Artificial neural networks (ANNs) have very successfully been used in numerical simulations for a series of computational problems ranging from image classification/image recognition, speech recognition, time series analysis, game intelligence, and computational advertising to numerical approximations of partial differential equations (PDEs). Such numerical simulations suggest that ANNs have the capacity to

We prove pathwise uniqueness for solutions of the nonlinear Schrödinger equation with conservative multiplicative noise on compact 3D manifolds. In particular, we generalize the result by Burq, Gérard and Tzvetkov,[11], to the stochastic setting. The proof is based on deterministic and stochastic Strichartz estimates and the Littlewood-Paley decomposition.

AbstractOne of the most challenging problems in applied mathematics is the approximate solution of nonlinear partial differential equations (PDEs) in high dimensions. Standard deterministic approximation methods like finite differences or finite elements suffer from the curse of dimensionality in the sense that the computational effort grows exponentially in the dimension. In this work we overcome this difficulty in the case of reaction-diffusion type PDEs with a locally Lipschitz continuous coervice nonlinearity (such as Allen–Cahn PDEs) by introducing and analyzing truncated variants of the recently introduced full-history recursive multilevel Picard approximation schemes.

Over the last few years deep artificial neural networks (DNNs) have very successfully been used in numerical simulations for a wide variety of computational problems including computer vision, image classification, speech recognition, natural language processing, as well as computational advertisement. In addition, it has recently been proposed to approximate solutions of partial differential equations (PDEs) by means of stochastic learning problems involving DNNs. There are now also a few rigorous mathematical results in the scientific literature which provide error estimates for such deep learning based approximation methods for PDEs. All of these articles provide spatial error estimates for neural network approximations for PDEs but do not provide error estimates for the entire space-time error for the 1 considered neural network approximations. It is the subject of the main result of this article to provide space-time error estimates for DNN approximations of Euler approximations of certain perturbed differential equations. Our proof of this result is based (i) on a certain artificial neural network (ANN) calculus and (ii) on ANN approximation results for products of the form r0, T s ˆRd Q pt, xq Þ Ñ tx P R d where T P p0, 8q, d P N, which we both develop within this article.

ABSTRACT. We consider the stochastic NLS with linear multiplicative noise in L 2 (R d ) and prove the existence and uniqueness of a global solution in the subcritical and a local solution in the critical case, respectively. In particular, we relax the regularity assumptions on the noise from Barbu, Röckner and Zhang (