Hybrid deep method improves solving elliptic interface problems
A Hybrid Iterative Deep Ritz Method for Elliptic Interface Problems
Machine Learning
Summary
Elliptic interface problems are mathematical challenges where a solution behaves differently across distinct regions separated by an interface. The authors developed a new hybrid method combining deep neural networks and iterative solving steps to handle these problems more effectively. Their approach uses neural networks that represent the interface in a smooth way, avoids tricky sampling on the interface, and solves several easier problems one after another. Tests show this method works better than others for complicated shapes and higher dimensions.
What this means in practice
- •For computational engineers: Improve numerical simulations involving materials with complex internal boundaries using efficient neural solvers.
- •For software developers in scientific computing: Create software tools that solve interface problems in high-dimensional spaces without explicit interface sampling.
Authors
Tianhao Hu, Bangti Jin, Fengru Wang, Yifeng Xu
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.