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Partial Differential Equations - Science topic
Explore the latest publications in Partial Differential Equations, and find Partial Differential Equations experts.
Publications related to Partial Differential Equations (10,000)
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Multiphysics simulation aims to predict and understand interactions between multiple physical phenomena, aiding in comprehending natural processes and guiding engineering design. The system of Partial Differential Equations (PDEs) is crucial for representing these physical fields, and solving these PDEs is fundamental to such simulations. However,...
Symmetric coin cell cycling is an important tool for the analysis of battery materials, enabling the study of electrode/electrolyte systems under realistic operating conditions. In the case of metal lithium SEI growth and shape changes, cycling studies are especially important to assess the impact of the alternation of anodic–cathodic polarization...
In this paper we consider the estimation of unknown parameters in Bayesian inverse problems. In most cases of practical interest, there are several barriers to performing such estimation, This includes a numerical approximation of a solution of a differential equation and, even if exact solutions are available, an analytical intractability of the m...
This paper addresses the challenge of establishing rigorous error bounds for zero-trace Rectified Linear Unit (ReLU) Neural Networks (NNs). We derive theoretical results to provide insights into the accuracy of these networks in approximating continuous functions, focusing on the influence of network architecture, such as the number of layers and n...
Deep learning has shown promise in solving partial differential equations (PDEs) in computational fluid dynamics, particularly for enhancing solutions from coarse-mesh simulations. However, integrating deep learning with traditional PDE solvers requires these solvers to support automatic differentiation, a feature often unavailable in existing blac...
Physics-Informed Neural Networks (PINNs) have emerged as a promising approach for solving Partial Differential Equations (PDEs) by incorporating physical constraints into deep learning models. However, standard PINNs often require a large number of training samples to achieve high accuracy, leading to increased computational costs. To address this...
This paper addresses the portfolio optimisation problem within the jump-diffusion stochastic differential equations (SDEs) framework. We begin by recalling a fundamental theoretical result concerning the existence of solutions to the Black–Scholes–Merton partial differential equation (PDE), which serves as the cornerstone for subsequent analysis. T...
The Burgers hierarchy consists of nonlinear evolutionary partial differential equations (PDEs) with progressively higher-order dispersive and nonlinear terms. Notable members of this hierarchy are the Burgers equation and the Sharma-Tasso-Olver equation, which are widely applied in fields such as plasma physics, fluid mechanics, optics, and biophys...
This article investigates the non-stationary reaction-diffusion-advection equation, emphasizing solutions with internal layers and the associated inverse problems. We examine a nonlinear singularly perturbed partial differential equation (PDE) within a bounded spatial domain and an infinite temporal domain, subject to periodic temporal boundary con...
We give an alternative proof of the Schoen–Simon–Yau curvature estimates and associated Bernstein-type theorems (Schoen et al. in Acta Math. 134:275–288, 1975), and extend the original result by including the case of 6-dimensional (stable minimal) immersions. The key step is an ε-regularity theorem, that assumes smallness of the scale-invariant \do...
The application of deep learning for partial differential equation (PDE)-constrained control is gaining increasing attention. However, existing methods rarely consider safety requirements crucial in real-world applications. To address this limitation, we propose Safe Diffusion Models for PDE Control (SafeDiffCon), which introduce the uncertainty qu...
This article deals with systems of ordinary differential equations (ODEs). New differential integrals are found for general ODEs systems. With the help of them, the Cauchy problem for an ODEs system is reduced to a Cauchy problem for a single linear PDE of the first order, and as well transformed to a Cauchy problem for an overdetermined universal...
p dir="ltr"> The availability of high quality image data urges the development of new conceptual techniques to realise cancer accurate prognostic and treatment. In this study we propose a boundary value problem to model tumor elasticity impacting various types of mechanical stresses on cancer cell properties, assuming isotropic and inhomogeneous co...
This article discusses integrals for systems of partial differential equations. A new
method for finding integrals for arbitrary PDE systems is proposed. It is shown that,
if they exist, then their appearance is ultimately determined from the solutions of
parametric ordinary differential equations. And they can also be found in analytical
form from...
In this paper, we address the exact solution of the general first-order systems of linear hyperbolic partial differential equations using the Fourier transformation technique. This transformation converts the system from the physical domain into a system of first-order ordinary differential equations in the frequency domain. Utilizing this method,...
This talk concerns the efficiency of algorithms and their implementation. When developing a new solver, your goal will naturally be to create an efficient method. This is alone because your prospects for publication would be greatly diminished if you were to advertise your new method as inherently inefficient. Efficiency, therefore, is central to r...
The Korteweg de Vries-Burgers (KdV-B) (1+1) equation
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{\partial \psi }{\partial t} +a \, \psi \frac{...
Although deep models have been widely explored in solving partial differential equations (PDEs), previous works are primarily limited to data only with up to tens of thousands of mesh points, far from the million-point scale required by industrial simulations that involve complex geometries. In the spirit of advancing neural PDE solvers to real ind...
Multi-task learning through composite loss functions is fundamental to modern deep learning, yet optimizing competing objectives remains challenging. We present new theoretical and practical approaches for addressing directional conflicts between loss terms, demonstrating their effectiveness in physics-informed neural networks (PINNs) where such co...
In this article, a new way was found to unify large classes of systems of differential
equations by introducing additional parameters-variables, which also allows to
overdetermine these new unified systems of equations. The Cauchy problem for the
original PDE systems can be solved in a general form by solving the special Cauchy
problem for the univ...
In this note we provide a new and efficient approach to uniform estimates for solutions to complex Monge-Ampere equations, as well as for solutions to geometric PDE's that satisfy a determinantal majorization.
In this article, by introducing additional unknowns and deriving new differential
relations from the a priori existence of integrals in the original PDE system, a new
method for overdetermination of PDE systems of a general form is proposed. Via
the Cauchy problem, additional parameters and new unknowns are introduced, and
the original system of eq...
In this paper, we study a nonlinearly coupled initial-boundary value problem describing the evolution of brain tumor growth including lactate metabolism. In our modeling approach, we also take into account the viscoelastic properties of the tissues as well as the reversible damage effects that could occur, possibly caused by surgery. After introduc...
In this paper, ODE systems of normal form are considered. It is shown that the
Cauchy problem for these systems of equations, if it has a unique solution, can be
reduced to the Cauchy problem for systems of universal equations, but with a larger
dimension (more variables). It is enough to solve the special Cauchy problem for a
universal overdetermi...
Photovoltaic (PV) power is generated by two common types of solar components that are primarily affected by fluctuations and development in cloud structures as a result of uncertain and chaotic processes. Local PV forecasting is unavoidable in supply and load planning necessary in integration of smart systems into electrical grids. Intra- or day-ah...
We propose a new financial model called the generalized fractional Brownian motion Heston exponential Hull–White model, which has stochastic volatility and interest rate, long memory, and heavy tail distribution. Based on the market price of the volatility and delta hedging strategies, we propose a partial differential equation (PDE) to obtain the...
The present paper highlights an exotic universe having a metric which is partially suggested by the metric of an universe without time seen in [2]. The new metric is globally defined and depends on an exotic matter created by some classical waves. The massless scalar field (MSF) of this universe is also analyzed. Its existence depends on some const...
Idiopathic pulmonary fibrosis (IPF) is a chronic lung disease characterized by excessive scarring and fibrosis due to the abnormal accumulation of extracellular matrix components, primarily collagen. This study aims to design and solve an optimal control problem to regulate M2 macrophage activity in IPF, thereby preventing fibrosis formation by con...
We introduce the Neural Preconditioning Operator (NPO), a novel approach designed to accelerate Krylov solvers in solving large, sparse linear systems derived from partial differential equations (PDEs). Unlike classical preconditioners that often require extensive tuning and struggle to generalize across different meshes or parameters, NPO employs...
Installment options, as path‐dependent contingent claims, involve paying the premium discretely or continuously in installments, rather than as a lump sum at the time of purchase. In this paper, we applied the PDE approach to price European continuous‐installment option and consider Heston stochastic volatility model for the dynamics of the underly...
While recent AI-for-math has made strides in pure mathematics, areas of applied mathematics, particularly PDEs, remain underexplored despite their significant real-world applications. We present PDE-Controller, a framework that enables large language models (LLMs) to control systems governed by partial differential equations (PDEs). Our approach en...
This paper extends a Finite Difference model reduction method to the Euler-Bernoulli beam equation with fully clamped boundary conditions. The corresponding partial differential equation (PDE) is exactly observable in the energy space with a single boundary observer in arbitrarily short observation times. However, standard Finite Difference spatial...
Experience replay is a foundational technique in reinforcement learning that enhances learning stability by storing past experiences in a replay buffer and reusing them during training. Despite its practical success, its theoretical properties remain underexplored. In this paper, we present a theoretical framework that models experience replay usin...
This book is devoted on some recent investigations of some classes partial differential equations in Sobolev and analytic spaces. The book contains twelve chapters.
Chapter 1 is entirely devoted to the presentation of definitions and results necessary as a result of this work. We first recall a few basic results on the linear, metric,
normed and Ba...
The construction of invariant solutions is a key application of Lie symmetry analysis in studying partial differential equations. The generalised double reduction method, which uses both symmetries and conservation laws of a PDE or system of PDEs, provides a powerful framework for constructing such solutions. This paper contributes to the applicati...
The dynamic partial differential equation (PDE) model governing longitudinal oscillations in magnetizable piezoelectric beams exhibits exponentially stable solutions when subjected to two boundary state feedback controllers. An analytically established exponential decay rate by the Lyapunov approach ensures rapid stabilization of the system to equi...
Physics-informed neural operators (PINOs) have emerged as powerful tools for learning solution operators of partial differential equations (PDEs). Recent research has demonstrated that incorporating Lie point symmetry information can significantly enhance the training efficiency of PINOs, primarily through techniques like data, architecture, and lo...
Learning the solution operators of PDEs on arbitrary domains is challenging due to the diversity of possible domain shapes, in addition to the often intricate underlying physics. We propose an end-to-end graph neural network (GNN) based neural operator to learn PDE solution operators from data on point clouds in arbitrary domains. Our multi-scale m...
Physics-Informed Neural Networks (PINNs) are a kind of deep-learning-based numerical solvers for partial differential equations (PDEs). Existing PINNs often suffer from failure modes of being unable to propagate patterns of initial conditions. We discover that these failure modes are caused by the simplicity bias of neural networks and the mismatch...
In this paper we study the dynamics of a fast-slow Fokker-Planck partial differential equation (PDE) viewed as the evolution equation for the density of a multiscale planar stochastic differential equation (SDE). Our key focus is on the existence of a slow manifold on the PDE level, which is a crucial tool from the geometric singular perturbation t...
XNet is a single-layer neural network architecture that leverages Cauchy integral-based activation functions for high-order function approximation. Through theoretical analysis, we show that the Cauchy activation functions used in XNet can achieve arbitrary-order polynomial convergence, fundamentally outperforming traditional MLPs and Kolmogorov-Ar...
This paper presents an implicit solution formula for the Hamilton-Jacobi partial differential equation (HJ PDE). The formula is derived using the method of characteristics and is shown to coincide with the Hopf and Lax formulas in the case where either the Hamiltonian or the initial function is convex. It provides a simple and efficient numerical a...
In this article, we proposed a single layer feedforward polynomial neural network for numerically solving the passport option pricing problem. The passport option can be valued by solving a nonlinear backward pricing PDE and it is difficult to find analytical solution of this PDE. Laguerre, Hermite and Legendre polynomials are employed separately a...
We introduce BCAT, a PDE foundation model designed for autoregressive prediction of solutions to two dimensional fluid dynamics problems. Our approach uses a block causal transformer architecture to model next frame predictions, leveraging previous frames as contextual priors rather than relying solely on sub-frames or pixel-based inputs commonly u...
Resumo O alcance de pedestres é um dos principais elementos a se considerar na definição da abrangência das áreas de adensamento urbano a partir de uma centralidade. O objetivo do artigo é analisar a definição da abrangência de áreas de adensamento urbano no contexto do Desenvolvimento Orientado ao Transporte (DOT), mediante aplicação de um raio si...
Differential equation models are crucial to scientific processes across many disciplines, and the values of model parameters are important for analyzing the behaviour of solutions. Identifying these values is known as a parameter estimation, a type of inverse problem, which has applications in areas that include industry, finance and biomedicine. A...
Autoregressive and recurrent networks have achieved remarkable progress across various fields, from weather forecasting to molecular generation and Large Language Models. Despite their strong predictive capabilities, these models lack a rigorous framework for addressing uncertainty, which is key in scientific applications such as PDE solving, molec...
This talk presents research on massively parallel solution algorithms for elliptic PDE. An old 1982 publication reminds us that multigrid methods can solve the Poisson equation with only 30 N FLOPS on a grid with N unknowns. They can thus produce the solution not only with asymp- totically optimal complexity, but also with a surprisingly low absolu...
Many physics-informed machine learning methods for PDE-based problems rely on Gaussian processes (GPs) or neural networks (NNs). However, both face limitations when data are scarce and the dimensionality is high. Although GPs are known for their robust uncertainty quantification in low-dimensional settings, their computational complexity becomes pr...
With the advancement of machine learning and deep learning, Physics-Informed Neural Networks (PINNs) have emerged as a prominent approach for solving partial differential equation (PDE) problems. In this article, we introduce a novel distillation framework specifically designed for PINNs, termed Self-Knowledge Distillation for PINNs (SKD-PINNs). Wi...
Poisson’s equation is an elliptic partial
differential equation of broad utility in
theoretical physics. For example, the solution to
poisons’ equation is the potential field caused
by a given electric charge or mass density
distribution; with the potential field known, one
can then calculate electrostatic or gravitational
force field.
In t...
We propose a method for obtaining statistically guaranteed confidence bands for functional machine learning techniques: surrogate models which map between function spaces, motivated by the need build reliable PDE emulators. The method constructs nested confidence sets on a low-dimensional representation (an SVD) of the surrogate model's prediction...
ICH Q3D (R2): Guidelines for Elemental Impurities
The ICH Q3D (R2) Guidelines are a critical framework established by the International Council for Harmonisation of Technical Requirements for Pharmaceuticals for Human Use (ICH) to regulate and control elemental impurities in pharmaceutical products. These guidelines ensure that elemental impurities...
Transformers, which are state-of-the-art in most machine learning tasks, represent the data as sequences of vectors called tokens. This representation is then exploited by the attention function, which learns dependencies between tokens and is key to the success of Transformers. However, the iterative application of attention across layers induces...
PDE-based group convolutional neural networks (PDE-G-CNNs) use solvers of evolution PDEs as substitutes for the conventional components in G-CNNs. PDE-G-CNNs can offer several benefits simultaneously: fewer parameters, inherent equivariance, better accuracy, and data efficiency. In this article, we focus on Euclidean equivariant PDE-G-CNNs where th...
We consider partial differential equations (PDEs) characterized by an upper barrier that depends on the solution itself and a fixed lower barrier, while accommodating a non-local driver. First, we show a Feynman–Kac representation for the PDE when the driver is local. Specifically, we relate the non-linear Snell envelope, arising from an optimal st...
We introduce a new technique for the fields of harmonic analysis and PDE, simultaneous saturation. Simultaneous saturation is a framework for controlling the size of a set where each element of the set is large. In this paper we apply this framework to multilinear restriction type estimates. Here the elements of the set are $Tf(p)$ where $T$ is an...
ICH Q3C (R9) – Guidelines on Residual Solvents in Pharmaceuticals
Overview:
The International Council for Harmonisation of Technical Requirements for Pharmaceuticals for Human Use (ICH) developed the ICH Q3C guideline to establish permissible limits for residual solvents in pharmaceutical products. Residual solvents are volatile organic chemicals t...
Many physical systems are represented by Partial Differential Equations (PDEs), and the study of chaotic dynamics in these systems is interesting and challenging. In this paper, the Li–Yorke chaos of PDEs is studied, and the Li–Yorke chaos is observable in several classes of PDEs, including systems with or without energy injection. For the PDEs wit...
Diffusion models have become the de facto framework for generating new datasets. The core of these models lies in the ability to reverse a diffusion process in time. The goal of this manuscript is to explain, from a PDE perspective, how this method works and how to derive the PDE governing the reverse dynamics as well as to study its solution analy...
The main goal of this article is to establish H\"older stability estimates for the Calder\'on problem related to a relativistic wave equation. The principal novelty of this article is that the partial differential equation (PDE) under consideration depends on three unknown potentials, namely a temporal dissipative potential $A_0$, a spatial vector...
Many inverse problems are naturally formulated as a PDE-constrained optimization problem. These non-linear, large-scale, constrained optimization problems know many challenges, of which the inherent non-linearity of the problem is an important one. In this paper, we focus on a relaxed formulation of the PDE-constrained optimization problem and prov...
In this paper, we extend the classical SIRS (Susceptible-Infectious-Recovered-Susceptible) model from mathematical epidemiology by incorporating a vaccinated compartment, V, accounting for an imperfect vaccine with waning efficacy over time. The SIRSV-model divides the population into four compartments and introduces periodic re-vaccination for wan...
On the occasion of the 80th birthday of Paolo Emilio Ricci, we highlight his important contributions to our Antwerp-Rome-Tbilisi program. It concerns 1) the geometric and analytic description of domains, 2) natural curvature conditions (the interplay between extrinsic and intrinsic characteristics), and 3) Solutions and behavior of solutions BVP of...
The definition of a non-trivial space of generalized functions of a complex variable allowing to consider derivatives of continuous functions is a non-obvious task, e.g. because of Morera theorem, because distributional Cauchy-Riemann equations implies holomorphicity and of course because including Dirac delta seems incompatible with the identity t...
Multiple time scales problems are investigated by combining geometrical and analytical approaches. More precisely, for fast-slow reaction-diffusion systems, we first prove the existence of slow manifolds for the abstract problem under the assumption that cross diffusion is small. This is done by extending the Fenichel theory to an infinite dimensio...
We solve the global asymptotic stability problem of an unstable reaction-diffusion Partial Differential Equation (PDE) subject to input delay and state quantization developing a switched predictor-feedback law. To deal with the input delay, we reformulate the problem as an actuated transport PDE coupled with the original reaction-diffusion PDE. The...
Suppose a system of partial differential equations with constant coefficients describes a classical field theory. Einstein proposed a definition of the strength of such a theory and its degrees of freedom (DoF) based on the asymptotic number of free Taylor series coefficients of bounded degree in the general solution of the system. however, direct...
We present a novel class of Physics-Informed Neural Networks that is formulated based on the principles of Evidential Deep Learning, where the model incorporates uncertainty quantification by learning parameters of a higher-order distribution. The dependent and trainable variables of the PDE residual loss and data-fitting loss terms are recast as f...
Phosphodiesterase-5 (PDE5) is a potent therapeutic target for the treatment of male erectile dysfunction and pulmonary arterial hypertension with several drugs available on the market. However, most of the reported PDE5 inhibitors lack specificity over PDE6, a holoenzyme in eleven PDE families, which may cause various adverse effects. Targeting a u...
Trough this work we investigate the existence of periodic solutions for the following partial differential equations with infinite delay of the form $\dot{\textit{w}}(t) = \mathcal{L}\textit{w}(t) + \mathcal{D}(\textit{w}_{t}) + \mathcal{H}(t)$. We assume that the operator $(\mathcal{L},\mathscr{D}(\mathcal{L}))$ is generally nondensely defined ope...
In this study, we investigate numerical simulation models for water flow in variably saturated (unsaturated) soils. These models are crucial for addressing soil-related challenges and analyzing water-related risks, particularly in the context of water resource management, soil water-induced disasters, and the agricultural impacts of global environm...
In this paper, we propose a general meshless structure-preserving Galerkin method for solving dissipative PDEs on surfaces. By posing the PDE in the variational formulation and simulating the solution in the finite-dimensional approximation space spanned by (local) Lagrange functions generated with positive definite kernels, we obtain a semi-discre...
Digital communication systems inherently operate through physical media governed by partial differential equations (PDEs). In this paper, we introduce a physics-aware decoding framework that integrates gradient descent-based error correcting algorithms with PDE-based channel modeling using differentiable PDE solvers. At the core of our approach is...
In this work, we present a novel iterative deep Ritz method (IDRM) for solving a general class of elliptic problems. It is inspired by the iterative procedure for minimizing the loss during the training of the neural network, but at each step encodes the geometry of the underlying function space and incorporates a convex penalty to enhance the perf...
In recent years we have witnessed a growth in mathematics for deep learning, which has been used to solve inverse problems of partial differential equations (PDEs). However, most deep learning-based inversion methods either require paired data or necessitate retraining neural networks for modifications in the conditions of the inverse problem, sign...