Homotopy-based training of NeuralODEs for accurate dynamics discovery

Conceptually, Neural Ordinary Differential Equations (NeuralODEs) pose an attractive way to extract dynamical laws from time series data, as they are natural extensions of the traditional differential equation-based modeling paradigm of the physical sciences. In practice, NeuralODEs display long training times and suboptimal results, especially for longer duration data where they may fail to fit the data altogether. While methods have been proposed to stabilize NeuralODE training, many of these involve placing a strong constraint on the functional form the trained NeuralODE can take that the actual underlying governing equation does not guarantee satisfaction. In this work, we present a novel NeuralODE training algorithm that leverages tools from the chaos and mathematical optimization communities - synchronization and homotopy optimization - for a breakthrough in tackling the NeuralODE training obstacle. We demonstrate architectural changes are unnecessary for effective NeuralODE training. Compared to the conventional training methods, our algorithm achieves drastically lower loss values without any changes to the model architectures. Experiments on both simulated and real systems with complex temporal behaviors demonstrate NeuralODEs trained with our algorithm are able to accurately capture true long term behaviors and correctly extrapolate into the future.Comment: 12 pages, 6 figures, submitted to ICLR202

Reproducible Nanostructure Fabrication Using Atomic Force Microscopy Indentation with Minimal Tip Damage

A uniform pattern of quantum dots and nanowires were reproducibly fabricated by creating holes in a two-layer structure using atomic force microscopy (AFM) indentation, dry-etching of polymer resists, and metal deposition through the indentation holes. The two-layer structure was created by depositing a thin gold layer onto a polymethyl methacrylate (PMMA) layer on a silicon substrate. The indentation depth was set so that the AFM tip penetrated the thin gold layer without the tip contacting the silicon substrate. This two-layer indentation was used to create a pattern of holes in the thin gold layer. Then, the PMMA was exposed to an isotropic O2 plasma etchant through the holes in the indentation pattern to form an undercut between the substrate and the gold layer. Quantum dots were subsequently created through the deposition of gold on the exposed silicon substrate through the indentation holes. Gold nanowires were also fabricated by creating indentation holes consecutively and optimizing the distance between the holes using the same two-layer indentation method. The topographic and electrical measurements of the fabricated gold nanowires suggest that our method is capable of making uniform and reproducible nanowires. The scanning electron microscopy images of the tips confirmed that the consecutive-hole-indentation method is less invasive than the conventional ploughing method, where constant tip contact occurs with the substrate during the formation of nanowires

Velocity tuning of friction with two trapped atoms

Our ability to control friction remains modest, as our understanding of the underlying microscopic processes is incomplete. Atomic force experiments have provided a wealth of results on the dependence of nanofriction on structure velocity and temperature but limitations in the dynamic range, time resolution, and control at the single-atom level have hampered a description from first principles. Here, using an ion-crystal system with single-atom, single-substrate-site spatial and single-slip temporal resolution we measure the friction force over nearly five orders of magnitude in velocity, and contiguously observe four distinct regimes, while controlling temperature and dissipation. We elucidate the interplay between thermal and structural lubricity for two coupled atoms, and provide a simple explanation in terms of the Peierls–Nabarro potential. This extensive control at the atomic scale enables fundamental studies of the interaction of many-atom surfaces, possibly into the quantum regime