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

The lacking of analytic solutions of diverse partial differential equations(PDEs) gives birth to series of computational techniques for numericalsolutions. In machine learning, numerous latest advances of solver designs areaccomplished in developing neural operators, a kind of mesh-free approximatorsof the infinite-dimensional operators that map between differentparameterization spaces of equation solutions. Although neural operatorsexhibit generalization capacities for learning an entire PDE familysimultaneously, they become less accurate and explainable while learninglong-term behaviours of non-linear PDE families. In this paper, we proposeKoopman neural operator (KNO), a new neural operator, to overcome thesechallenges. With the same objective of learning an infinite-dimensional mappingbetween Banach spaces that serves as the solution operator of target PDEfamily, our approach differs from existing models by formulating a non-lineardynamic system of equation solution. By approximating the Koopman operator, aninfinite-dimensional linear operator governing all possible observations of thedynamic system, to act on the flow mapping of dynamic system, we canequivalently learn the solution of an entire non-linear PDE family by solvingsimple linear prediction problems. In zero-shot prediction and long-termprediction experiments on representative PDEs (e.g., the Navier-Stokesequation), KNO exhibits notable advantages in breaking the tradeoff betweenaccuracy and efficiency (e.g., model size) while previous state-of-the-artmodels are limited. These results suggest that more efficient PDE solvers canbe developed by the joint efforts from physics and machine learning.

Koopman neural operator as a mesh-free solver of non-linear partial differential equations