Optimizing neural network surrogate models: Application to black hole merger remnants

We present the first surrogate model for gravitational waveforms from the coalescence of precessing binary black holes. We call this surrogate model NRSur4d2s. Our methodology significantly extends recently introduced reduced-order and surrogate modeling techniques, and is capable of directly modeling numerical relativity waveforms without introducing phenomenological assumptions or approximations to general relativity. Motivated by GW150914, LIGO's first detection of gravitational waves from merging black holes, the model is built from a set of $276$ numerical relativity (NR) simulations with mass ratios $q \leq 2$, dimensionless spin magnitudes up to $0.8$, and the restriction that the initial spin of the smaller black hole lies along the axis of orbital angular momentum. It produces waveforms which begin $\sim 30$ gravitational wave cycles before merger and continue through ringdown, and which contain the effects of precession as well as all $\ell \in \{2, 3\}$ spin-weighted spherical-harmonic modes. We perform cross-validation studies to compare the model to NR waveforms \emph{not} used to build the model, and find a better agreement within the parameter range of the model than other, state-of-the-art precessing waveform models, with typical mismatches of $10^{-3}$. We also construct a frequency domain surrogate model (called NRSur4d2s_FDROM) which can be evaluated in $50\, \mathrm{ms}$ and is suitable for performing parameter estimation studies on gravitational wave detections similar to GW150914.

Advanced LIGO and Advanced Virgo are detecting a large number of binary stellar origin black hole (BH) mergers. A promising channel for accelerated BH merger lies in active galactic nucleus (AGN) discs of gas around supermasssive BHs. Here, we investigate the relative number of compact object (CO) mergers in AGN disc models, including BH, neutron stars (NS), and white dwarfs, via Monte Carlo simulations. We find the number of all merger types in the bulk disc grows ∝ t1/3 which is driven by the Hill sphere of the more massive merger component. Median mass ratios of NS–BH mergers in AGN discs are $\tilde{q}=0.07\pm 0.06(0.14\pm 0.07)$ for mass functions (MF) M−1(− 2). If a fraction fAGN of the observed rate of BH–BH mergers (RBH–BH) come from AGN, the rate of NS–BH (NS–NS) mergers in the AGN channel is ${R}_{\mathrm{ BH}\!-\!\mathrm{ NS}} \sim f_{\mathrm{ AGN}}[10,300]\, \rm {Gpc}^{-3}\, \rm {yr}^{-1},({\mathit{ R}}_{NS\!-\!NS} \le \mathit{ f}_{AGN}400\, \rm {Gpc}^{-3}\, \rm {yr}^{-1}$). Given the ratio of NS–NS/BH–BH LIGO search volumes, from preliminary O3 results the AGN channel is not the dominant contribution to observed NS–NS mergers. The number of lower mass gap events expected is a strong function of the nuclear MF and mass segregation efficiency. CO merger ratios derived from LIGO can restrict models of MF, mass segregation, and populations embedded in AGN discs. The expected number of electromagnetic (EM) counterparts to NS–BH mergers in AGN discs at z < 1 is $\sim [30,900]\, {\rm {yr}}^{-1}(f_{\mathrm{ AGN}}/0.1)$. EM searches for flaring events in large AGN surveys will complement LIGO constraints on AGN models and the embedded populations that must live in them.

We present techniques for successfully performing numerical relativity simulations of binary black holes with fourth-order accuracy. Our simulations are based on a new coding framework which currently supports higher-order finite differencing for the Baumgarte-Shapiro-Shibata-Nakamura formulation of Einstein's equations, but which is designed to be readily applicable to a broad class of formulations. We apply our techniques to a standard set of numerical relativity test problems, demonstrating the fourth-order accuracy of the solutions. Finally we apply our approach to binary black hole head-on collisions, calculating the waveforms of gravitational radiation generated and demonstrating significant improvements in waveform accuracy over second-order methods with typically achievable numerical resolution.

A surrogate model assisted with a subdomain model for permanent-magnet machine

With the goal of bringing theory, particularly numerical relativity, to bear on an astrophysical problem of critical interest to gravitational wave observers we introduce a model for coalescence radiation from binary black hole systems. We build our model using the Lazarus approach, a technique that bridges far and close limit approaches with full numerical relativity to solve Einstein equations applied in the truly nonlinear dynamical regime. We specifically study the post-orbital radiation from a system of equal-mass non-spinning black holes, deriving waveforms which indicate strongly circularly polarized radiation of roughly 3% of the system's total energy and 12% of its total angular momentum in just a few cycles. To support this result we first establish the reliability of the late-time part of our model, including the numerical relativity and close-limit components, with a thorough study of waveforms from a sequence of black hole configurations that varies from previously treated head-on collisions to a representative target for ``ISCO'' data corresponding to the end of the inspiral period. We then complete our model with a simple treatment for the early part of the spacetime based on a standard family of initial data for binary black holes in circular orbit. A detailed analysis shows strong robustness in the results as the initial separation of the black holes is increased from $5.0$ to $7.8M$ supporting our waveforms as a suitable basic description of the astrophysical radiation from this system. Finally, a simple fitting of the plunge waveforms is introduced as a first attempt to facilitate the task of analyzing data from gravitational wave detectors.

We study the compact binary population in star clusters, focusing on binaries containing black holes, using a self-consistent Monte Carlo treatment of dynamics and full stellar evolution. We find that the black holes experience strong mass segregation and become centrally concentrated. In the core the black holes interact strongly with each other and black hole-black hole binaries are formed very efficiently. The strong interactions, however, also destroy or eject the black hole-black hole binaries. We find no black hole-black hole mergers within our simulations but produce many hard escapers that will merge in the galactic field within a Hubble time. We also find several highly eccentric black hole-black hole binaries that are potential LISA sources, suggesting that star clusters are interesting targets for space-based detectors. We conclude that star clusters must be taken into account when predicting compact binary population statistics.

This research work deals with aspects concerned with delamination detection in composite structures as revealed by an approach based on vibration measurements. Variations in vibration characteristics generated in composite laminates indicate the existence of delaminations because degradation due to delamination causes reduction in flexural stiffness and strength of the material and as a result vibration parameters like natural frequency responses are changed. Hence, it is possible to monitor the variation in natural frequencies to identify the presence of delamination, and assess its size and location for online structural health monitoring (SHM). The approach to this paper, therefore, typically depends on undertaking the analysis of structural models implemented by finite element analysis (FEA). The numerical solutions using FE models known as the simulator computes the natural frequencies for the delaminated and undelaminated specimens of composite laminates. However, these FE models are computationally expensive, and surrogate (approximation) models are introduced to curtail the computational expense. The simulator is employed to solve the inverse problem using algorithms based on computational intelligence concepts. An artificial neural network (ANN) model is developed to also solve the inverse problem for delamination detection directly and to provide surrogate models integrated with optimization algorithms (the gradient-based local search and non-dominated sorting genetic algorithm-II) to contain the computationally expensive simulations by FEA. This approach is termed as surrogate assisted optimization and it is seen that the engagement of surrogate models in lieu of the FE models in the optimization loop greatly enhances the accuracy of delamination detection results within an affordable computational cost and provides control over handling different variables. Meanwhile, to aid with the building of effective surrogate models using substantial number of training datasets, K-means clustering algorithm is harnessed and this effectively reduces the large training datasets usually required for ANN training. This paper demonstrated that ANN and optimization algorithms with surrogates show immense potentialities for use in delamination damage detection scenarios. Prediction errors of the algorithms were quantified and they were shown to be satisfactory when applied to previously experimental data. The algorithms in their inverse formulations are capable of predicting accurately delamination parameters. Hence, these algorithms should be employed for application in the domain of SHM where their small computational requirements could be exploited for online damage detection.

Lithium-ion batteries (LIBs) have found widespread application in energy storage due to their high energy density, low self-discharge and low maintenance characteristics. However, the stability and longevity of the batteries under extreme operating conditions is yet to be fully understood. Numerous experimental and simulation studies have been performed to elucidate the effect of temperature, current, depth of discharge, and the number of cycles [1,2,3] on the performance of LIBs. These studies show that the solid-electrolyte interface (SEI) layer and gas formation accelerate at high C-rates due to increased electrolyte decomposition [4,5]. Consequently, the cell's internal resistance increases, resulting in ever-increasing irreversible heat generation inside the cell [6]. This heat generation leads to high internal temperature and a corresponding reduction in cell capacity. The data of cell temperature for a wide range of ambient temperatures and cycling rates is limited because cell performance is typically reported at room temperature. The design of battery packs for electric vehicles(EVs) is also based on test data taken at room conditions. Given that the cell's capacity, specific energy, maximum power output is bound to depend on C-rate and ambient temperature, it is necessary to test and design battery packs based on these parameters.The thermal characteristics of a lithium-ion cell can be predicted using a physics-based thermal model or data-driven methods (DDM) relying on empirical data. Experimental determination of internal cell parameters required in physics-based thermal models is challenging and time-consuming for commercial cells. On the other hand, DDM requires only cell cycling data at specific operating conditions, thereby eliminating the need for internal parameter estimation. This model predicts the experimental input and output data pattern and fits the response surface to estimate the unknown output. In this study, surrogate modelling has been employed to estimate the surface temperature, capacity, energy, and average power of commercial lithium-ion cells for different discharge currents and ambient temperatures. Surrogate modelling, a popular data analysis and reduced-order modelling technique, aims to find a global minimum of a particular objective function using a few objective function evaluations [7]. The surrogate-based model divides experimental data into training and test data sets. The training data set is employed to train the algorithm. After that, the testing data set is used to validate the model's accuracy. Experiments were performed to develop a surrogate model, with the number of experiments decided using the design of experiments[8] for a current range of 0.5C to 3C-rate and ambient temperature range of 0°C to 45°C . The cycling of cells was performed using a high current battery cycler (Arbin), and the ambient temperature was maintained using a thermal chamber (Cincinnati). The surface temperature of commercial 18650 (NMC811) lithium-ion cells was recorded using T-type thermocouples and a National Instruments DAQ module. A polynomial response surface was fitted using surrogate modelling on experimentally obtained data. The response surface shown in figure 1. contains nine data points for the preliminary study, sub-divided into a set of seven training and two testing data. The prediction error sum of squares (PRESS) is currently bounded within 10% due to the limited training data set availability and is expected to be within a band of 1% after adding data from ongoing experiments. The estimation of temperature, capacity, and specific energy can be used to optimize and develop an improved thermal management system for electric vehicles that works under various operating conditions. References Waldmann, T., Wilka, M., Kasper, M., Fleischhammer, M., & Wohlfahrt-Mehrens, M. (2014). Journal of Power Sources, 262, 129-135.Guan, T., Sun, S., Yu, F., Gao, Y., Fan, P., Zuo, P., Du,C., & Yin, G. (2018). Electrochimica Acta, 279, 204-212.Simolka, M., Heger, J. F., Traub, N., Kaess, H., & Friedrich, K. A. (2020). Journal of The Electrochemical Society, 167(11), 110502.Xu, B., Diao, W., Wen, G., Choe, S. Y., Kim, J., & Pecht, M. (2021). Journal of Power Sources, 510, 230390.Rashid, M., & Gupta, A. (2017). Electrochimica Acta, 231, 171-184.Leng, F., Tan, C. M., & Pecht, M. (2015). Scientific reports, 5(1), 1-12.Queipo, N. V., Haftka, R. T., Shyy, W., Goel, T., Vaidyanathan, R., & Tucker, P. K. (2005). Progress in aerospace sciences, 41(1), 1-28.Li, W., Xiao, M., Peng, X., Garg, A., & Gao, L. (2019). Applied Thermal Engineering, 147, 90-100. Figure 1

A waveform model for the eccentric binary black holes named SEOBNRE has been used to analyze the LIGO-Virgo's gravitational wave data by several groups. The accuracy of this model has been validated by comparing it with numerical relativity. However, SEOBNRE is a time-domain model, and the efficiency for generating waveforms is a bottleneck in data analysis. To overcome this disadvantage, we offer a reduced-order surrogate model for eccentric binary black holes based on the SEOBNRE waveforms. This surrogate model (SEOBNRE_S) can simulate the complete inspiral-merger-ringdown waves with enough accuracy, covering eccentricities from 0 to 0.25 (0.1), and mass ratio from $1\ensuremath{\mathbin:}1$ to $5\ensuremath{\mathbin:}1$ ($2\ensuremath{\mathbin:}1$) for nonspinning (spinning) binaries. The speed of waveform generation is accelerated about ${10}^{2}--{10}^{3}$ times more than the original SEOBNRE model. Therefore SEOBNRE_S could be helpful in the analysis of LIGO data to find potential eccentricities.

ABSTRACTWe present the evolution of black holes (BHs) and their relationship with their host galaxies in Astrid, a large-volume cosmological hydrodynamical simulation with box size 250 h−1Mpc containing 2 × 55003 particles evolved to z = 3. Astrid statistically models BH gas accretion and active galactic nucleus (AGN) feedback to their environments, applies a power-law distribution for BH seed mass $\, M_{\rm sd}$, uses a dynamical friction model for BH dynamics, and executes a physical treatment of BH mergers. The BH population is broadly consistent with empirical constraints on the BH mass function, the bright end of the luminosity functions, and the time evolution of BH mass and accretion rate density. The BH mass and accretion exhibit a tight correlation with host stellar mass and star formation rate. We trace BHs seeded before z > 10 down to z = 3, finding that BHs carry virtually no imprint of the initial $\, M_{\rm sd}$ except those with the smallest $\, M_{\rm sd}$, where less than 50 per cent of them have doubled in mass. Gas accretion is the dominant channel for BH growth compared to BH mergers. With dynamical friction, Astrid predicts a significant delay for BH mergers after the first encounter of a BH pair, with a typical elapse time of about 200 Myr. There are in total 4.5 × 105 BH mergers in Astrid at z > 3, ∼103 of which have X-ray detectable EM counterparts: a bright $\, {\rm kpc}$ scale dual AGN with LX > 1043 erg s−1. BHs with $\, M_{\rm BH}\sim 10^{7\!-\!8} \, M_{\odot }$ experience the most frequent mergers. Galaxies that host BH mergers are unbiased tracers of the overall $\, M_{\rm BH}\!-\! M_*$ relation. Massive ($\gt 10^{11}\, M_{\odot }$) galaxies have a high occupation number (≳10) of BHs, and hence host the majority of BH mergers.

We calculate the linear momentum flux from merging black holes (BHs) with arbitrary masses and spin orientations, using the effective-one-body (EOB) model. This model includes an analytic description of the inspiral phase, a short merger, and a superposition of exponentially damped quasi-normal ring-down modes of a Kerr BH. By varying the matching point between inspiral and ring-down, we can estimate the systematic errors generated with this method. Within these confidence limits, we find close agreement with previously reported results from numerical relativity. Using a Monte Carlo implementation of the EOB model, we are able to sample a large volume of BH parameter space and estimate the distribution of recoil velocities. For a range of mass ratios 1 ≤ m1/m2 ≤ 10, spin magnitudes of a1, 2 = 0.9, and uniform random spin orientations, we find that a fraction f500 = 0.12 of binaries have recoil velocities greater than 500 km s-1 and that a fraction f1000 = 0.027 of binaries have kicks greater than 1000 km s-1. These velocities likely are capable of ejecting the final BH from its host galaxy. Limiting the sample to comparable-mass binaries with m1/m2 ≤ 4, the typical kicks are even larger, with f500 = 0.31 and f1000 = 0.079.

Stellar black hole (BH) binaries are one of the most promising gravitational wave (GW) sources for GW detection by the ground-based detectors. Nuclear star clusters (NCs) located at the centre of galaxies are known to harbour massive black holes (MBHs) and to be bounded by a gravitational potential by other galactic components such as the galactic bulge. Such an environment of NCs provides a favourable conditions for the BH–BH binary formation by the gravitational radiation capture due to the high BH number density and velocity dispersion. We carried out detailed numerical study of the formation of BH binaries in the NCs using a series of N-body simulations for equal-mass cases. There is no mass segregation introduced. We have derived scaling relations of the binary formation rate with the velocity dispersion of the stellar system beyond the radius of influence and made estimates of the rate of formation of BH binaries per unit comoving volume and thus expected detection rate by integrating the binary formation rate over galaxy population within the detection distance of the advanced detectors. We find that the overall formation rates for BH–BH binaries per NC is ∼10−10 yr−1 for the Milky Way-like galaxies and weakly dependent on the mass of MBH as Γ ∝ |$M_{\rm MBH}^{3/28}$|⁠. We estimate the detection rate of 0.02–14 yr−1 for advanced LIGO/Virgo considering several factors such as the dynamical evolution of NCs, the variance of the number density of stars and the mass range of MBH giving uncertainties.

Accurate models of gravitational waves from merging black holes are necessary for detectors to observe as many events as possible while extracting the maximum science. Near the time of merger, the gravitational waves from merging black holes can be computed only using numerical relativity. In this paper, we present a major update of the Simulating eXtreme Spacetimes (SXS) Collaboration catalog of numerical simulations for merging black holes. The catalog contains 2018 distinct configurations (a factor of 11 increase compared to the 2013 SXS catalog), including 1426 spin-precessing configurations, with mass ratios between 1 and 10, and spin magnitudes up to 0.998. The median length of a waveform in the catalog is 39 cycles of the dominant gravitational-wave mode, with the shortest waveform containing 7.0 cycles and the longest 351.3 cycles. We discuss improvements such as correcting for moving centers of mass and extended coverage of the parameter space. We also present a thorough analysis of numerical errors, finding typical truncation errors corresponding to a waveform mismatch of ∼10−4. The simulations provide remnant masses and spins with uncertainties of 0.03% and 0.1% (90th percentile), about an order of magnitude better than analytical models for remnant properties. The full catalog is publicly available at www.black-holes.org/waveforms.

A new radio map of the Abell 85 Brightest Cluster Galaxy (BCG) was obtained with the Karl G. Jansky Very Large Array. With a resolution of 0.″02, this radio image shows two kiloparsec-scale bipolar active galactic nucleus jets emanating from the active galactic nucleus of the A85 BCG. The galaxy core appears as a single entity on the new radio map. It has been assumed that the A85 BCG contained a binary black hole in its core. However, Chandra X-ray data and the new high-resolution radio map show no evidence that the A85 BCG harbors a binary black hole. The assumption that this galaxy contains a binary black hole was based on the analysis of its optical surface brightness profile obtained under poor seeing conditions. We demonstrate how the well-known blurring effects of atmospheric seeing can mimic the effects of a binary supermassive black hole (SMBH). Likewise, SDSS J004150.75–091824.3 was postulated to be “a third” SMBH associated with the BCG. In the optical and X-rays, SDSS J004150.75–091824.3 is a point-like source located ∼14″ away from the nucleus of the A85 BCG. A new spectrum of SDSS J004150.75–091824.3, obtained with the 10.4 m Gran Telescopio Canarias, reveals that this source is a background quasar at a redshift of z = 1.5603 ± 0.003 and not associated in any way with the A85 cluster.

Coalescing binaries of neutron stars (NS) and black holes (BH) are one of the most important sources of gravitational waves for the upcoming network of ground based detectors. Detection and extraction of astrophysical information from gravitational-wave signals requires accurate waveform models. The Effective-One-Body and other phenomenological models interpolate between analytic results and $10-30$ orbit numerical relativity (NR) merger simulations. In this paper we study the accuracy of these models using new NR simulations that span $36-88$ orbits, with mass-ratios and black hole spins $(q,\chi_{BH}) = (7, \pm 0.4), (7, \pm 0.6)$, and $(5, -0.9)$. We find that: (i) the recently published SEOBNRv1 and SEOBNRv2 models of the Effective-One-Body family disagree with each other (mismatches of a few percent) for black hole spins $\geq 0.5$ or $\leq -0.3$, with waveform mismatch accumulating during early inspiral; (ii) comparison with numerical waveforms indicate that this disagreement is due to phasing errors of SEOBNRv1, with SEOBNRv2 in good agreement with all of our simulations; (iii) Phenomenological waveforms disagree with SEOBNRv2 over most of the NSBH binary parameter space; (iv) comparison with NR waveforms shows that most of the model's dephasing accumulates near the frequency interval where it switches to a phenomenological phasing prescription; and finally (v) both SEOBNR and post-Newtonian (PN) models are effectual for NSBH systems, but PN waveforms will give a significant bias in parameter recovery. Our results suggest that future gravitational-wave detection searches and parameter estimation efforts targeted at NSBH systems with $q\lesssim 7$ and $\chi_\mathrm{BH} \approx [-0.9, +0.6]$ will benefit from using SEOBNRv2 templates. For larger black hole spins and/or binary mass-ratios, we recommend the models be further investigated as suitable NR simulations become available.