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

Physics-informed Neural Networks (PINNs) have been shown as a promisingapproach for solving both forward and inverse problems of partial differentialequations (PDEs). Meanwhile, the neural operator approach, including methodssuch as Deep Operator Network (DeepONet) and Fourier neural operator (FNO), hasbeen introduced and extensively employed in approximating solution of PDEs.Nevertheless, to solve problems consisting of sharp solutions poses asignificant challenge when employing these two approaches. To address thisissue, we propose in this work a novel framework termed Operator LearningEnhanced Physics-informed Neural Networks (OL-PINN). Initially, we utilizeDeepONet to learn the solution operator for a set of smooth problems relevantto the PDEs characterized by sharp solutions. Subsequently, we integrate thepre-trained DeepONet with PINN to resolve the target sharp solution problem. Weshowcase the efficacy of OL-PINN by successfully addressing various problems,such as the nonlinear diffusion-reaction equation, the Burgers equation and theincompressible Navier-Stokes equation at high Reynolds number. Compared withthe vanilla PINN, the proposed method requires only a small number of residualpoints to achieve a strong generalization capability. Moreover, itsubstantially enhances accuracy, while also ensuring a robust training process.Furthermore, OL-PINN inherits the advantage of PINN for solving inverseproblems. To this end, we apply the OL-PINN approach for solving problems withonly partial boundary conditions, which usually cannot be solved by theclassical numerical methods, showing its capacity in solving ill-posed problemsand consequently more complex inverse problems.

Operator Learning Enhanced Physics-informed Neural Networks for Solving Partial Differential Equations Characterized by Sharp Solutions