A Hybrid Iterative Deep Ritz Method for Elliptic Interface Problems

Abstract:In this work, we propose a hybrid iterative deep Ritz method (H-IDRM) for a class of interface problems for second-order elliptic operators. It is based on a new mixed formulation of the problem and involves solving a sequence of convex minimization problems. We employ a level-set neural network architecture, featuring a level-set representation of the interface, to accommodate the piecewise smoothness of the solution and the flux. The approach involves only volumetric representations instead of duality pairing on the interface and avoids explicit interface sampling that is inconvenient for complex interface geometries. Further, we present an analysis of the method, including the errors arising from the neural network approximation, Monte Carlo approximation, iterative scheme, and penalty parameters. Numerical experiments indicate that the H-IDRM outperforms existing neural solvers on problems with high-dimensional domains, intricate interface geometries, and lower subdomain regularity.

Numerical Analysis (math.NA); Machine Learning (cs.LG)

arXiv:2609.27325 [math.NA]

(or arXiv:2609.27325v1 [math.NA] for this version)

https://doi.org/10.48550/arXiv.2609.27325 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.