TCCT: Tightly-Coupled Convolutional Transformer on Time Series Forecasting

Time series forecasting is essential for a wide range of real-world applications. Recent studies have shown the superiority of Transformer in dealing with such problems, especially long sequence time series input(LSTI) and long sequence time series forecasting(LSTF) problems. To improve the efficiency and enhance the locality of Transformer, these studies combine Transformer with CNN in varying degrees. However, their combinations are loosely-coupled and do not make full use of CNN. To address this issue, we propose the concept of tightly-coupled convolutional Transformer(TCCT) and three TCCT architectures which apply transformed CNN architectures into Transformer: (1) CSPAttention: through fusing CSPNet with self-attention mechanism, the computation cost of self-attention mechanism is reduced by 30% and the memory usage is reduced by 50% while achieving equivalent or beyond prediction accuracy. (2) Dilated causal convolution: this method is to modify the distilling operation proposed by Informer through replacing canonical convolutional layers with dilated causal convolutional layers to gain exponentially receptive field growth. (3) Passthrough mechanism: the application of passthrough mechanism to stack of self-attention blocks helps Transformer-like models get more fine-grained information with negligible extra computation costs. Our experiments on real-world datasets show that our TCCT architectures could greatly improve the performance of existing state-of-art Transformer models on time series forecasting with much lower computation and memory costs, including canonical Transformer, LogTrans and Informer.

Population properties of neutron stars in the coalescing compact binaries

We perform a hierarchical Bayesian inference to investigate the population properties of the coalesc- ing compact binaries involving at least one neutron star (NS). With the current observation data, we can not rule out either of the Double Gaussian, Single Gaussian and Uniform NS mass distribution models, although the mass distribution of the Galactic NSs is slightly preferred by the gravitational wave (GW) observations. The mass distribution of black holes (BHs) in the neutron star-black hole (NSBH) population is found to be similar to that for the Galactic X-ray binaries. Additionally, the ratio of the merger rate densities between NSBHs and BNSs is estimated to be about 3 : 7. The spin properties of the binaries, though constrained relatively poor, play nontrivial role in reconstructing the mass distribution of NSs and BHs. We find that a perfectly aligned spin distribution can be ruled out, while a purely isotropic distribution of spin orientation is still allowed.

On the Angular Momentum Transport Efficiency within the Star Constrained from Gravitational-wave Observations

The LIGO Scientific Collaboration and Virgo Collaboration (LVC) have recently reported in GWTC-2.1 eight additional candidate events with a probability of astrophysical origin greater than 0.5 in the LVC deeper search on O3a running. In GWTC-2.1, the majority of the effective inspiral spins ($chi_{rm eff}$) show magnitudes consistent with zero, while two (GW190403$_{-}$051519 and GW190805$_{-}$211137) of the eight new events have $chi_{rm eff}$ $> 0$ (at 90% credibility). We note that GW190403$_{-}$051519 was reported with $chi_{rm eff}$ = $0.70^{+0.15}_{-0.27}$ and mass ratio $q$ = $0.25^{+0.54}_{-0.11}$, respectively. Assuming a uniform prior probability between 0 and 1 for each black holes dimensionless spin magnitude, GW190403$_{-}$051519 was reported with the dimensionless spin of the more massive black hole, $chi_1$ = $0.92^{+0.07}_{-0.22}$. This is the fastest first-born black hole ever measured in all current gravitational-wave events. If GW190403$_{-}$051519 is formed through isolated binary evolution channel, this extremely high spin challenges, at least in that case, the existence of efficient angular momentum transport mechanism between the stellar core and the radiative envelope of massive stars, as for instance predicted by the Tayler-Spruit dynamo (Spruit 2002) or its revised version by Fuller et al. 2019..

Hamiltonian Operator Inference: Physics-preserving Learning of Reduced-order Models for Hamiltonian Systems

This work presents a nonintrusive physics-preserving method to learn reduced-order models (ROMs) of Hamiltonian systems. Traditional intrusive projection-based model reduction approaches utilize symplectic Galerkin projection to construct Hamiltonian reduced models by projecting Hamiltons equations of the full model onto a symplectic subspace. This symplectic projection requires complete knowledge about the full model operators and full access to manipulate the computer code. In contrast, the proposed Hamiltonian operator inference approach embeds the physics into the operator inference framework to develop a data-driven model reduction method that preserves the underlying symplectic structure. Our method exploits knowledge of the Hamiltonian functional to define and parametrize a Hamiltonian ROM form which can then be learned from data projected via symplectic projectors. The proposed method is `gray-box in that it utilizes knowledge of the Hamiltonian structure at the partial differential equation level, as well as knowledge of spatially local components in the system. However, it does not require access to computer code, only data to learn the models. Our numerical results demonstrate Hamiltonian operator inference on a linear wave equation, the cubic nonlinear Schr{o}dinger equation, and a nonpolynomial sine-Gordon equation. Accurate long-time predictions far outside the training time interval for nonlinear examples illustrate the generalizability of our learned models.

Black hole gravitational potential enhanced fallback accretion onto the nascent lighter compact object: tentative evidence in the O3 run data of LIGO/Virgo

In a binary system, the gravitational potential of the primary black hole may play an important role in enhancing the fallback accretion onto the lighter compact object newly formed in the second supernova explosion. As a result, the final masses of the binary compact objects would be correlated, as suggested recently by Safarzadeh & Wysocki (2021). In this work, we analyze the mass distribution of four gravitational wave events, which are characterized by both a small mass ratio and a low mass ($leq 5M_odot$) of the light component, and find tentative evidence for a mass correlation among the objects. To evaluate the feasibility of testing such a hypothesis with upcoming observations, we carry out simulations with a mock population and perform Bayesian hierarchical inference for the mass distribution. We find that with dozens of low mass ratio events, whether there exists correlation in the component mass distributions or not can be robustly tested and the correlation, if it exists, can be well determined.

Bilinear Control of Convection-Cooling: From Open-Loop to Closed-Loop

This paper is concerned with a bilinear control problem for enhancing convection-cooling via an incompressible velocity field. Both optimal open-loop control and closed-loop feedback control designs are addressed. First and second order optimality conditions for characterizing the optimal solution are discussed. In particular, the method of instantaneous control is applied to establish the feedback laws. Moreover, the construction of feedback laws is also investigated by directly utilizing the optimality system with appropriate numerical discretization schemes. Computationally, it is much easier to implement the closed-loop feedback control than the optimal open-loop control, as the latter requires to solve the state equations forward in time, coupled with the adjoint equations backward in time together with a nonlinear optimality condition. Rigorous analysis and numerical experiments are presented to demonstrate our ideas and validate the efficacy of the control designs.

A flexible Gaussian process reconstruction method and the mass function of the coalescing binary black hole systems

We develop a new method based on Gaussian process to reconstruct the mass distribution of binary black holes (BBHs). Instead of prespecifying the formalisms of mass distribution, we introduce a more flexible and nonparametric model with which the distribution can be mainly determined by the observed data. We first test our method with simulated data, and find that it can well recover the injected distribution. Then we apply this method to analyze the data of BBHs observations from LIGO-Virgo Gravitational-Wave Transient Catalog 2. By reconstructing the chirp mass distribution, we find that there is a peak or a platform locating at $20-30,M_{odot}$ rather than a single-power-law-like decrease from low mass to high mass. Moreover, one or two peaks in the chirp mass range of $mathcal{M}<20,M_{odot}$ may be favored by the data. Assuming a mass-independent mass ratio distribution of $p(q)propto q^{1.4}$, we further obtain a distribution of primary mass, and find that there is a feature locating in the range of $(30, 40),M_{odot}$, which can be related to textsc{Broken Power Law} and textsc{Power Law + Peak} distributions described in The LIGO Scientific Collaboration et al. (2020). Besides, the merger rate of BBHs is estimated to $mathcal{R} = 26.29^{+14.21}_{-8.96}~{rm Gpc^{-3}~yr^{-1}}$ supposing there is no redshift evolution.

Black Hole Mass Function of Coalescing Binary Black Hole Systems: Is There a Pulsational Pair Instability Mass Cutoff?

We analyze the LIGO/Virgo GWTC-2 catalog to study the primary mass distribution of the merging black holes. We perform hierarchical Bayesian analysis, and examine whether the mass distribution has a sharp cutoff for primary black hole masses below $65 M_odot$, as predicted in pulsational pair instability supernova model. We construct two empirical mass functions. One is a piece-wise function with two power-law segments jointed by a sudden drop. The other consists of a main truncated power-law component, a Gaussian component, and a third very massive component. Both models can reasonably fit the data and a sharp drop of the mass distribution is found at $sim 50M_odot$, suggesting that the majority of the observed black holes can be explained by the stellar evolution scenarios in which the pulsational pair-instability process takes place. On the other hand, the very massive sub-population, which accounts for at most several percents of the total, may be formed through hierarchical mergers or other processes.

Generalized Nearest Neighbor Decoding

It is well known that for linear Gaussian channels, a nearest neighbor decoding rule, which seeks the minimum Euclidean distance between a codeword and the received channel output vector, is the maximum likelihood solution and hence capacity-achieving. Nearest neighbor decoding remains a convenient and yet mismatched solution for general channels, and the key message of this paper is that the performance of the nearest neighbor decoding can be improved by generalizing its decoding metric to incorporate channel state dependent output processing and codeword scaling. Using generalized mutual information, which is a lower bound to the mismatched capacity under independent and identically distributed codebook ensemble, as the performance measure, this paper establishes the optimal generalized nearest neighbor decoding rule, under Gaussian channel input. Several suboptimal but reduced-complexity generalized nearest neighbor decoding rules are also derived and compared with existing solutions. The results are illustrated through several case studies for channels with nonlinear effects, and fading channels with receiver channel state information or with pilot-assisted training.

GW190521 and the GWTC-1 Events: Implication on the Black Hole Mass Function of Coalescing Binary Black Hole Systems

With the black hole mass function (BHMF; assuming an exponential cutoff at a mass of $sim 40,M_odot$) of coalescing binary black hole systems constructed with the events detected in the O1 run of the advanced LIGO/Virgo network, Liang et al.(2017) predicted that the birth of the lightest intermediate mass black holes (LIMBHs; with a final mass of $gtrsim 100,M_odot$) is very likely to be caught by the advanced LIGO/Virgo detectors in their O3 run. The O1 and O2 observation run data, however, strongly favor a cutoff of the BHMF much sharper than the exponential one. In this work we show that a power-law function followed by a sudden drop at $sim 40,M_odot$ by a factor of $sim $a few tens and then a new power-law component extending to $geq 100M_odot$ are consistent with the O1 and O2 observation run data. With this new BHMF, quite a few LIMBH events can be detected in the O3 observation run of advanced LIGO/Virgo. The first LIMBH born in GW190521, an event detected in the early stage of the O3 run of advanced LIGO/Virgo network, provides additional motivation for our hypothesis.