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Recently, researchers observed that gradient descent for deep neural networks operates in an “edge-of-stability” (EoS) regime: the sharpness (maximum eigenvalue of the Hessian) is often larger than stability threshold 2/\eta (where \eta is the step size). Despite this, the loss oscillates and converges in the long run, and the sharpness at the end is just slightly below 2/\eta . While many other well-understood nonconvex objectives such as matrix factorization or two-layer networks can also converge despite large sharpness, there is often a larger gap between sharpness of the endpoint and 2/\eta . In this paper, we study EoS phenomenon by constructing a simple function that has the same behavior. We give rigorous analysis for its training dynamics in a large local region and explain why the fnal converging point has sharpness close to 2/\eta . Globally we observe that the training dynamics for our example have an interesting bifurcating behavior, which was also observed in the training of neural nets. 

In this study, we used GROMACS, a versatile package for performing molecular dynamics to simulate the interactions between different nanoparticles and Dipalmitoyl Phosphatidyl Choline (DPPC) to understand the physical mechanisms that govern the interactions between nanoparticles and lipid membrane. Our simulations show the responses of the lipid bilayer to the nanoparticles, including the formation of an adsorbent layer on the nanoparticle surface, transmembrane ectopic movements, and inconspicuous endocytosis of the nanoparticle by the membrane. The effects of the size of the nanoparticles, structural shape, and charge state on the interaction and transport processes will be examined and summarized. 

A new approach is described for fabricating 3D poly(ε‐caprolactone) (PCL)/gelatin (1:1) nanofiber aerogels with patterned macrochannels and anisotropic microchannels by freeze‐casting with 3D‐printed sacrificial templates. Single layer or multiple layers of macrochannels are formed through an inverse replica of 3D‐printed templates. Aligned microchannels formed by partially anisotropic freezing act as interconnected pores between templated macrochannels. The resulting macro‐/microchannels within nanofiber aerogels significantly increase preosteoblast infiltration in vitro. The conjugation of vascular endothelial growth factor (VEGF)‐mimicking QK peptide to PCL/gelatin/gelatin methacryloyl (1:0.5:0.5) nanofiber aerogels with patterned macrochannels promotes the formation of a microvascular network of seeded human microvascular endothelial cells. Moreover, nanofiber aerogels with patterned macrochannels and anisotropic microchannels show significantly enhanced cellular infiltration rates and host tissue integration compared to aerogels without macrochannels following subcutaneous implantation in rats. Taken together, this novel class of nanofiber aerogels holds great potential in biomedical applications including tissue repair and regeneration, wound healing, and 3D tissue/disease modeling.