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 machine learning (PIML) has emerged as a promisingalternative to conventional numerical methods for solving partial differentialequations (PDEs). PIML models are increasingly built via deep neural networks(NNs) whose architecture and training process are designed such that thenetwork satisfies the PDE system. While such PIML models have substantiallyadvanced over the past few years, their performance is still very sensitive tothe NN's architecture and loss function. Motivated by this limitation, weintroduce kernel-weighted Corrective Residuals (CoRes) to integrate thestrengths of kernel methods and deep NNs for solving nonlinear PDE systems. Toachieve this integration, we design a modular and robust framework whichconsistently outperforms competing methods in solving a broad range ofbenchmark problems. This performance improvement has a theoreticaljustification and is particularly attractive since we simplify the trainingprocess while negligibly increasing the inference costs. Additionally, ourstudies on solving multiple PDEs indicate that kernel-weighted CoResconsiderably decrease the sensitivity of NNs to factors such as randominitialization, architecture type, and choice of optimizer. We believe ourfindings have the potential to spark a renewed interest in leveraging kernelmethods for solving PDEs.

Neural Networks with Kernel-Weighted Corrective Residuals for Solving Partial Differential Equations