Partial Differential Equations Research Articles

Multi-level physics informed deep learning for solving partial differential equations in computational structural mechanics

Physics-informed neural network has emerged as a promising approach for solving partial differential equations. However, it is still a challenge for the computation of structural mechanics problems since it involves solving higher-order partial differential equations as the governing equations are fourth-order nonlinear equations. Here we develop a multi-level physics-informed neural network framework where an aggregation model is developed by combining multiple neural networks, with each one involving only first-order or second-order partial differential equations representing different physics information such as geometrical, constitutive, and equilibrium relations of the structure. The proposed framework demonstrates a remarkable advancement over the classical neural networks in terms of the accuracy and computation time. The proposed method holds the potential to become a promising paradigm for structural mechanics computation and facilitate the intelligent computation of digital twin systems.

Diversified characteristics of the dissipative heat on the radiative micropolar hybrid nanofluid over a wedged surface: Gauss-Lobatto IIIA numerical approach

The present examination focuses on the Falkner-Skan flow of micropolar hybridized nanofluid via a wedge surface. The proposed study examines thermal radiative fluxing and heat dissipation in hybridized nanoparticle aqueous solutions. Simulations of uni-directional radiative transport in optically dense fluids use Rosseland's diffusion model. This study created a Cu-TiO2/water hybrid nanofluid by mixing Cu and TiO2 nanomolecules with H2O. Partial differential equations from Naiver-Stokes theory are used to generate the regulating flow phenomena, then convert them into ordinary differentiation equations using an appropriate similarity approach. Additionally, the three-stage Lobatto IIIA method is used to compute the formulae. Calculations are done using MATLAB's built-in bvp5c function. We found that increasing material characteristics slows fluid flow because micropolar nanofluids minimize drag. These alterations alter fluid flow and boost temperature. However, boosting thermal radiation and Eckert number slows heat movement but improves temperature profiles. Much prior research ignored thermal radiative fluxing and heat dissipation in the aqueous solution of hybrid nanomolecules (Cu and TiO2) in Falkner-Skan micropolar flow via a wedge surface. Tables show wall frictional factor and Nusselt quantity results. Micropolar fluid characteristic decreases velocity but increases micro-rotational velocity. Power-law parameters, volume fractions, radiation, heat source, and Eckert amount affect thermal contours.

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.

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%.

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.

On the Limited Role of Conservation Laws in the Double Reduction Routine for Partial Differential Equations

The double reduction method for finding invariant solutions of a given partial differential equation (PDE) provides for the reduction of a $ q $-th order PDE that admits a nontrivial Lie symmetry and an associated nontrivial conservation law to an ordinary differential equation (ODE) of order $ q-1. $ In all the articles we have seen where the method has been used, the algorithm has involved writing the conservation law in canonical variables determined by the associated symmetry. In this paper, we illustrate that it is not necessary to use or even have the associated conservation law. It is enough to know that there exists a conservation law associated with a given Lie symmetry. Canonical variables derived from the symmetry are sufficient to achieve double reduction. In the canonical variables, the PDE is transformed after routine calculations into an ODE of order one less than that of the PDE. We have outlined steps involved in this variation of the double reduction method and illustrated the routine using five PDEs.

Deep Learning for Solving Partial Differential Equations: A Review of Literature

Partial Differential Equations (PDEs) are fundamental in modeling various phenomena in physics, engineering, and finance. Traditional numerical methods for solving PDEs, such as finite element and finite difference methods, often face limitations when applied to high-dimensional and complex systems. In recent years, deep learning has emerged as a promising alternative for approximating solutions to PDEs, offering potential improvements in both efficiency and scalability. This paper provides a comprehensive review of the literature on deep learning-based methods for solving PDEs, focusing on key approaches such as Physics-Informed Neural Networks (PINNs), deep Galerkin methods, and neural operators. These methods leverage the expressiveness of neural networks to capture underlying physics while avoiding the curse of dimensionality associated with classical techniques. We explore the theoretical foundations, advantages, and limitations of these deep learning models, along with their applications in diverse fields like fluid dynamics, quantum mechanics, and financial modeling. Additionally, this review examines recent advancements in hybrid models that combine traditional numerical methods with deep learning approaches to enhance accuracy and stability. Through this review, we highlight key trends and open challenges in the field, paving the way for future research at the intersection of deep learning and computational mathematics.

Multidimensional backward stochastic differential equations with rough drifts

In this paper, we study a multidimensional backward stochastic differential equation (BSDE) with an additional rough drift (rough BSDE), and give the existence and uniqueness of the adapted solution, either when the terminal value and the geometric rough path are small, or when each component of the rough drift only depends on the corresponding component of the first unknown variable (but we drop the one-dimensional assumption of Diehl and Friz [Ann. Probab. 40 (2012), pp. 1715–1758]). We also introduce a new notion of the p p -rough stochastic integral for p ∈ [ 2 , 3 ) p \in \left [2, 3\right ) , and then succeed in giving—through a fixed-point argument—a general existence and uniqueness result on a multidimensional rough BSDE with a general square-integrable terminal value, allowing the rough drift to be random and time-varying but having to be linear; furthermore, we connect it to a system of rough partial differential equations.

η̈-Ricci Soliton and Its Applications on φ-Recurrent LP Sasakian Manifold

The objective of this paper is to study the φ-recurrent nature and application of LP-Sasakian manifold associated with η̈ -Ricci soliton. Initially, we introduce an example to show the existance of LP Sasakian manifold equipped with η̈ -Ricci soliton. The idea of Ricci soliton recognized as a generalization of an Einstein metric and governing as the solution of partial differential equations representing Ricci flow. Some conditions have been obtained representing the nature of soliton (expanding, unchanged and shrinkingness) in pseudo projective, Weyl projective and semi generlized curvature with φ-recurrent condition. The application of η̈ -Ricci Soliton on spacetime also discussed.

Employing the Sadik Residual Power Series Method to Analyze a System of Nonlinear Caputo Time-Fractional Partial Differential Equations

The present study proposes an approximate analytical solution to a nonlinear system of time-fractional partial differential equations. The Sadik residual power series method, which integrates the two-part Sadik integral transform with the residual power series technique, is employed to solve the fractional differential equation in the Caputo sense. Nonlinear problems with known and unknown solutions are examined to demonstrate the capacity of the technique. Numerical simulations and 3D visualizations are conducted for various values of the fractional order to further understand the solution’s behavior. Additionally, the results are validated against exact solutions or existing methodologies to ensure their reliability and accuracy. A key advantage of the proposed method is its ability to generate results without the need for Adomian polynomials, perturbation techniques, discretization, or linearization, enabling a more efficient.

Stability and convergence computational analysis of a new semi analytical-numerical method for fractional order linear inhomogeneous integro-partial-differential equations

Abstract The motive behind this research work, is to originate a semi-analytical-numerical approach for the solution of fractional order linear integro partial differential equations (FOLIPDEs), especially in-homogeneous FOLIPDEs of different types, like fractional version of Fredholm and Volterra type IEs etc. To attain the said purpose, we will study some fractional formulations of linear model integral equations from the existing literature. Then we will describe the proposed semi analytical-numerical procedure alongwith its stability analysis and convergence properties. We will then prove with the aid of some particular numerical examples that the said approach is not only lucid and efficient but also precise. The outcomes of the proposed technique will show that this scheme has notable prospectives for solving a large class of FOLIEs. At the end, the contribution of this work in the advancements of semi analytical-numerical approaches for
the solution of fractional order linear integro partial differential equations will be laid down and discussion for future research in this field.

Enhanced Adomian Decomposition Method for Accurate Numerical Solutions of PDE Systems

This research addresses critical challenges in numerical solutions, which are vital for various engineering and physical fields. The Modified Adomian Decomposition Method (MADM) is proposed as a novel approach for solving linear and nonlinear partial differential equations (PDEs). MADM builds upon the Adomian Decomposition Method (ADM) by incorporating a new integral operator that significantly improves convergence rates and accuracy. Numerical examples demonstrate the effectiveness of MADM in handling complex nonlinear PDEs. Compared to traditional ADM, MADM consistently achieves more accurate and rapidly converging solutions. This enhancement is attributed to the novel integral operator, which addresses the limitations of ADM for intricate nonlinear problems. The paper outlines the application of MADM, its solution procedure, and its effectiveness through numerical examples. Comparisons with standard ADM solutions and exact solutions validate MADM's accuracy and superiority. The results suggest that MADM is a promising tool for expanding the applicability of Adomian methods in the field of solving PDEs.

Fast simulation of coronary in‐stent restenosis: A non‐intrusive data‐driven reduced order surrogate model

AbstractModeling and simulation of coronary artery disease (CAD) is of great importance for supporting and predicting the outcome of percutaneous coronary intervention (PCI). However, an in silico model generally requires heavy computational resources. An effective reduced order surrogate model is indispensable in this context. This study aims to develop a non‐intrusive data‐driven reduced order surrogate model for coronary in‐stent restenosis (ISR) incorporating anti‐inflammatory drugs embedded in the drug‐eluting stents. The constitutive model includes a detailed multiphysics approach based on partial differential equations (PDEs), which include descriptions of platelet aggregation, growth‐factor release, cellular motility and drug deposition. Dimensionality reduction is carried out based on a 3D convolutional autoencoder, which comprises an encoder and decoder. The former condenses the full‐order solution into a lower‐dimensional latent space, while the latter recovers the full solution from the latent space. Special attention is paid to handle the multidimensional outputs and network architecture.

An Effective PDE-based Thresholding for MRI Image Denoising and H-FCM-based Segmentation

Image denoising and segmentation play a crucial role in computer graphics and computer vision. A good image-denoising method must effectively remove noise while preserving important boundaries. Various image-denoising techniques have been employed to remove noise, but complete elimination is often impossible. In this paper, we utilize Partial Differential Equation (PDE) and generalised cross-validation (GCV) within Adaptive Haar Wavelet Transform algorithms to effectively denoise an image, with the digital image serving as the input. After denoising, the image is segmented using the Histon-related fuzzy c-means algorithm (H-FCM), with the processed image serving as the output. The proposed method is tested on images exposed to varying levels of noise. The performance of image denoising and segmentation techniques is evaluated using metrics such as Peak Signal-to-Noise Ratio (PSNR) of 77.42, Mean Squared Error (MSE) of 0.0011, and Structural Similarity Index (SSIM) of 0.7848. Additionally, segmentation performance is measured with a sensitivity of 99%, specificity of 98%, and an accuracy of 98%. The results demonstrate that the proposed methods outperform conventional approaches in these metrics. The implementation of the proposed methods is carried out on the MATLAB platform.

Identification of moment equations via data-driven approaches in nonlinear Schrödinger models

IntroductionThe moment quantities associated with the nonlinear Schrödinger equation offer important insights into the evolution dynamics of such dispersive wave partial differential equation (PDE) models. The effective dynamics of the moment quantities are amenable to both analytical and numerical treatments.MethodsIn this paper, we present a data-driven approach associated with the “Sparse Identification of Nonlinear Dynamics” (SINDy) to capture the evolution behaviors of such moment quantities numerically.Results and DiscussionOur method is applied first to some well-known closed systems of ordinary differential equations (ODEs) which describe the evolution dynamics of relevant moment quantities. Our examples are, progressively, of increasing complexity and our findings explore different choices within the SINDy library. We also consider the potential discovery of coordinate transformations that lead to moment system closure. Finally, we extend considerations to settings where a closed analytical form of the moment dynamics is not available.

Joint Effect of Velocity Slip and Joule Heating MHD Casson-Williamson Nanofluid Passes Through the Stretching Porous Medium

The objective of this study is to examine the heat and mass transport characteristics of a non-Newtonian Casson-Williamson nanofluid flow over a porous stretching sheet. The viscoelastic characteristic of a fluid is obtained by combining Casson and Williamson fluids. It is anticipated that the porous media through which the non-Newtonian fluid flows will adhere to Darcy's law. The effects of magnetic and electric fields are taken into account. The mathematical modeling of this physical problem involves a set of nonlinear partial differential equations that are mass, energy and momentum, together with corresponding boundary conditions, these PDEs are transformed into dimensionless ODE by appropriate similarity transformations and solved by R-K method. The numerical analysis is subsequently presented in a visual format to illustrate the influence of different controlling parameters on velocity, temperature, and concentration. Moreover, the analysis gives higher values of magnetic, viscous dissipation and joule heating parameters leads the temperature and Nusselt number. Conversely an increasing in mixed convection parameter result in a depreciation in the temperature. The skin-friction coefficient exhibited an upward trend with an increase in the porosity parameter. The rate of heat transfer demonstrated a rise under the Joule heating conditions. These findings are compared to other recorded results for a specific situation, then displayed graphically and analyzed in terms of engineering and industrial implications. Novelty this paper is by adding joule heating to nanofluid control over heat dissipation, thermal stability, or enhanced efficiency in heat exchange applications.

Controllability of some nonlocal‐impulsive Volterra evolution systems via measures of noncompactness

This study investigates the controllability of a Volterra evolution equation with impulsive terms and nonlocal initial conditions. With the aid of the resolvent operator generated by the linear part of the equation, mild solutions can be defined. Notably, the resolvent operator lacks compactness and equicontinuity. Additionally, the compactness of the impulsive and nonlocal functions is not required. Sufficient conditions for controllability are obtained through measures of noncompactness in Banach spaces. Functional differential equations and hyperbolic partial differential equations can be solved with these results. An example is given to illustrate the validity of the presented results.

Accelerating wavepacket propagation with machine learning.

In this work, we discuss the use of a recently introduced machine learning (ML) technique known as Fourier neural operators (FNO) as an efficient alternative to the traditional solution of the time-dependent Schrödinger equation (TDSE). FNOs are ML models which are employed in the approximated solution of partial differential equations. For a wavepacket propagating in an anharmonic potential and for a tunneling system, we show that the FNO approach can accurately and faithfully model wavepacket propagation via the density. Additionally, we demonstrate that FNOs can be a suitable replacement for traditional TDSE solvers in cases where the results of the quantum dynamical simulation are required repeatedly such as in the case of parameter optimization problems (e.g., control). The speed-up from the FNO method allows for its combination with the Markov-chain Monte Carlo approach in applications that involve solving inverse problems such as optimal and coherent laser control of the outcome of dynamical processes.