English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

Nonlinear partial differential equations (PDEs) are used to model dynamicalprocesses in a large number of scientific fields, ranging from finance tobiology. In many applications standard local models are not sufficient toaccurately account for certain non-local phenomena such as, e.g., interactionsat a distance. In order to properly capture these phenomena non-local nonlinearPDE models are frequently employed in the literature. In this article wepropose two numerical methods based on machine learning and on Picarditerations, respectively, to approximately solve non-local nonlinear PDEs. Theproposed machine learning-based method is an extended variant of a deeplearning-based splitting-up type approximation method previously introduced inthe literature and utilizes neural networks to provide approximate solutions ona subset of the spatial domain of the solution. The Picard iterations-basedmethod is an extended variant of the so-called full history recursivemultilevel Picard approximation scheme previously introduced in the literatureand provides an approximate solution for a single point of the domain. Bothmethods are mesh-free and allow non-local nonlinear PDEs with Neumann boundaryconditions to be solved in high dimensions. In the two methods, the numericaldifficulties arising due to the dimensionality of the PDEs are avoided by (i)using the correspondence between the expected trajectory of reflectedstochastic processes and the solution of PDEs (given by the Feynman-Kacformula) and by (ii) using a plain vanilla Monte Carlo integration to handlethe non-local term. We evaluate the performance of the two methods on fivedifferent PDEs arising in physics and biology. In all cases, the methods yieldgood results in up to 10 dimensions with short run times. Our work extendsrecently developed methods to overcome the curse of dimensionality in solvingPDEs.

Deep learning approximations for non-local nonlinear PDEs with Neumann boundary conditions