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 Plus annual

Presents the access of premium content as premium feature

Premium Content

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 Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We consider the approximation of initial/boundary value problems involving,possibly high-dimensional, dissipative evolution partial differential equations(PDEs) using a deep neural network framework. More specifically, we firstpropose discrete gradient flow approximations based on non-standard Dirichletenergies for problems involving essential boundary conditions posed on boundedspatial domains. The imposition of the boundary conditions is realized weaklyvia non-standard functionals; the latter classically arise in the constructionof Galerkin-type numerical methods and are often referred to as "Nitsche-type"methods. Moreover, inspired by the seminal work of Jordan, Kinderleher, andOtto (JKO) \cite{jko}, we consider the second class of discrete gradient flowsfor special classes of dissipative evolution PDE problems with non-essentialboundary conditions. These JKO-type gradient flows are solved via deep neuralnetwork approximations. A key, distinct aspect of the proposed methods is thatthe discretization is constructed via a sequence of residual-type deep neuralnetworks (DNN) corresponding to implicit time-stepping. As a result, a DNNrepresents the PDE problem solution at each time node. This approach offersseveral advantages in the training of each DNN. We present a series ofnumerical experiments which showcase the good performance of Dirichlet-typeenergy approximations for lower space dimensions and the excellent performanceof the JKO-type energies for higher spatial dimensions.

Discrete Gradient Flow Approximations of High Dimensional Evolution Partial Differential Equations via Deep Neural Networks