相关论文: Partial and complete linearization of PDEs based o…
Many physical processes such as weather phenomena or fluid mechanics are governed by partial differential equations (PDEs). Modelling such dynamical systems using Neural Networks is an active research field. However, current methods are…
Solving inverse and optimization problems over solutions of nonlinear partial differential equations (PDEs) on complex spatial domains is a long-standing challenge. Here we introduce a method that parameterizes the solution using spectral…
In this paper, we construct a semi-implicit finite difference method for the time dependent Poisson-Nernst-Planck system. Although the Poisson-Nernst-Planck system is a nonlinear system, the numerical method presented in this paper only…
We introduce a simple, rigorous, and unified framework for solving nonlinear partial differential equations (PDEs), and for solving inverse problems (IPs) involving the identification of parameters in PDEs, using the framework of Gaussian…
Online parameter identification is of importance, e.g., for model predictive control. Since the parameters have to be identified simultaneously to the process of the modeled system, dynamical update laws are used for state and parameter…
Physics-informed neural networks (PINNs) incorporate physical laws into their training to efficiently solve partial differential equations (PDEs) with minimal data. However, PINNs fail to guarantee adherence to conservation laws, which are…
Carleman linearization is a technique that embeds systems of ordinary differential equations with polynomial nonlinearities into infinite dimensional linear systems in a procedural way. In this paper we generalize the method for systems of…
The Partial Integral Equation (PIE) framework was developed to computationally analyze linear Partial Differential Equations (PDEs) where the PDE is first converted to a PIE and then the analysis problem is solved by solving operator-valued…
We introduce a methodology for seeking conservation laws within a Hamiltonian dynamical system, which we term ``neural deflation''. Inspired by deflation methods for steady states of dynamical systems, we propose to {iteratively} train a…
Identifying parameters in partial differential equations (PDEs) represents a very broad class of applied inverse problems. In recent years, several unsupervised learning approaches using (deep) neural networks have been developed to solve…
We study higher-order conservation laws of the non-linearizable elliptic Poisson equation $ \frac{{\partial}^2 u}{\partial z \partial \bar{z}} = -f(u) $ as elements of the characteristic cohomology of the associated exterior differential…
Using the maximal Lie algebra of point symmetries of a system of nonlinear equations used in geophysical fluid dynamics, two conservation laws are found in addition to the conservation of energy.
The purpose of this article is to provide a perspective -- admittedly, a rather subjective one -- of recent developments at the interface of machine learning/data-driven methods and nonlinear wave studies. We review some recent pillars of…
A class of partial differential equations (a conservation law and four balance laws), with four independent variables and involving sixteen arbitrary continuously differentiable functions, is considered in the framework of equivalence…
The solutions to a large class of semi-linear parabolic PDEs are given in terms of expectations of suitable functionals of a tree of branching particles. A sufficient, and in some cases necessary, condition is given for the integrability of…
A system of partial differential equations (PDEs) is derived to compute the full-field stress from an observed kinematic field when the flow rule governing the plastic deformation is unknown. These equations generalize previously proposed…
Partial differential equation (PDE) models are widely used in engineering and natural sciences to describe spatio-temporal processes. The parameters of the considered processes are often unknown and have to be estimated from experimental…
The compact finite difference method is a powerful tool for discretizing conservation laws, owing to its inherent flexibility in developing high-resolution and highly stable schemes. In this paper, we propose a framework for the design of…
The first part of this paper introduces sufficient conditions to determine conservation laws of diffusion equations of arbitrary fractional order in time. Numerical methods that satisfy a discrete analogue of these conditions have…
Many inverse and parameter estimation problems can be written as PDE-constrained optimization problems. The goal, then, is to infer the parameters, typically coefficients of the PDE, from partial measurements of the solutions of the PDE for…