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1. Machine learning moment closure models for the radiative transfer equation II: enforcing global hyperbolicity in gradient based closures

   Juntao Huang, Yingda Cheng (January 2022, Multiscale modeling simulation)

   This is the second paper in a series in which we develop machine learning (ML) moment closure models for the radiative transfer equation (RTE). In our previous work \cite{huang2021gradient}, we proposed an approach to directly learn the gradient of the unclosed high order moment, which performs much better than learning the moment itself and the conventional PN closure. However, the ML moment closure model in \cite{huang2021gradient} is not able to guarantee hyperbolicity and long time stability. We propose in this paper a method to enforce the global hyperbolicity of the ML closure model. The main idea is to seek a symmetrizer (a symmetric positive definite matrix) for the closure system, and derive constraints such that the system is globally symmetrizable hyperbolic. It is shown that the new ML closure system inherits the dissipativeness of the RTE and preserves the correct diffusion limit as the Knunsden number goes to zero. Several benchmark tests including the Gaussian source problem and the two-material problem show the good accuracy, long time stability and generalizability of our globally hyperbolic ML closure model. 

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2. Machine learning-based statistical closure models for turbulent dynamical systems

   https://doi.org/10.1098/rsta.2021.0205  

   Qi, Di; Harlim, John (August 2022, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences)

   We propose a machine learning (ML) non-Markovian closure modelling framework for accurate predictions of statistical responses of turbulent dynamical systems subjected to external forcings. One of the difficulties in this statistical closure problem is the lack of training data, which is a configuration that is not desirable in supervised learning with neural network models. In this study with the 40-dimensional Lorenz-96 model, the shortage of data is due to the stationarity of the statistics beyond the decorrelation time. Thus, the only informative content in the training data is from the short-time transient statistics. We adopt a unified closure framework on various truncation regimes, including and excluding the detailed dynamical equations for the variances. The closure framework employs a Long-Short-Term-Memory architecture to represent the higher-order unresolved statistical feedbacks with a choice of ansatz that accounts for the intrinsic instability yet produces stable long-time predictions. We found that this unified agnostic ML approach performs well under various truncation scenarios. Numerically, it is shown that the ML closure model can accurately predict the long-time statistical responses subjected to various time-dependent external forces that have larger maximum forcing amplitudes and are not in the training dataset. This article is part of the theme issue ‘Data-driven prediction in dynamical systems’. 

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3. Long-Term Loop Closure Detection through Visual-Spatial Information Preserving Multi-Order Graph Matching

   https://doi.org/10.1609/aaai.v34i06.6604  

   Gao, Peng; Zhang, Hao (June 2020, Proceedings of the AAAI Conference on Artificial Intelligence)

   Loop closure detection is a fundamental problem for simultaneous localization and mapping (SLAM) in robotics. Most of the previous methods only consider one type of information, based on either visual appearances or spatial relationships of landmarks. In this paper, we introduce a novel visual-spatial information preserving multi-order graph matching approach for long-term loop closure detection. Our approach constructs a graph representation of a place from an input image to integrate visual-spatial information, including visual appearances of the landmarks and the background environment, as well as the second and third-order spatial relationships between two and three landmarks, respectively. Furthermore, we introduce a new formulation that formulates loop closure detection as a multi-order graph matching problem to compute a similarity score directly from the graph representations of the query and template images, instead of performing conventional vector-based image matching. We evaluate the proposed multi-order graph matching approach based on two public long-term loop closure detection benchmark datasets, including the St. Lucia and CMU-VL datasets. Experimental results have shown that our approach is effective for long-term loop closure detection and it outperforms the previous state-of-the-art methods. 

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4. Effects of Different Closure Choices in Core-collapse Supernova Simulations

   https://doi.org/10.3847/1538-4357/aca75c  

   Wang, Tianshu; Burrows, Adam (January 2023, The Astrophysical Journal)

   Abstract The two-moment method is widely used to approximate the full neutrino transport equation in core-collapse supernova (CCSN) simulations, and different closures lead to subtle differences in the simulation results. In this paper, we compare the effects of closure choices on various physical quantities in 1D and 2D time-dependent CCSN simulations with our multigroup radiation hydrodynamics code Fornax. We find that choices of the third-order closure relations influence the time-dependent simulations only slightly. Choices of the second-order closure relation have larger consequences than choices of the third-order closure, but these are still small compared to the remaining variations due to ambiguities in some physical inputs such as the nuclear equation of state. We also find that deviations in Eddington factors are not monotonically related to deviations in physical quantities, which means that simply comparing the Eddington factors does not inform one concerning which closure is better. 

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5. High-order moment closure models with random batch method for efficient computation of multiscale turbulent systems

   https://doi.org/10.1063/5.0160057  

   Qi, Di; Liu, Jian-Guo (October 2023, Chaos: An Interdisciplinary Journal of Nonlinear Science)

   We propose a high-order stochastic–statistical moment closure model for efficient ensemble prediction of leading-order statistical moments and probability density functions in multiscale complex turbulent systems. The statistical moment equations are closed by a precise calibration of the high-order feedbacks using ensemble solutions of the consistent stochastic equations, suitable for modeling complex phenomena including non-Gaussian statistics and extreme events. To address challenges associated with closely coupled spatiotemporal scales in turbulent states and expensive large ensemble simulation for high-dimensional systems, we introduce efficient computational strategies using the random batch method (RBM). This approach significantly reduces the required ensemble size while accurately capturing essential high-order structures. Only a small batch of small-scale fluctuation modes is used for each time update of the samples, and exact convergence to the full model statistics is ensured through frequent resampling of the batches during time evolution. Furthermore, we develop a reduced-order model to handle systems with really high dimensions by linking the large number of small-scale fluctuation modes to ensemble samples of dominant leading modes. The effectiveness of the proposed models is validated by numerical experiments on the one-layer and two-layer Lorenz ‘96 systems, which exhibit representative chaotic features and various statistical regimes. The full and reduced-order RBM models demonstrate uniformly high skill in capturing the time evolution of crucial leading-order statistics, non-Gaussian probability distributions, while achieving significantly lower computational cost compared to direct Monte-Carlo approaches. The models provide effective tools for a wide range of real-world applications in prediction, uncertainty quantification, and data assimilation. 

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