General Partial Differential Equations Research Articles

Mathematical modeling of insulating coating of thermal conductivity including body’s own radiation and non-local spatial effects

Abstract In this paper thermal conductivity analysis of insulating coating is discussed. The general partial differential thermal conductivity equation is used with consideration to nonlocal boundary effects. We relying on the Eringen’s approach involves heat-flow vector transformation using nonlocal influence function and weight coefficients. Also presented model includes own surface radiation of the coating.

A robust collocation method for time fractional PDEs based on mean value theorem and cubic B-splines

This paper explains and applies a numerical technique utilizing the cubic B-spline functions and the mean value theorem (MVT) to solve a general time fractional partial differential equation (FPDE). The MVT for integrals enables us to approximate the time fractional derivatives in an appropriate simple form. We use the cubic B-spline functions to construct the numerical solution and its spatial derivatives. The great advantage of our technique is that it enables us to approximate solutions of many time FPDEs for several choices of F(x, t, u, ux, uxx). Two numerical examples have been included to emphasize the accuracy and efficiency of the method. It is demonstrated that the numerical method is unconditionally stable by employing the Von Neumann method (VNM).

Quantum algorithms for nonlinear partial differential equations

Computations of nonlinear partial differential equations (PDEs) are some of the most important problems in science and engineering, but come with tremendous costs. Quantum algorithms have the potential to speed up these computations, but current methods are bound to weak or polynomial nonlinearity, short time of validity, or display no quantum advantage otherwise. We construct new quantum algorithms–based on level set methods – for Hamilton-Jacobi and scalar hyperbolic PDEs that generate algorithms with advantages with respect to critical numerical parameters, even for computing the physical observables. These algorithms hold for arbitrary nonlinearity for the equations under study and are valid globally in time. In addition, we also introduce a new quantum encoding, called level set encoding, that allows the efficient computation of physical observables. These PDEs are important for many applications like geometric optics, semi-classical limit of the Schrödinger equations, and high-frequency limit of symmetric linear hyperbolic systems (for example the elastic wave and Maxwell equations). It can display up to exponential advantage –depending on the initial conditions and domain of the relevant observables– in both the dimension of the PDE and the error in computing its observables with respect to classical finite difference methods for these PDEs. Compared to classical finite difference methods, our algorithm also has an advantage with respect to the number of initial data, which has potential important applications in Monte Carlo simulations and uncertainty quantification. For more general PDEs, while an advantage with respect to a very large number of initial data is possible, this is mostly unrealistic for problems of practical interest, and new methods are required.

Sparse Cholesky factorization for solving nonlinear PDEs via Gaussian processes

In recent years, there has been widespread adoption of machine learning-based approaches to automate the solving of partial differential equations (PDEs). Among these approaches, Gaussian processes (GPs) and kernel methods have garnered considerable interest due to their flexibility, robust theoretical guarantees, and close ties to traditional methods. They can transform the solving of general nonlinear PDEs into solving quadratic optimization problems with nonlinear, PDE-induced constraints. However, the complexity bottleneck lies in computing with dense kernel matrices obtained from pointwise evaluations of the covariance kernel, and its partial derivatives, a result of the PDE constraint and for which fast algorithms are scarce. The primary goal of this paper is to provide a near-linear complexity algorithm for working with such kernel matrices. We present a sparse Cholesky factorization algorithm for these matrices based on the near-sparsity of the Cholesky factor under a novel ordering of pointwise and derivative measurements. The near-sparsity is rigorously justified by directly connecting the factor to GP regression and exponential decay of basis functions in numerical homogenization. We then employ the Vecchia approximation of GPs, which is optimal in the Kullback-Leibler divergence, to compute the approximate factor. This enables us to compute ϵ \epsilon -approximate inverse Cholesky factors of the kernel matrices with complexity O ( N log d ⁡ ( N / ϵ ) ) O(N\log ^d(N/\epsilon )) in space and O ( N log 2 d ⁡ ( N / ϵ ) ) O(N\log ^{2d}(N/\epsilon )) in time. We integrate sparse Cholesky factorizations into optimization algorithms to obtain fast solvers of the nonlinear PDE. We numerically illustrate our algorithm’s near-linear space/time complexity for a broad class of nonlinear PDEs such as the nonlinear elliptic, Burgers, and Monge-Ampère equations. In summary, we provide a fast, scalable, and accurate method for solving general PDEs with GPs and kernel methods.

A gradient-enhanced physics-informed neural networks method for the wave equation

Physics-informed neural networks (PINNs) have been proven to be a useful tool for solving general partial differential equations (PDEs), which is meshless and dimensionally free compared with traditional numerical solvers. Based on PINNs, gradient-enhanced physics-informed neural networks (gPINNs) add the partial derivative loss term of the independent variable and the physical constraint term, which improves the accuracy of network training. In this paper, the gPINNs method is proposed to solve the wave problem. The equation boundary value of the wave problem is added in the network construction, thus the network is forced to satisfy the equation boundary value during each training process. Two examples of wave equations are given in our paper, which are solved numerically by PINNs and gPINNs, respectively. It is found that using fewer data sets, gPINNs can learn data features more fully than PINNs and better results of gPINNs can be obtained.

Tackling the curse of dimensionality with physics-informed neural networks

The curse-of-dimensionality taxes computational resources heavily with exponentially increasing computational cost as the dimension increases. This poses great challenges in solving high-dimensional partial differential equations (PDEs), as Richard E. Bellman first pointed out over 60 years ago. While there has been some recent success in solving numerical PDEs in high dimensions, such computations are prohibitively expensive, and true scaling of general nonlinear PDEs to high dimensions has never been achieved. We develop a new method of scaling up physics-informed neural networks (PINNs) to solve arbitrary high-dimensional PDEs. The new method, called Stochastic Dimension Gradient Descent (SDGD), decomposes a gradient of PDEs’ and PINNs’ residual into pieces corresponding to different dimensions and randomly samples a subset of these dimensional pieces in each iteration of training PINNs. We prove theoretically the convergence and other desired properties of the proposed method. We demonstrate in various diverse tests that the proposed method can solve many notoriously hard high-dimensional PDEs, including the Hamilton–Jacobi-Bellman (HJB) and the Schrödinger equations in tens of thousands of dimensions very fast on a single GPU using the PINNs mesh-free approach. Notably, we solve nonlinear PDEs with nontrivial, anisotropic, and inseparable solutions in less than one hour for 1000 dimensions and in 12 h for 100,000 dimensions on a single GPU using SDGD with PINNs. Since SDGD is a general training methodology of PINNs, it can be applied to any current and future variants of PINNs to scale them up for arbitrary high-dimensional PDEs.

Structural identifiability analysis of linear reaction–advection–diffusion processes in mathematical biology

Effective application of mathematical models to interpret biological data and make accurate predictions often requires that model parameters are identifiable. Approaches to assess the so-called structural identifiability of models are well established for ordinary differential equation models, yet there are no commonly adopted approaches that can be applied to assess the structural identifiability of the partial differential equation (PDE) models that are requisite to capture spatial features inherent to many phenomena. The differential algebra approach to structural identifiability has recently been demonstrated to be applicable to several specific PDE models. In this brief article, we present general methodology for performing structural identifiability analysis on partially observed reaction–advection–diffusion PDE models that are linear in the unobserved quantities. We show that the differential algebra approach can always, in theory, be applied to such models. Moreover, despite the perceived complexity introduced by the addition of advection and diffusion terms, consideration of spatial analogues of non-spatial models cannot exacerbate structural identifiability. We conclude by discussing future possibilities and the computational cost of performing structural identifiability analysis on more general PDE models.

How Can We Use Mathematical Modeling of Amyloid-β in Alzheimer's Disease Research and Clinical Practices?

The accumulation of amyloid-β (Aβ) plaques in the brain is considered a hallmark of Alzheimer's disease (AD). Mathematical modeling, capable of predicting the motion and accumulation of Aβ, has obtained increasing interest as a potential alternative to aid the diagnosis of AD and predict disease prognosis. These mathematical models have provided insights into the pathogenesis and progression of AD that are difficult to obtain through experimental studies alone. Mathematical modeling can also simulate the effects of therapeutics on brain Aβ levels, thereby holding potential for drug efficacy simulation and the optimization of personalized treatment approaches. In this review, we provide an overview of the mathematical models that have been used to simulate brain levels of Aβ (oligomers, protofibrils, and/or plaques). We classify the models into five categories: the general ordinary differential equation models, the general partial differential equation models, the network models, the linear optimal ordinary differential equation models, and the modified partial differential equation models (i.e., Smoluchowski equation models). The assumptions, advantages and limitations of these models are discussed. Given the popularity of using the Smoluchowski equation models to simulate brain levels of Aβ, our review summarizes the history and major advancements in these models (e.g., their application to predict the onset of AD and their combined use with network models). This review is intended to bring mathematical modeling to the attention of more scientists and clinical researchers working on AD to promote cross-disciplinary research.

Some new methods for studying boundary value problems for general partial differential equations

Рассматриваются методы изучения граничных задач для линейных дифференциальных уравнений с частными производными в области независимо от типа уравнения. Предлагается несколько методов изучения граничных задач, которые, как правило, основываются на формуле Грина. Этим методам были посвящены ранее вышедшие публикации автора, а в настоящей статье эти результаты представлены в собранном виде и в краткой форме.

Some new methods for studying boundary value problems for general partial differential equations

Some new methods for studying boundary value problems for general partial differential equations

Existence of solutions of fractal fractional partial differential equations through different contractions

<abstract><p>In the past, the existence and uniqueness of the solutions of fractional differential equations have been investigated by many researchers theoretically in various approaches in the literature. In this paper, there is no discussion of the existence of solutions for the nonlinear differential equations with fractal fractional operators. The objective of this work is to present novel contraction approaches, notably the $ \varpropto $-$ \psi $-contraction $ \varpropto $-type of the $ \tilde{\texttt{F}} $-contraction, within the context of $ \hat{F} $-metric and orbital metric spaces. The aim of this study is to illustrate certain fixed point theorems that offer a new and direct approach to establish the existence and uniqueness of the solution to the general partial differential equations by employing the fractal fractional operators.</p></abstract>

Is the neural tangent kernel of PINNs deep learning general partial differential equations always convergent?

In this paper, we study the neural tangent kernel (NTK) for general partial differential equations (PDEs) based on physics-informed neural networks (PINNs). As we all know, the training of an artificial neural network can be converted to the evolution of NTK. We analyze the initialization of NTK and the convergence conditions of NTK during training for general PDEs. The theoretical results show that the homogeneity of differential operators plays a crucial role for the convergence of NTK. Moreover, based on the PINNs, we validate the convergence conditions of NTK using the initial value problems of the sine–Gordon equation and the initial–boundary value problem of the KdV equation.

Convergent evolving finite element approximations of boundary evolution under shape gradient flow

Abstract As a specific type of shape gradient descent algorithm, shape gradient flow is widely used for shape optimization problems constrained by partial differential equations. In this approach, the constraint partial differential equations could be solved by finite element methods on a domain with a solution-driven evolving boundary. Rigorous analysis for the stability and convergence of such finite element approximations is still missing from the literature due to the complex nonlinear dependence of the boundary evolution on the solution. In this article, rigorous analysis of numerical approximations to the evolution of the boundary in a prototypical shape gradient flow is addressed. First-order convergence in time and $k$th order convergence in space for finite elements of degree $k\geqslant 2$ are proved for a linearly semi-implicit evolving finite element algorithm up to a given time. The theoretical analysis is consistent with the numerical experiments, which also illustrate the effectiveness of the proposed method in simulating two- and three-dimensional boundary evolution under shape gradient flow. The extension of the formulation, algorithm and analysis to more general shape density functions and constraint partial differential equations is also discussed.

Polynomial inclusions: Definitions, applications, and open problems

Predictive modeling in physical science and engineering is mostly based on solving certain partial differential equations where the complexity of solutions is dictated by the geometry of the domain. Motivated by the broad applications of explicit solutions for spherical and ellipsoidal domains, in particular, the Eshelby’s solution in elasticity, we propose a generalization of ellipsoidal shapes called polynomial inclusions. A polynomial inclusion (or p-inclusion for brevity) of degree k is defined as a smooth, connected and bounded body whose Newtonian potential is a polynomial of degree k inside the body. From this viewpoint, ellipsoids are identified as the only p-inclusions of degree two; many fundamental problems in various physical settings admit simple closed-form solutions for general p-inclusions as for ellipsoids. Therefore, we anticipate that p-inclusions will be useful for applications including predictive materials models, optimal designs, and inverse problems. However, the existence of p-inclusions beyond degree two is not obvious, not to mention their explicit algebraic parameterizations.In this work, we explore alternative definitions and properties of p-inclusions in the context of potential theory. Based on the theory of variational inequalities, we show that p-inclusions do exist for certain polynomials, though a complete characterization remains open. We reformulate the determination of surfaces of p-inclusions as nonlocal geometric flows which are convenient for numerical simulations and studying geometric properties of p-inclusions. In two dimensions, by the method of conformal mapping we find an explicit algebraic parameterization of p-inclusions. We also propose a few open problems whose solution will deepen our understanding of relations between domain geometry, Newtonian potentials, and solutions to general partial differential equations. We conclude by presenting examples of applications of p-inclusions in the context of Eshelby inclusion problems and magnet designs.

Neural network approximation of optimal controls for stochastic reaction-diffusion equations.

We present a numerical algorithm that allows the approximation of optimal controls for stochastic reaction-diffusion equations with additive noise by first reducing the problem to controls of feedback form and then approximating the feedback function using finitely based approximations. Using structural assumptions on the finitely based approximations, rates for the approximation error of the cost can be obtained. Our algorithm significantly reduces the computational complexity of finding controls with asymptotically optimal cost. Numerical experiments using artificial neural networks as well as radial basis function networks illustrate the performance of our algorithm. Our approach can also be applied to stochastic control problems for high dimensional stochastic differential equations and more general stochastic partial differential equations.

Seq-SVF: An unsupervised data-driven method for automatically identifying hidden governing equations

In this work, an unsupervised data-driven method based on sequential singular value filtering (Seq-SVF) is proposed to simultaneously identify multiple partial differential equations from observed data considering potential noises. This method is aimed to extend the Sparse Identification of Nonlinear Dynamics (SINDy) to the identification of general nonlinear partial differential equations by transforming the paradigm based on regression to an unsupervised paradigm. To discover the complex coupled equations of vector or tensor forms without prior knowledge, the techniques of singular value decomposition (SVD) and strong rank-revealing QR factorization (sRRQR) are applied to the data matrix, which ensures that the method can automatically identify the number and the corresponding linearly independent terms as the left-hand terms of governing equations. To balance the complexity and the precision of modeling, a strategy for filtering singular values is designed to determine the sparse structure of governing equations from a large number of nonlinear basis functions. We show the success of the method to extract explicit and succinct models from many complex linear, nonlinear, and multiphysics mechanical systems, and the examples show more accuracy compared with using traditional sparse learning methods.

A New Complex Double Integral Transform and It’s Applications in Partial Differential Equations

In this paper, we present a new complex double integral transform namely “Complex Double Sadik Transform”, to solve general linear partial differential equations. Several functions are used (applied) to show the usefulness of this new double transformation.

Strong convergence of a fractional exponential integrator scheme for finite element discretization of time-fractional SPDE driven by fractional and standard Brownian motions

The aim of this work is to provide the first strong convergence result of a numerical approximation of a general time-fractional second order stochastic partial differential equation involving a Caputo derivative in time of order α∈(12,1) and driven simultaneously by a multiplicative standard Brownian motion and additive fBm with Hurst parameter H∈(12,1), more realistic to model the random effects on transport of particles in medium with thermal memory. We prove the existence and uniqueness results, and perform the spatial discretization using the standard finite element and the temporal discretization based on a generalized exponential time differencing method (GETD). We provide the temporal and spatial convergence proofs for our fully discrete scheme and the result shows that the convergence orders depend on the regularity of the initial data, the power of the fractional derivative, and the Hurst parameter H. Numerical results are provided to illustrate our theoretical results.

Adaptive FEM with quasi-optimal overall cost for nonsymmetric linear elliptic PDEs

Abstract We consider a general nonsymmetric second-order linear elliptic partial differential equation in the framework of the Lax–Milgram lemma. We formulate and analyze an adaptive finite element algorithm with arbitrary polynomial degree that steers the adaptive meshrefinement and the inexact iterative solution of the arising linear systems. More precisely, the iterative solver employs, as an outer loop, the so-called Zarantonello iteration to symmetrize the system and, as an inner loop, a uniformly contractive algebraic solver, for example, an optimally preconditioned conjugate gradient method or an optimal geometric multigrid algorithm. We prove that the proposed inexact adaptive iteratively symmetrized finite element method leads to full linear convergence and, for sufficiently small adaptivity parameters, to optimal convergence rates with respect to the overall computational cost, i.e., the total computational time. Numerical experiments underline the theory.

Hyers-Ulam stability of a general linear partial differential equation

In this paper we will study Hyers-Ulam stability for a general linear partial differential equation of first order in a Banach space.