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/

Zendy Pro

Presents the zaia usage as premium feature

ZAIA

Presents AI Discovery search usage as a premium feature

AI Discovery

Presents AI Discovery explain and summary usage as a premium feature

Content Analysis

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Presents the pdf analysis as premium feature

PDF Analysis

1500 Top-up

3000 Top-up

Zendy Basic

2500 Top-up

Zendy Unlimited

Presents the access of premium content as premium feature

Premium Content

We present PFNN, a penalty-free neural network method, to efficiently solve aclass of second-order boundary-value problems on complex geometries. To reducethe smoothness requirement, the original problem is reformulated to a weak formso that the evaluations of high-order derivatives are avoided. Two neuralnetworks, rather than just one, are employed to construct the approximatesolution, with one network satisfying the essential boundary conditions and theother handling the rest part of the domain. In this way, an unconstrainedoptimization problem, instead of a constrained one, is solved without addingany penalty terms. The entanglement of the two networks is eliminated with thehelp of a length factor function that is scale invariant and can adapt withcomplex geometries. We prove the convergence of the PFNN method and conductnumerical experiments on a series of linear and nonlinear second-orderboundary-value problems to demonstrate that PFNN is superior to severalexisting approaches in terms of accuracy, flexibility and robustness.

PFNN: A Penalty-Free Neural Network Method for Solving a Class of Second-Order Boundary-Value Problems on Complex Geometries