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 paper proposes a deep learning method specifically dealing with theforward and inverse problem of variable coefficient partial differentialequations -- Variable Coefficient Physical Information Neural Network(VC-PINN). The shortcut connections (ResNet structure) introduced into thenetwork alleviates the "Vanishing gradient" and unifies the linear andnonlinear coefficients. The developed method was applied to four equationsincluding the variable coefficient Sine-Gordon (vSG), the generalized variablecoefficient Kadomtsev-Petviashvili equation (gvKP), the variable coefficientKorteweg-de Vries equation (vKdV), the variable coefficient Sawada-Koteraequation (vSK). Numerical results show that VC-PINN is successful in the caseof high dimensionality, various variable coefficients (polynomials,trigonometric functions, fractions, oscillation attenuation coefficients), andthe coexistence of multiple variable coefficients. We also conducted anin-depth analysis of VC-PINN in a combination of theory and numericalexperiments, including four aspects, the necessity of ResNet, the relationshipbetween the convexity of variable coefficients and learning, anti-noiseanalysis, the unity of forward and inverse problems/relationship with standardPINN.

VC-PINN: Variable Coefficient Physical Information Neural Network For Forward And Inverse PDE Problems with Variable Coefficient