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Learning the solution of partial differential equations (PDEs) with a neural network is an attractive alternative to traditional solvers due to its elegance, greater flexibility and the ease of incorporating observed data. However, training…
The distance between the solutions to the integrable Korteweg-de Vries (KdV) equation and a broad class of non-integrable generalized KdV (gKdV) equations is estimated in appropriate Sobolev spaces. This family of equations includes, as…
The lacking of analytic solutions of diverse partial differential equations (PDEs) gives birth to a series of computational techniques for numerical solutions. Although numerous latest advances are accomplished in developing neural…
In this work, we prove the existence of wave operator for the following generalized derivative nonlinear Schr\"odinger equation \begin{align*} i\partial_t u+\partial_x^2 u +i |u|^{2\sigma}\partial_x u=0, \end{align*} with…
Many problems in science and engineering can be represented by a set of partial differential equations (PDEs) through mathematical modeling. Mechanism-based computation following PDEs has long been an essential paradigm for studying topics…
Neural networks have shown promising potential in accelerating the numerical simulation of systems governed by partial differential equations (PDEs). Different from many existing neural network surrogates operating on high-dimensional…
Recent advances in nonlinear dynamical systems theory provide a new insight into numerical properties of discrete algorithms developed to solve nonlinear initial value problems. Basic features like accuracy and stability are well pointed…
Neural ordinary differential equations are an attractive option for modelling temporal dynamics. However, a fundamental issue is that the solution to an ordinary differential equation is determined by its initial condition, and there is no…
Neural ordinary differential equations (neural ODEs) are a popular type of deep learning model that operate with continuous-depth architectures. To assess how well such models perform on unseen data, it is crucial to understand their…
Deep Learning Reduced Order Models (ROMs) are becoming increasingly popular as surrogate models for parametric partial differential equations (PDEs) due to their ability to handle high-dimensional data, approximate highly nonlinear…
We develop and evaluate a method for learning solution operators to nonlinear problems governed by partial differential equations (PDEs). The approach is based on a finite element discretization and aims at representing the solution…
Recent progress in AI has established neural operators as powerful tools that can predict the evolution of partial differential equations, such as the Navier-Stokes equations. Some complex problems rely on sophisticated algorithms to deal…
Evolution PDEs for dispersive waves are considered in both linear and nonlinear integrable cases, and initial-boundary value problems associated with them are formulated in spectral space. A method of solution is presented, which is based…
We consider linear and nonlinear hyperbolic SPDEs with mixed derivatives with additive space-time Gaussian white noise of the form $Y_{xt}=F(Y) + \sigma W_{xt}.$ Such equations, which transform to linear and nonlinear wave equations,…
Physics-informed neural networks (PINNs) [31] use automatic differentiation to solve partial differential equations (PDEs) by penalizing the PDE in the loss function at a random set of points in the domain of interest. Here, we develop a…
In the past, we have presented a systematic computational framework for analyzing self-similar and traveling wave dynamics in nonlinear partial differential equations (PDEs) by dynamically factoring out continuous symmetries such as…
Frequency-domain simulation of seismic waves plays an important role in seismic inversion, but it remains challenging in large models. The recently proposed physics-informed neural network (PINN), as an effective deep learning method, has…
Deep neural networks (DNN) have been used to model nonlinear relations between physical quantities. Those DNNs are embedded in physical systems described by partial differential equations (PDE) and trained by minimizing a loss function that…
We consider the problem of forecasting complex, nonlinear space-time processes when observations provide only partial information of on the system's state. We propose a natural data-driven framework, where the system's dynamics are modelled…
Physics-informed neural networks (PINNs) have emerged as a promising approach to solving partial differential equations (PDEs) using neural networks, particularly in data-scarce scenarios, due to their unsupervised training capability.…