On the Geometry Transferability of the Hybrid Iterative Numerical Solver for Differential Equations

TL;DR

This paper investigates the geometry transferability of the Hybrid Iterative Numerical Transferable Solver (HINTS), demonstrating its effectiveness and improved convergence when applied to different geometries and with transfer learning enhancements.

Contribution

The paper introduces the analysis of HINTS' ability to transfer across geometries and integrates transfer learning to enhance its convergence on new geometries.

Findings

HINTS can be directly applied to different geometries with accurate results.

Transfer learning significantly improves HINTS convergence speed.

HINTS outperforms traditional solvers in geometry transfer scenarios.

Abstract

The discovery of fast numerical solvers prompted a clear and rapid shift towards iterative techniques in many applications, especially in computational mechanics, due to the increased necessity for solving very large linear systems. Most numerical solvers are highly dependent on the problem geometry and discretization, facing issues when any of these properties change. The newly developed Hybrid Iterative Numerical Transferable Solver (HINTS) combines a standard solver with a neural operator to achieve better performance, focusing on a single geometry at a time. In this work, we explore the "T" in HINTS, i.e., the geometry transferability properties of HINTS. We first propose to directly employ HINTS built for a specific geometry to a different but related geometry without any adjustments. In addition, we propose the integration of an operator level transfer learning with HINTS to even further improve the convergence of HINTS on new geometries and discretizations. We conduct numerical experiments for a Darcy flow problem and a plane-strain elasticity problem. The results show that both the direct application of HINTS and the transfer learning enhanced HINTS are able to accurately solve these problems on different geometries. In addition, using transfer learning, HINTS is able to converge to machine zero even faster than the direct application of HINTS.