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Understanding the loss landscapes of neural networks (NNs) is critical for optimizing model performance. Previous research has identified the phenomenon of mode connectivity on curves, where two well-trained NNs can be connected by a continuous path in parameter space where the path maintains nearly constant loss. In this work, we extend the concept of mode connectivity to explore connectivity on surfaces, significantly broadening its applicability and unlocking new opportunities. While initial attempts to connect models via linear surfaces in parameter space were unsuccessful, we propose a novel optimization technique that consistently discovers Bézier surfaces with low-loss and high-accuracy connecting multiple NNs in a nonlinear manner. We further demonstrate that even without optimization, mode connectivity exists in certain cases of Bézier surfaces, where the models are carefully selected and combined linearly. This approach provides a deeper and more comprehensive understanding of the loss landscape and offers a novel way to identify models with enhanced performance for model averaging and output ensembling. We demonstrate the effectiveness of our method on CIFAR-10, CIFAR-100, and Tiny-ImageNet datasets using VGG16, ResNet18, and ViT architectures. 

Abstract The far-from-equilibrium solidification during additive manufacturing often creates large residual stresses that induce solid-state cracking. Here we present a strategy to suppress solid-state cracking in an additively manufactured AlCrFe₂Ni₂high-entropy alloy via engineering phase transformation pathway. We investigate the solidification microstructures formed during laser powder-bed fusion and directed energy deposition, encompassing a broad range of cooling rates. At high cooling rates (10⁴−10⁶ K/s), we observe a single-phase BCC/B2 microstructure that is susceptible to solid-state cracking. At low cooling rates (10²−10⁴ K/s), FCC phase precipitates out from the BCC/B2 matrix, resulting in enhanced ductility (~10 %) and resistance to solid-state cracking. Site-specific residual stress/strain analysis reveals that the ductile FCC phase can largely accommodate residual stresses, a feature which helps relieve residual strains within the BCC/B2 phase to prevent cracking. Our work underscores the value of exploiting the toolbox of phase transformation pathway engineering for material design during additive manufacturing.