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Publications related to Partial Differential Equations (10,000)
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Many Hamiltonian systems can be recast in multi-symplectic form. We develop a reduced-order model (ROM) for multi-symplectic Hamiltonian partial differential equations (PDEs) that preserves the global energy. The full-order solutions are obtained by finite difference discretization in space and the global energy preserving average vector field (AVF...
Article
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Traditionally, partial differential equation (PDE) problems are solved numerically through a discretization process. Iterative methods are then used to determine the algebraic system generated by this process. Recently, scientists have emerged artificial neural networks (ANNs), which solve PDE problems without a discretization process. Therefore, i...
Article
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The question about behavior of gaps between zeros of polynomials under differentiation is classical and goes back to Marcel Riesz. Recently, Stefan Steinerberger [42] formally derived a nonlocal nonlinear partial differential equation which models dynamics of roots of polynomials under differentiation. In this paper, we connect rigorously solutions...
Article
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A problem of motion of a piezoelectric actuator in contact with an elasto-plastic obstacle is reformulated as a PDE in one spatial dimension with hysteresis in the bulk and on the contact boundary. The model is shown to dissipate energy in agreement with the principles of thermodynamics. The main result includes existence, uniqueness, and continuou...
Article
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This is a follow-up of our paper (Klainerman and Szeftel in Construction of GCM spheres in perturbations of Kerr, Accepted for publication in Annals of PDE) on the construction of general covariant modulated (GCM) spheres in perturbations of Kerr, which we expect to play a central role in establishing their nonlinear stability. We reformulate the m...
Article
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Nowadays, in the Scientific Machine Learning (SML) research field, the traditional machine learning (ML) tools and scientific computing approaches are fruitfully intersected for solving problems modelled by Partial Differential Equations (PDEs) in science and engineering applications. Challenging SML methodologies are the new computational paradigm...
Article
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Physics informed neural networks (PINNs) are capable of finding the solution for a given boundary value problem. Here, the training of the network is equivalent to the minimization of a loss function that includes the governing (partial differential) equations (PDE) as well as initial and boundary conditions. We employ several ideas from the finite...
Article
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We prove homogenization for a class of nonconvex (possibly degenerate) viscous Hamilton-Jacobi equations in stationary ergodic random environments in one space dimension. The results concern Hamiltonians of the form G(p)+V(x,ω), where the nonlinearity G is the minimum of two or more convex functions with the same absolute minimum, and the potential...
Preprint
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Most problems in uncertainty quantification, despite its ubiquitousness in scientific computing, applied mathematics and data science, remain formidable on a classical computer. For uncertainties that arise in partial differential equations (PDEs), large numbers M>>1 of samples are required to obtain accurate ensemble averages. This usually involve...
Article
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This paper deals with a control constrained optimization problem governed by a nonsmooth elliptic PDE in the presence of a non-differentiable objective. The nonsmooth non-linearity in the state equation is locally Lipschitz continuous and directionally differentiable, while one of the nonsmooth terms appearing in the objective is convex. Since thes...
Preprint
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In this paper, we demonstrate and investigate several challenges that stand in the way of tackling complex problems using physics-informed neural networks. In particular, we visualize the loss landscapes of trained models and perform sensitivity analysis of backpropagated gradients in the presence of physics. Our findings suggest that existing meth...
Article
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We analyse parallel overlapping Schwarz domain decomposition methods for the Helmholtz equation, where the exchange of information between subdomains is achieved using first-order absorbing (impedance) transmission conditions, together with a partition of unity. We provide a novel analysis of this method at the PDE level (without discretization). F...
Preprint
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Diameter estimates for K\"ahler metrics are established which require only an entropy bound and no lower bound on the Ricci curvature. The proof builds on recent PDE techniques for $L^\infty$ estimates for the Monge-Amp\`ere equation, with a key improvement allowing degeneracies of the volume form of codimension strictly greater than one. As a cons...
Preprint
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In this paper, an exponentially fitted finite difference method is developed to solve singularly perturbed delay parabolic partial differential equations having a large delay on the spatial variable with an integral boundary condition on the right side of the domain. The problem's solution exhibits an interior layer and a parabolic boundary layer o...
Article
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In this paper, we analyze the regularity in time of solutions for nonlinear fractional reaction–subdiffusion equations (FrRSEs). By constituting regularity estimates of resolvent operator and employing fixed point theorems, we establish some results on the global existence and regularity in time of solutions to FrRSE in two different cases of nonli...
Preprint
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The box-ball system (BBS) is a cellular automaton that is an ultradiscrete analogue of the Korteweg--de Vries equation, a non-linear PDE used to model water waves. In 2001, Hikami and Inoue generalised the BBS to the general linear Lie superalgebra $\mathfrak{gl}(m|n)$. We further generalise the Hikami--Inoue BBS to column tableaux using the Kirill...
Preprint
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This paper is concerned with an optimal control problem for a forward-backward stochastic differential equation (FBSDE, for short) with a recursive cost functional determined by a backward stochastic Volterra integral equation (BSVIE, for short). It is found that such an optimal control problem is time-inconsistent in general, even if the cost func...
Article
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Physics-informed neural network (PINN) models are developed in this work for solving highly anisotropic diffusion equations. Compared to traditional numerical discretization schemes such as the finite volume method and finite element method, PINN models are meshless and, therefore, have the advantage of imposing no constraint on the orientations of...
Preprint
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We compare the solutions of two systems of partial differential equations (PDE), seen as two different interpretations of the same model that describes formation of complex biological networks. Both approaches take into account the time evolution of the medium flowing through the network, and we compute the solution of an elliptic-parabolic PDE sys...
Article
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This paper addresses the numerical resolution of controllability problems for partial differential equations (PDEs) by using physics-informed neural networks. Error estimates for the generalization error for both state and control are derived from classical observability inequalities and energy estimates for the considered PDE. These error bounds,...
Preprint
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The intrusive (sample-free) spectral stochastic finite element method (SSFEM) is a powerful numerical tool for solving stochastic partial differential equations (PDEs). However, it is not widely adopted in academic and industrial applications because it demands intrusive adjustments in the PDE solver, which require substantial coding efforts compar...
Article
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The paper first provides an iterated quasi-interpolation scheme for function approximation over periodic domain and then attempts its applications to solve time-dependent surface partial differential equations (PDEs). The key feature of our scheme is that it gives an approximation directly just by taking a weighted average of the available discrete...
Preprint
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Classical-quantum hybrid algorithms have recently garnered significant attention, which are characterized by combining quantum and classical computing protocols to obtain readout from quantum circuits of interest. Recent progress due to Lubasch et al in a 2019 paper provides readout for solutions to the Schrodinger and Inviscid Burgers equations, b...
Article
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The topic of this paper is to study the numerical solution of the fourth-order partial differential equation and analyze its visual application in software simulation. Therefore, for the initial circular domain, the expansion should be transformed into a fourth-order problem in the plane dimension. Then, we introduced appropriate-weighted Sobolev s...
Preprint
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Neural diffusion on graphs is a novel class of graph neural networks that has attracted increasing attention recently. The capability of graph neural partial differential equations (PDEs) in addressing common hurdles of graph neural networks (GNNs), such as the problems of over-smoothing and bottlenecks, has been investigated but not their robustne...
Cover Page
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Partial differential equations (PDEs) have been applied successfully to formulate some dynamical phenomena in many engineering domains, since these equations model continuous change. Thus, in the last 35 years, PDEs have been used to solve many challenges in various image and video processing and analysis and computer vision areas, including image...
Article
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Dynamical systems often exhibit limit cycle oscillations (LCOs), self-sustaining oscillations of limited amplitude. LCOs can be supercritical or subcritical. The supercritical response is benign, while the subcritical response can be bi-stable and exhibit a hysteretic response. Subcritical responses can be avoided in design optimization by enforcin...
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In this paper, aiming at attitude control of asymmetric flexible spacecraft, the rigid-flexible coupled dynamic model is established, and a distributed control scheme for attitude maneuver and synchronous vibration suppression is proposed. First, based on the Hamiltonian principle, a distributed parameter model represented by coupled partial differ...
Preprint
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We introduce an agent-based model for co-evolving opinion and social dynamics, under the influence of multiplicative noise. In this model, every agent is characterized by a position in a social space and a continuous opinion state variable. Agents' movements are governed by positions and opinions of other agents and similarly, the opinion dynamics...
Article
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Invasive pest establishment is a pervasive threat to global ecosystems, agriculture, and public health. The recent establishment of the invasive spotted lanternfly in the northeastern United States has proven devastating to farms and vineyards, necessitating urgent development of population dynamical models and effective control practices. In this...
Presentation
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Artificial intelligence techniques are widely utilized in various engineering sectors thanks to the abundance of digital data, growing computing power, and advanced algorithms. A neural network as a data simulation method is considered a significantly valuable tool to approximate governing functions that map from the inputs to the outputs of a spec...
Preprint
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Least squares regression is a ubiquitous tool for building emulators (a.k.a. surrogate models) of problems across science and engineering for purposes such as design space exploration and uncertainty quantifi-cation. When the regression data are generated using an experimental design process (e.g., a quadrature grid) involving computationally expen...
Conference Paper
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Executable Digital Twins (xDTs) assist maintenance engineers by calculating current and future health status of physical assets based on operational conditions. In xDTs, the simulation describes the ideal physical behavior of the system, whereas the real data from sensors assist in updating the xDTs following the changes to the physical asset. We i...
Article
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Detuned resonance, that is, resonance with some nonzero frequency mismatch, is a topic of widespread multidisciplinary interest describing many physical, mechanical, biological, and other evolutionary dispersive PDE systems. In this paper, we attempt to introduce some systematic terminology to the field, and we also point out some counter-intuitive...
Article
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In this paper, the finite difference scheme of the spatiotemporal fractional convection-diffusion equation is established, and its stability and convergence are proved. Furthermore, this discrete technique is extended to solve nonlinear spatiotemporal fractional convection-diffusion equations. By using the Krylov subspace method to solve the discre...
Article
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Cyclic nucleotides (cAMP, cGMP) play a major role in normal and pathologic signaling. Beyond receptors, cyclic nucleotide phosphodiesterases; (PDEs) rapidly convert the cyclic nucleotide in its respective 5′-nucleotide to control intracellular cAMP and/or cGMP levels to maintain a normal physiological state. However, in many pathologies, dysregulat...
Article
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We consider the uniqueness of ground states for combined power-type nonlinear scalar field equations involving the Sobolev critical exponent at high frequencies. The uniqueness of ground states at high frequencies in five and higher dimensions has been proved in Akahori et al. (Calc. Var. Partial Differential Equations 58:32, 2019). Moreover, that...
Preprint
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Neurological studies show that injured neurons can regain their functionality with therapeutics such as Chondroitinase ABC (ChABC). These therapeutics promote axon elongation by manipulating the injured neuron and its intercellular space to modify tubulin protein concentration. This fundamental protein is the source of axon elongation, and its spat...
Article
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In this work, the fractional novel analytic method (FNAM) is successfully implemented on some well-known, strongly nonlinear fractional partial differential equations (NFPDEs), and the results show the approach’s efficiency. The main purpose is to show the method’s strength on FPDEs by minimizing the calculation effort. The novel numerical approach...
Preprint
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In this paper we investigate a generalisation of a Boltzmann mean field game (BMFG) for knowledge growth, originally introduced by the economists Lucas and Moll [23]. In BMFG the evolution of the agent density with respect to their knowledge level is described by a Boltzmann equation. Agents increase their knowledge through binary interactions with...
Article
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Singularly perturbed 2D parabolic delay differential equations with the discontinuous source term and convection coefficient are taken into consideration in this paper. For the time derivative, we use the fractional implicit Euler method, followed by the fitted finite difference method with bilinear interpolation for locally one-dimensional problem...
Article
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In this paper, we obtain existence and uniqueness of strong solutions to the inhomogenous Neumann initial-boundary problem for a parabolic PDE which arises as a generalization of the time-dependent minimal surface equation. Existence and regularity in time of the solution are proved by means of a suitable pseudoparabolic relaxed approximation of th...
Article
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We investigate the repeated averaging model for money exchanges: two agents picked uniformly at random share half of their wealth to each other. It is intuitively convincing that a Dirac distribution of wealth (centered at the initial average wealth) will be the long time equilibrium for this dynamics. In other words, the Gini index should converge...
Thesis
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In this study, Lie symmetry analysis is used to investigate invariance properties of some nonlinear time and time-space conformable fractional partial differential equations with two and three independent variables. The efficiency of the method is illustrated by its applications to conformable time and time-space fractional Korteweg-de Vries, modif...
Conference Paper
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In this article, using the simplicial Bernstein form of polynomials on a simplex, we provide positivity certificates for the approximate solution of a linear elliptic PDE obtained by simplicial Bernstein Bubnov-Galerkin method. Particularly, we show how to obtain a simplicial Bernstein certificate of positivity for the approximate solution, althoug...
Article
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This study addresses CuO-TiO2/CMC-water hybridnano-liquid in the influence of mixed convection flow and thermal radiative flow past a stretchable vertical surface. Cross nanofluid containing Titanium dioxide (TiO2), and Copper Oxide (CuO) are scattered in a base fluid of kind CMC water. In addition, theirreversibility analysis is also examined in t...
Article
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We study the diffusion equation with an appropriate change of variables. This equation is, in general, a partial differential equation (PDE). With the self-similar and related Ansatz, we transform the PDE of diffusion to an ordinary differential equation. The solutions of the PDE belong to a family of functions which are presented for the case of i...
Article
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Understanding the glassy nature of neural networks is pivotal both for theoretical and computational advances in Machine Learning and Theoretical Artificial Intelligence. Keeping the focus on dense associative Hebbian neural networks (i.e. Hopfield networks with polynomial interactions of even degree \(P >2\)), the purpose of this paper is twofold:...
Preprint
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In this paper, we have constructed two dimensional (2D ) hyperbolic quasi-linear partial differential equations (HQLPDE) by considering two cases. The Cauchy data is given in the form of exponential curve in the 1st case. The initial value problems (IVP) are mentioned as sin x in the 2nd case where as all the cases are associated with time. We have...
Preprint
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Partial differential equations (PDEs) see widespread use in sciences and engineering to describe simulation of physical processes as scalar and vector fields interacting and coevolving over time. Due to the computationally expensive nature of their standard solution methods, neural PDE surrogates have become an active research topic to accelerate t...
Preprint
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In this paper, we have constructed one dimensional (1D) hyperbolic quasi-linear partial differential equations (HQLPDE) considering three cases. In the first two cases, we have considered the Cauchy data in the form of exponential curve. The initial value problem (IVP) is mentioned as sin x in the third case. All the three cases are associated with...
Preprint
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In this work, we develop an efficient solver based on deep neural networks for the Poisson equation with variable coefficients and singular sources expressed by the Dirac delta function $\delta(\mathbf{x})$. This class of problems covers general point sources, line sources and point-line combinations, and has a broad range of practical applications...
Preprint
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Physics Informed Neural Networks (PINNs) are shown to be a promising method for the approximation of Partial Differential Equations (PDEs). PINNs approximate the PDE solution by minimizing physics-based loss functions over a given domain. Despite substantial progress in the application of PINNs to a range of problem classes, investigation of error...
Preprint
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Planning for the impacts of climate change requires accurate projections by Earth System Models (ESMs). ESMs, as developed by many research centres, estimate changes to weather and climate as atmospheric Greenhouse Gases (GHGs) rise, and they inform the influential Intergovernmental Panel on Climate Change (IPCC) reports. ESMs are advancing the und...
Article
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This paper presents an in-depth study and analysis of the mathematical control of space-based kinetic energy weapons and the evaluation of the damaging effect by partial differential equations. The spectral element discrete format of the optimal control problem is constructed, the a priori error estimate of the control problem solution is proved th...
Article
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We consider a system of charged particles moving on the real line driven by electrostatic interactions. Since we consider charges of both signs, collisions might occur in finite time. Upon collision, some of the colliding particles are effectively removed from the system (annihilation). The two applications we have in mind are vortices and dislocat...
Preprint
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Using Riemann-Hilbert problem methods, we establish a Tracy-Widom like formula for generating function of occupancy numbers of the Pearcey process. This formula is linked to a coupled vector differential equation of order 3. We also obtained a non linear coupled heat equation. Combining these two equations leads to a PDE for the logarithm of the Fr...
Article
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In this investigation, we perform a systematic computational search for potential singularities in 3D Navier–Stokes flows based on the Ladyzhenskaya–Prodi–Serrin conditions. They assert that if the quantity \(\int _0^T \Vert \textbf {u}(t) \Vert _{L^q(\Omega )}^p \, dt\), where \({2/p+3/q = 1}\), \(q > 3\), is bounded, then the solution \(\textbf {...
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Synthetically modified fluorescent nucleotides (SFNs) are highly popular in a variety of experiments to explore biochemistry in molecular imaging, but their photodynamics and quenching mechanisms remain relatively unstudied computationally. We combine various levels of theory, including classical force field dynamics and excited state QM/MM Born-Op...
Article
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A Hermite fitted block integrator (HFBI) for numerically solving second-order anisotropic elliptic partial differential equations (PDEs) was developed, analyzed, and implemented in this study. The method was derived through collocation and interpolation techniques using the Hermite polynomial as the basis function. The Hermite polynomial was interp...
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We propose a new method for the evaluation of intersection numbers for twisted meromorphic $n$-forms, through Stokes' theorem in $n$ dimensions. It is based on the solution of an $n$-th order partial differential equation and on the evaluation of multivariate residues. We also present an algebraic expression for the contribution from each multivari...
Article
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Distributional extensions of the benchmark AK endogenous growth model and of the Ramsey model are presented in this paper. The resulting geographic growth model - a forward-backward parabolic partial differential equation (PDE) over a bounded spatial domain - is governed by two main driving forces: a spatial friction in the real- location of physic...
Article
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An improved neural networks method based on domain decomposition is proposed to solve partial differential equations, which is an extension of the physics informed neural networks (PINNs). Although recent research has shown that PINNs perform effectively in solving partial differential equations, they still have difficulties in solving large-scale...
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In this paper, we study a learning problem in which a forecaster only observes partial information. By properly rescaling the problem, we heuristically derive a limiting PDE on Wasserstein space which characterizes the asymptotic behavior of the regret of the forecaster. Using a verification type argument, we show that the problem of obtaining regr...
Article
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Almost sophisticated physical phenomena and computational problems arise as variational problems. Recently, the development of neural networks (NNs), which has accomplished unbelievable success in many fields, especially in scientific computational fields. And almost sophisticated computational problems of physical phenomena can be viewed as a vari...
Article
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We consider operator preconditioning B−1A, which is employed in the numerical solution of boundary value problems. Here, the self-adjoint operators A,B:H01(Ω)→H−1(Ω) are the standard integral/functional representations of the partial differential operators −∇⋅ (k(x)∇u) and −∇⋅ (g(x)∇u), respectively, and the scalar coefficient functions k(x) and g(...