HRGrad: Conflict-Aware Harmonized Rotational Gradient for Multiscale Kinetic Problems

Summary

HRGrad is a harmonized rotational gradient method proposed by Zhangyong Liang for simultaneously solving multiscale time-dependent kinetic problems that involve varying small parameters. Such problems exhibit asymptotic transitions from microscopic to macroscopic physics, and solving across all asymptotic regimes at once forms a challenging multi-task learning problem where gradient conflicts between tasks can cause training failure. HRGrad addresses this by explicitly encoding a hidden representation of the small parameters so that the corresponding solving tasks are serialized during simultaneous training. It further mitigates gradient conflicts by segmenting prediction results to construct task-specific losses and introducing a novel gradient alignment metric, which guarantees that the final update direction has a positive dot product with every task-specific gradient. The paper provides a mathematical proof of convergence for the method. The work is available as arXiv preprint 2504.20638 (April 2025) and is relevant to machine learning for scientific computing, multiscale modeling, and multi-task optimization.