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1. SGOOP-d: Estimating kinetic distances and reaction coordinate dimensionality for rare event systems from biased/unbiased simulations

   Tsai, Sun-Ting; Smith, Zachary; Tiwary, Pratyush (April 2021, ArXivorg)

   null (Ed.)

   Understanding kinetics including reaction pathways and associated transition rates is an important yet difficult problem in numerous chemical and biological systems especially in situations with multiple competing pathways. When these high-dimensional systems are projected on low-dimensional coordinates often needed for enhanced sampling or for interpretation of simulations and experiments, one often ends up losing the kinetic connectivity of the underlying high-dimensional landscape. Thus in the low-dimensional projection metastable states might appear closer or further than they actually are. To deal with this issue, in this work we develop a formalism that learns a multi-dimensional yet minimally complex reaction coordinate (RC) for generic high-dimensional systems, such that when projected along this RC, all possible kinetically relevant pathways can be demarcated and the true high-dimensional connectivity is maintained. One of the defining attributes of our method lies in that it can work on long unbiased simulations as well as biased simulations often needed for rare event systems. We demonstrate the utility of the method by studying a range of model systems including conformational transitions in a small peptide Ace-Ala3-Nme, where we show how SGOOPderived two-dimensional and three-dimensional reaction coordinate can capture the kinetics for 23 and all 28 out of the 28 dominant state-to-state-transitions respectively. 

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2. Nonlocal topological insulators: Deterministic aperiodic arrays supporting localized topological states protected by nonlocal symmetries

   https://doi.org/10.1073/pnas.2100691118  

   Chen, Kai; Weiner, Matthew; Li, Mengyao; Ni, Xiang; Alù, Andrea; Khanikaev, Alexander B. (August 2021, Proceedings of the National Academy of Sciences)

   The properties of topological systems are inherently tied to their dimensionality. Indeed, higher-dimensional periodic systems exhibit topological phases not shared by their lower-dimensional counterparts. On the other hand, aperiodic arrays in lower-dimensional systems (e.g., the Harper model) have been successfully employed to emulate higher-dimensional physics. This raises a general question on the possibility of extended topological classification in lower dimensions, and whether the topological invariants of higher-dimensional periodic systems may assume a different meaning in their lower-dimensional aperiodic counterparts. Here, we demonstrate that, indeed, for a topological system in higher dimensions one can construct a one-dimensional (1D) deterministic aperiodic counterpart which retains its spectrum and topological characteristics. We consider a four-dimensional (4D) quantized hexadecapole higher-order topological insulator (HOTI) which supports topological corner modes. We apply the Lanczos transformation and map it onto an equivalent deterministic aperiodic 1D array (DAA) emulating 4D HOTI in 1D. We observe topological zero-energy zero-dimensional (0D) states of the DAA—the direct counterparts of corner states in 4D HOTI and the hallmark of the multipole topological phase, which is meaningless in lower dimensions. To explain this paradox, we show that higher-dimension invariant, the multipole polarization, retains its quantization in the DAA, yet changes its meaning by becoming a nonlocal correlator in the 1D system. By introducing nonlocal topological phases of DAAs, our discovery opens a direction in topological physics. It also unveils opportunities to engineer topological states in aperiodic systems and paves the path to application of resonances associates with such states protected by nonlocal symmetries. 

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3. On Deep Generative Models for Approximation and Estimation of Distributions on Manifolds

   Dahal, Biraj; Havrilla, Alexander; Chen, Minshuo; Zhao, Tuo; Liao, Wenjing (December 2022, Advances in Neural Information Processing Systems)

   Deep generative models have experienced great empirical successes in distribution learning. Many existing experiments have demonstrated that deep generative networks can efficiently generate high-dimensional complex data from a low-dimensional easy-to-sample distribution. However, this phenomenon can not be justified by existing theories. The widely held manifold hypothesis speculates that real-world data sets, such as natural images and signals, exhibit low-dimensional geometric structures. In this paper, we take such low-dimensional data structures into consideration by assuming that data distributions are supported on a low-dimensional manifold. We prove approximation and estimation theories of deep generative networks for estimating distributions on a low-dimensional manifold under the Wasserstein-1 loss. We show that the Wasserstein-1 loss converges to zero at a fast rate depending on the intrinsic dimension instead of the ambient data dimension. Our theory leverages the low-dimensional geometric structures in data sets and justifies the practical power of deep generative models. We require no smoothness assumptions on the data distribution which is desirable in practice. 

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4. On computational simulations of dynamic stall and its three-dimensional nature

   https://doi.org/10.1063/5.0170251  

   Khalifa, Nabil M; Rezaei, Amirsaman; Taha, Haithem E (October 2023, Physics of Fluids)

   In this paper, we investigate the three-dimensional nature of dynamic stall. Conducting the investigation, the flow around a harmonically pitching National Advisory Committee for Aeronautics (NACA) 0012 airfoil is numerically simulated using Unsteady-Reynolds-Averaged Navier–Stokes (URANS) and multiple detached eddy simulation (DES) solvers: the Delayed-DES (DDES) and the Improved-DDES (IDDES). Two- and three-dimensional simulations are performed for each solver, and the results are compared against experimental measurements in the literature. The results showed that three-dimensional simulations surpass two-dimensional ones in capturing the stages of dynamic stall and predicting the lift coefficient values, with a distinguished performance of the DES solvers over the URANS ones. For instance, the IDDES simulations, as an inherently three-dimensional solver, predicted the necessary cascaded amalgamation process of vortices to form the adequate strength of the dynamic stall vortex. This vortex size and timing provided accurate and sufficient suction that resulted in identical matching of the numerical and experimental lift coefficients at the peak value. Hence, the hypothesis that dynamic stall has a three-dimensional nature is supported by the superiority of the three-dimensional simulation in all aspects. In conclusion, it is found that dynamic stall is intrinsically a three-dimensional phenomenon. 

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5. Neural Set Function Extensions: Learning with Discrete Functions in High Dimensions

   Nikolaos Karalias, Joshua Robinson (November 2022, Neural Information Processing Systems (NeurIPS))

   Integrating functions on discrete domains into neural networks is key to develop- ing their capability to reason about discrete objects. But, discrete domains are (I) not naturally amenable to gradient-based optimization, and (II) incompatible with deep learning architectures that rely on representations in high-dimensional vector spaces. In this work, we address both difficulties for set functions, which capture many important discrete problems. First, we develop a framework for extending set functions onto low-dimensional continuous domains, where many extensions are naturally defined. Our framework subsumes many well-known extensions as special cases. Second, to avoid undesirable low-dimensional neural network bottlenecks, we convert low-dimensional extensions into representations in high-dimensional spaces, taking inspiration from the success of semidefinite programs for combinatorial optimization. Empirically, we observe benefits of our extensions for unsupervised neural combinatorial optimization, in particular with high-dimensional representations. 

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