A multi-grid, single-mesh online learning framework for efficient large-scale topology optimization

Abstract Recent advances in topology optimization (TO) have increasingly incorporated machine learning techniques to enhance computational efficiency. Among them, online learning methods emerge as promising approaches for large-scale TO designs. Unlike off-line methods requiring pre-generated training data, online learning frameworks adaptively train surrogate models during the optimization process itself using data from earlier iterations. Current state-of-the-art approaches typically employ a two-scale localized training paradigm, where machine learning models iteratively learn mappings between coarse-scale sensitivity fields and fine-scale structural responses within localized patches. While this localized strategy improves computational scalability and provides implicit data augmentation, it faces a limitation: the exclusion of global structural information often leads to underdetermined mappings and compromised prediction accuracy, as fine-scale sensitivities inherently depend on structural context beyond individual patches. To address this issue, we propose a multi-grid decomposition strategy and integrate it within the online learning framework. This strategy hierarchically incorporates global structural information into local training patches to enhance mapping uniqueness. Additionally, our framework implements online learning entirely within a single mesh resolution, bypassing costly coarse-scale simulations while maintaining accuracy. We employ Fourier neural operators as our surrogate model of choice, leveraging their superior capability in capturing structural sensitivity patterns directly from spatial information The proposed method is validated through 2D and 3D compliance design examples, demonstrating significant computational savings by reducing high-dimensional numerical simulations. The influence of key hyperparameters on the framework’s performance is also investigated.