reconstruction models.

Non-stationary spatio-temporal modeling using the stochastic advection-diffusion equation

We construct flexible spatio-temporal models through stochastic partial differential equations (SPDEs) where both diffusion and advection can be spatially varying. Computations are done through a Gaussian Markov random field approximation of the solution of the SPDE, which is constructed through a finite volume method. The new flexible non-separable model is compared to a flexible separable model both for reconstruction and forecasting, and evaluated in terms of root mean square errors and continuous rank probability scores. A simulation study demonstrates that the non-separable model performs better when the data is simulated from a non-separable model with diffusion and advection. Further, we estimate surrogate models for emulating the output of a ocean model in Trondheimsfjorden, Norway, and simulate observations of autonomous underwater vehicles. The results show that the flexible non-separable model outperforms the flexible separable model for real-time prediction of unobserved locations.

Variational inequalities of multilayer elastic contact systems with interlayer friction: Existence and uniqueness of solution and convergence of numerical solution

Inspired by the layered structure models in pavement mechanics research, in this study, a class of multilayer elastic contact systems with interlayer frictional contact conditions and deformable supporting frictional contact conditions on the foundation has been constructed. Based on the nonlinear elastic constitutive equations, the corresponding system of partial differential equations and variational inequalities are respectively introduced. Under the framework of variational inequalities, the existence and uniqueness of solutions for such models, along with the approximation properties of finite element numerical solutions, are proven and analyzed. The aforementioned conclusions provide fundamental and broadly applicable theoretical support for addressing mechanical problems in multilayer elastic contact systems within the framework of variational inequalities. Finally, the numerical experimental results based on the mixed finite element method also substantiate our theoretical conclusions.

Artificial neural network-based computational heat transfer analysis of Carreau fluid over a rotating cone

Heat transport in a dynamically rotating cone immersed in a Carreau fluid is the subject of this investigation. The fluid is non-Newtonian, admired for its characteristics, and extensively utilized in numerous industrial domains. The study investigates the interplay between buoyancy and centrifugal forces within an analytical framework. The study employs sophisticated mathematical methods, including similarity transformations, to convert governing partial differential equations into nonlinear ordinary differential equations. These equations are then solved using the shooting method, a numerical technique that solves a boundary value problem by iteratively adjusting the initial conditions until the boundary conditions are satisfied. We employ an artificial neural network algorithm with backpropagation Levenberg–Marquardt scheme to analyze the heat transfer mechanism quantitatively. In conjunction with the shooting mechanism, we will use numerical simulation with an artificial neural network algorithm, namely the backpropagation Levenberg–Marquardt scheme. The results prove the enormous influence of centrifugation and buoyancy on complex fluid dynamics and heat exchange processes. Some critical parameters that govern the convective heat transport process are the Nusselt number, the Reynolds number, the Grashof number, and the fluid and cone rotational velocities. The research validates the requirement of considering non-Newtonian complexity and viscous dissipation when investigating heat transfer dynamics and fluid flow, facilitating more accurate expectations and improved efficiency in various industrial processes.

Solving Viscous Burgers’ Equation: Hybrid Approach Combining Boundary Layer Theory and Physics-Informed Neural Networks

In this paper, we develop a hybrid approach to solve the viscous Burgers’ equation by combining classical boundary layer theory with modern Physics-Informed Neural Networks (PINNs). The boundary layer theory provides an approximate analytical solution to the equation, particularly in regimes where viscosity dominates. PINNs, on the other hand, offer a data-driven framework that can address complex boundary and initial conditions more flexibly. We demonstrate that PINNs capture the key dynamics of the Burgers’ equation, such as shock wave formation and the smoothing effects of viscosity, and show how the combination of these methods provides a powerful tool for solving nonlinear partial differential equations.

Surrogate modeling of multi-dimensional premixed and non-premixed combustion using pseudo-time stepping physics-informed neural networks

Physics-informed neural networks (PINNs) have emerged as a promising alternative to conventional computational fluid dynamics (CFD) approaches for solving and modeling multi-dimensional flow fields. They offer instant inference speed and cost-effectiveness without the need for training datasets. However, compared to common data-driven methods, purely learning the physical constraints of partial differential equations and boundary conditions is much more challenging and prone to convergence issues leading to incorrect local optima. This training robustness issue significantly increases the difficulty of fine-tuning PINNs and limits their widespread adoption. In this work, we present improvements to the prior field-resolving surrogate modeling framework for combustion systems based on PINNs. First, inspired by the time-stepping schemes used in CFD numerical methods, we introduce a pseudo-time stepping loss aggregation algorithm to enhance the convergence robustness of the PINNs training process. This new pseudo-time stepping PINNs (PTS-PINNs) method is then tested in non-reactive convection–diffusion problem, and the results demonstrated its good convergence capability for multi-species transport problems. Second, the effectiveness of the PTS-PINNs method was verified in the case of methane–air premixed combustion, and the results show that the L2 norm relative error of all variables can be reduced within 5%. Finally, we also extend the capability of the PTS-PINNs method to address a more complex methane–air non-premixed combustion problem. The results indicate that the PTS-PINNs method can still achieve commendable accuracy by reducing the relative error to within 10%. Overall, the PTS-PINNs method demonstrates the ability to rapidly and accurately identify the convergence direction of the model, surpassing traditional PINNs methods in this regard.

A spectral dominance approach to large random matrices: Part II

This paper is the second of a series devoted to the study of the dynamics of the spectrum of large random matrices. We precise and extend some results of the first part. We study general extensions of the partial differential equation arising to characterize the limit spectral measure of the Dyson Brownian motion. We provide a regularizing result for those generalizations. We also show that several results of part I extend to cases in which there is no spectral dominance property. We then provide several modeling extensions of such models as well as several identities for the Dyson Brownian motion.

The Heat Transfer in Skin Tissues Under The General Two-Temperature Three-Phase-Lag Model of Heat Conduction with a Comparative Study

In this paper, a new and more general model of heat conduction that depends on the drift velocity due to thermomass motion assumption will be established and will be applied to the skin tissue. Four different heat conduction models will be incorporated into a unified equation of heat conduction: the Pennes, Vernotte-Cattaneo, dual-phase-lag of Tzou, and the general two-temperature three-phase-lag of Youssef. The governing partial differential equations of the general two-temperature three-phase-lag model of bioheat conduction will be implemented and solved directly in the domain of the Laplace transformation. The numerical solutions of the Laplace transform will be calculated by executing the Tzou iteration formula. The ramp-type heat on the surface of the skin tissue will be considered as thermal loading. The conductive and dynamic temperature increment reactions have been studied and discussed with different values of ramp-time heat, characteristic length, and drift velocity parameters. The novelty of this work is to introduce some comparisons of the four under-studied bioheat conduction models and show the differences between them in the figures. The numerical results show that the ramp-time heat, drift velocity, and characteristic length parameters have major impacts on the increment of both dynamical and conductive temperature distributions.