I Introduction
The first detection (GW150914) of gravitational waves (GWs), from the merger of two black holes (BHs), with the advanced Laser Interferometer Gravitational-wave Observatory (LIGO) The LIGO Scientific Collaboration et al. (2015) has set in motion a scientific revolution Abbott et al. (2016a) leading to the Nobel prize in Physics in 2017. This and subsequent groundbreaking discoveries Abbott et al. (2016b, c, 2017); The LIGO Scientific Collaboration et al. (2017) were brought to fruition by a trans-disciplinary research program at the interface of experimental and theoretical physics, computer science and engineering as well as the exploitation of high-performance computing (HPC) for numerical relativity simulations Einstein (1915); Abbott et al. (2016d); Usman et al. (2016) and high-throughput computing facilities for data analysis Huerta et al. (2017a); Weitzel et al. (2017).
The recent detection of the binary black hole (BBH) merger (GW170814) with a three-detector network enabled new phenomenological tests of general relativity regarding the nature of GW polarizations, while significantly improving the sky localization of this GW transient The LIGO Scientific Collaboration et al. (2017). This enhanced capability to localize GW transients provided critical input for the first detection of GWs from the merger of two neutron stars (NSs) and in conjunction with follow-up observations across the electromagnetic (EM) spectrum The LIGO Scientific Collaboration and The Virgo Collaboration (2017). This multimessenger event has finally confirmed that NS mergers are the central engines of short gamma ray bursts Eichler et al. (1989); Paczynski (1986); Narayan et al. (1992); Kochanek and Piran (1993); Ott (2009); Phinney (2009).
Matched-filtering, the most sensitive GW detection algorithm used by LIGO, currently targets a 3D parameter space (compact binary sources with spin-aligned components on quasi-circular orbits) Gerosa et al. (2013); Rodriguez et al. (2016, 2015)—a subset of the 8D parameter space available to GW detectors Antonini et al. (2016); Naoz et al. (2013); Samsing et al. (2014); Huerta et al. (2017b); Lehner and Pretorius (2014). Recent studies also indicate that these searches may miss GWs generated by compact binary populations formed in dense stellar environments Klimenko et al. (2016); Huerta et al. (2017b, 2014); Huerta and Brown (2013). Extending these template-matching searches to target spin-precessing, quasi-circular or eccentric BBHs is computationally prohibitive Harry et al. (2016).
Based on the aforementioned considerations, we need a new paradigm to overcome the limitations and computational challenges of existing GW detection algorithms. An ideal candidate would be the rapidly advancing field called Deep Learning, which is a highly scalable machine learning technique that can learn directly from raw data, without any manual feature engineering, by using deep hierarchical layers of “artificial neurons”, called neural networks, in combination with optimization techniques based on back-propagation and gradient descent
Lecun et al. (2015); Goodfellow et al. (2016). Deep learning, especially with the aid of GPU computing, has recently achieved immense success in both commercial applications and artificial intelligence (AI) research
Lecun et al. (2015); Schmidhuber (2015); Esteva et al. (2017); Silver et al. (2017); Moravčík et al. (2017); van den Oord et al. (2016); O’Shea et al. (2016).Our technique, Deep Filtering George and Huerta (2017), employs a system of two deep convolution neural networks (CNNs LeCun et al. (1998)) that directly take time-series inputs for both classification and regression. In our foundational article George and Huerta (2017), we provided a comprehensive introduction to the fundamental concepts of deep learning and CNNs along with a detailed description of this method. Our previous results showed that CNNs can outperform traditional machine learning methods, reaching sensitivities comparable to matched-filtering for directly processing highly noisy time-series data streams to detect weak GW signals and estimate the parameters of their source in real-time, using GW signals injected into simulated LIGO noise.
In this article, we extend Deep Filtering to analyze GW signals in real LIGO noise. We demonstrate, for the first time, that Deep Learning can be used for both signal detection and multiple-parameter estimation directly from extremely weak time-series signals embedded in highly non-Gaussian and non-stationary noise, once trained with some templates of the expected signals. Our results show that deep CNNs achieve performance comparable to matched-filtering methods, while being several orders of magnitude faster and far more resilient to transient noise artifacts such as glitches. We also illustrate how Deep Filtering can deal with data streams of arbitrary length from multiple detectors. Most importantly, this article shows for the very first time that machine learning can successfully detect and recover true parameters of real GW signals observed by LIGO. Furthermore, we show that after a single training process, Deep Filtering can automatically generalize to noise having new Power Spectral Densities (PSDs) from different LIGO events, without re-training.
Our results indicate that Deep Filtering
can interpolate between templates,
generalize to new classes of signals beyond the training data, and, surprisingly, detect GW signals and measure their parameters even when they are contaminated by glitches. We present experiments demonstrating the robustness of Deep Filtering in the presence of glitches, which indicate its applicability in the future for glitch classification and clustering efforts essential for GW detector characterization. Deep learning, in principle, can learn characteristics of noise in the LIGO detectors and develop better strategies than matched-filtering, which is known to be only optimal for Gaussian noise. Since all the intensive computation is diverted to the one-time training stage of the CNNs, template banks of practically any size may be used for training after which continuous data streams can be analyzed in real-time with a single CPU, while very intensive searches can be rapidly carried out using a single GPU. Deep Filtering can also be used to instantly narrow down the parameter space of GW detections, which can then be followed up with existing pipelines using a few templates around the predicted parameters, thus accelerating GW analysis with minimal computational resources across the full parameter space of signals.Ii Methods
Deep Filtering
consists of two steps, involving a classifier CNN and a predictor CNN, with similar architectures, as described in our previous article
George and Huerta (2017). The classifier has an additional softmax layer which returns probabilities for True or False depending on whether a signal is present. The classifier is first applied to the continuous data stream via a sliding window. If the classifier returns higher probability for True, the predictor is applied to the same input to determine the parameters of the source. In a multi-detector scenario, the
Deep Filtering CNNs may be applied separately to each data stream and the coincidence of detections with similar parameters would strengthen the confidence of a true detection, which can then be verified quickly by matched-filtering with the predicted templates.In this work, we have used injections of GW templates originating from quasi-circular, non-spinning, stellar-mass BBH systems, which LIGO/Virgo is expected to detect with the highest rate Belczynski et al. (2016). We assumed the source is optimally oriented with respect to the detectors which reduces our parameter-space to the two individual masses of the BBH system, which we restricted in the range to such that their mass-ratios were between 1 and 10. In the same manner as before, we fixed the input duration to 1 second, and a sampling rate of 8192Hz, which is more than sufficient for the events we are considering. These are arbitrary choices, as the input size of the CNNs can be easily modified to take inputs with any duration or sampling rate from any number of detectors.
The datasets of waveform templates used to train and test our CNNs were obtained using the open-source, effective-one-body (EOB) code Taracchini et al. (2014). Our training set contained about 2500 templates, with BBHs component masses sampled in the range to in steps of . The testing dataset also contained approximately 2500 templates with intermediate component masses separated from the training set by each. Subsequently, we produced copies of each signal by shifting the location of their peaks randomly within the final 0.2 seconds to make the CNNs more resilient to time translations. This means that in practice, our algorithm will be applied to the continuous data stream using a 1-second sliding window with offsets of 0.2 seconds.
We obtained real LIGO data from the LIGO Open Science Center (LOSC) around the first 3 GW events, namely, GW150914, LVT151012, and GW151226. Each event contained 4096 seconds of real data from each detector. We used noise sampled from GW151226 and LVT151012 for training and validation of our model and noise from GW150914 was used for testing. These tests ensure that our method is able to generalize to different noise distributions, also in the presence of transient glitches, since it is well known that the PSD of LIGO is highly non-stationary, varying widely with time. Therefore, if Deep Filtering performs well on these test sets, it would also perform well on data from future time periods, without being re-trained.
Next, we superimposed different realizations of noise randomly sampled from the training set of real LIGO noise from the two events GW151226 and LVT151012 and injected signals over multiple iterations, thus amplifying the size of the training datasets. The power of the noise was adjusted according to the desired optimal matched-filter Signal-to-Noise Ratio (SNR Owen and Sathyaprakash (1999)) for each training round. The inputs were then whitened with the average PSD of the real noise measured at that time-period. We also scaled and mixed different samples of LIGO noise together to artificially produce more training data and various levels of Gaussian noise was also added to augment the training process. However, the testing results were measured using only pure LIGO noise not used in training with true GW signals or with signals injected from the unaltered test sets (see Fig. 1).
We used similar hyperparameters to our original CNNs
George and Huerta (2017)with a slightly deeper architecture. There were 4 convolution layers with the filter sizes to 64, 128, 256, and 512 respectively and 2 fully connected layers with sizes 128 and 64. The standard ReLU activation function, max
, was used throughout as the non-linearity between layers. We used kernel sizes of 16, 16, 16, and 32 for the convolutional layers and 4 for all the (max) pooling layers. Stride was chosen to be 1 for all the convolution layers and 4 for all the pooling layers. We observed that using dilations
Yu and Koltun (2016) of 1, 2, 2, and 2 in the corresponding convolution layers improved the performance. The final layout of our predictor CNN is shown in Fig. 3.We had originally optimized this CNN architecture to deal with only Gaussian noise having a flat PSD. However, we later found that this model also obtained the best performance with noise having the colored PSD of LIGO, among all the models we tested. This indicates that our architecture is robust to a wide range of noise distributions. Furthermore, pre-training the CNNs on Gaussian noise (transfer learning) before fine-tuning on the limited amount of real noise prevented over-fitting, i.e., memorizing only the training data without generalizing to new inputs. We used the Wolfram Language neural network functionality, based on the open-source MXNet framework
Chen et al. (2015), that uses the cuDNN library Chetlur et al. (2014) for accelerating the training with NVIDIA GPUs. The learning algorithm was again set to ADAM Kingma and Ba (2014) and other details were the same as before George and Huerta (2017).For training, we used the curriculum learning strategy in our first article George and Huerta (2017) to improve the performance and reduce training times of the CNNs while retaining performance at very high SNR. By starting off training inputs having high SNR () and then gradually increasing the noise in each subsequent training session until a final SNR distributed in the range 4 to 15, we found that the performance of prediction can be quickly maximized for low SNR while retaining performance at high SNR. We first trained the predictor on the datasets labeled with the BBH masses and then copied the weights of this network to initialize the classifier and then trained it on datasets having 90% pure random noise inputs, after adding a softmax layer. This transfer learning procedure, similar to multi-task learning, decreases the training time for the classifier and improves its sensitivity.
Iii Results
The sensitivity (probability of detecting a true signal) of the classifier as a function of SNR is shown in Fig. 4. We achieved 100% sensitivity when SNR is greater than 10. The false alarm rate was tuned to be less than 1%, i.e., 1 per 100 seconds of noise in our test set was classified as signals. Given independent noise from multiple detectors, this implies our 2-detector false alarm rate would be less than 0.01%, when the classifier is applied independently to each detector and coincidence is enforced. Although the false alarm rate can be further decreased by tuning the fraction of noise used for training or by checking that the predicted parameters are consistent, this may not be necessary since running matched-filtering pipelines with a few templates close to our predicted parameters can quickly eliminate these false alarms.
Our predictor was able to successfully measure the component masses given noisy GW signals, that were not used for training, with an error lower than the spacing between templates for optimal matched-filter SNR . The variation in relative error against SNR is shown in Fig. 5
. We observed that the errors follow a Gaussian distribution for each region of the parameter space for SNR greater than 10. For high SNR, our predictor achieved mean relative error less than 10%, whereas matched-filtering with the same template bank always has error greater than 10%. This implies that Deep Filtering is capable of interpolating between templates seen in the training data.
Although, we trained only on simulated quasi-circular non-spinning GW injections, we applied Deep Filtering to the LIGO data streams containing a true GW signal, GW150914, using a sliding window of 1s width with offsets of 0.2s through the data around each event from each detector. This signal was correctly identified by the classifier at the true position in time and each of the predicted component masses were within the published error bars Abbott et al. (2016a). There were zero false alarms after enforcing the constraint that the detection should be made simultaneously in multiple detectors. This shows that deep learning is able to generalize to real GW signals after being trained only with simulated GW templates injected into LIGO noise from other events with different PSDs. A demo showing the application of Deep Filtering to GW150914 can be found here: tiny.cc/CNN.
The data from the first LIGO event, that was used for testing, contained a large number of non-Gaussian transient noise called glitches. Some of these can be seen in Fig. 2. Therefore, our results demonstrate that the Deep Filtering method can automatically recognize these glitches and classify them as noise. This suggests that by adding additional neurons for each “glitch” class, Deep Filtering could serve as an alternative to glitch classification algorithms based on two-dimensional CNNs applied to spectrograms of LIGO Zevin et al. (2016); George et al. (2017) or machine learning methods based on manually engineered features Mukund et al. (2017); Powell et al. (2017, 2015).
Furthermore, we conducted some experiments to show the resilience of Deep Filtering to transient disturbances, with a simulated set of sine-Gaussian glitches, which cover a broad range of morphologies found in real LIGO glitches, following Powell et al. (2017) (see Fig. 6 for some examples). We ensured that a different set of frequencies, amplitudes, peak positions, and widths were used for training and testing. We then injected some of these glitches into the training process and found that the classifier CNN was able to easily distinguish new glitches from true signals, with a false alarm rate less than 1%. When we applied the standard naive matched-filtering algorithm to the same test set of glitches, approximately 30% of glitches were classified as signals due to their high SNR. Note that signal consistency tests and coherence across detectors can be enforced to decrease this false alarm rate for both methods.
We then tested the performance of Deep Filtering, when a signal happens to occur in coincidence with a glitch, i.e., the signal is superimposed with both a glitch and real LIGO noise. We trained the network by injecting glitches from the training set into the training process and measured the sensitivity of the classifier on the test set signals superimposed with glitches sampled from the test set of glitches. We found that over 80% of the signals with SNR of 10 were detected, and their parameters estimated with less than 30% relative error, even after they were superimposed with glitches. These results are very promising, motivating further in-depth investigation, since we may be able to automatically detect GW signals that occur during periods of bad data quality in the detectors using Deep Filtering, whereas currently such periods are often vetoed and left out of the analysis by LIGO pipelines.
Another important experiment that we carried out was to inject waveforms obtained from simulations of eccentric BBH systems, with eccentricities between 0.1 and 0.2 when entering the LIGO band, that we performed using the open-source Einstein Toolkit Löffler et al. (2012) as well as waveforms from spin-precessing binaries from the public SXS catalog Chu et al. (2016). We found that these signals were detected with the same sensitivity as the original test set of quasi-circular BBH waveforms by our classifier, thus demonstrating its ability to automatically generalize to new classes of GW signals.
Both our CNNs are only 23MB in size each, yet encodes all the relevant information from about 2,500 GW templates (~300MB) of templates and several GB of noise used to generate the training data. The time-domain matched-filtering algorithm used for comparison required over 2s to analyze 1s inputs on our CPU. The average time taken for evaluating each of our CNNs per second of data is approximately 85 milliseconds and 540 microseconds using a single CPU core and GPU respectively, thus enabling analysis even faster than real-time. While the computational cost of matched-filtering grows exponentially with the number of parameters, the Deep Filtering algorithm requires only a one-time training process, after which the analysis can be performed in constant time. Therefore, we expect the speed-up compared to matched-filtering to further increase by several orders of magnitude when the analysis is extended to larger number of parameters.
Iv Conclusion
In this article, we have shown for the very first time that CNNs can be used for both detection and parameter estimation of GW signals in LIGO data. This new paradigm for real-time GW analysis may overcome outstanding challenges regarding the extension of established GW detection algorithms to higher dimensions for targeting a deeper parameter space of astrophysically motivated GW sources. The results of Deep Filtering can be quickly verified via matched-filtering with a small set of templates in the predicted region of parameter space. Therefore, by combining Deep Filtering with well-established GW detection algorithms we may be able to accelerate multimessenger campaigns, pushing the frontiers of GW astrophysics and fully realize its potential for scientific discovery.
The intrinsic scalability of deep learning can overcome the curse of dimensionality, and take advantage of terabytes of training data. This ability could enable simultaneous GW searches covering millions or billions of templates over the full range of parameter-space that is beyond the reach of existing algorithms. Extending
Deep Filtering to predict any number of parameters such as spins, eccentricities, etc., or additional classes of signals or noise, is as simple as adding an additional neuron for each new parameter, or class, to the final layer and training with noisy waveforms with the corresponding labels. Furthermore, the input dimensions of the CNNs can be enlarged to take time-series inputs from multiple detectors, thus allowing coherent searches and measurements of parameters such as sky locations.The results presented in this article provide a strong incentive to extend Deep Filtering to cover the parameter space of spin-aligned BBHs on quasi-circular orbits and beyond. This study is underway, and will be described in a subsequent article. In addition to our primary results, we have also presented experiments exhibiting the remarkable resilience of this method for detection in periods of bad data quality, even when GW signals are contaminated with non-Gaussian transients. This motivates including additional classes of real glitches, e.g., from the Gravity Spy project Zevin et al. (2016), to the training process to automatically classify and cluster glitches directly from the time-series inputs. Therefore, a single, robust, and efficient data analysis pipeline for GW detectors, based on Deep Filtering, that unifies detection and parameter estimation along with glitch classification and clustering in real-time with very low computational overhead may potentially be built in the near future.
Furthermore, Deep Filtering can be used to accelerate Bayesian parameter estimation methods by constraining the parameter space of new GW detections and provide instant alerts with accurate parameters for EM follow-up campaigns. As deep CNNs excel at image processing, applying the same approach to analyze raw telescope data may also accelerate the subsequent search for transient EM counterparts. Our results also suggest that, given templates of expected signals, Deep Filtering can be used as a generic tool for efficiently detecting and estimating properties of highly noisy time-domain signals embedded in Gaussian noise or non-stationary non-Gaussian noise, even in the presence of transient disturbances. Therefore, we anticipate that the techniques we have developed for analyzing weak signals hidden in complex noise backgrounds could be useful in many other domains of science and technology.
V Acknowledgements
This research is part of the Blue Waters sustained-petascale computing project, supported by the National Science Foundation (awards OCI-0725070 and ACI-1238993) and the state of Illinois. Blue Waters is a joint effort of the University of Illinois at Urbana-Champaign and its National Center for Supercomputing Applications. The eccentric numerical relativity simulations used in this article were generated with the open source, community software, the Einstein Toolkit on the Blue Waters petascale supercomputer and XSEDE (TG-PHY160053). We express our gratitude to Gabrielle Allen, Ed Seidel, Roland Haas, Miguel Holgado, Haris Markakis, Zhizhen Zhao and other members of the NCSA Gravity Group along with Prannoy Mupparaju for comments and interactions and to the many others who reviewed our manuscript, including Jin Li and Kai Staats. We acknowledge the LIGO/Virgo collaboration, especially the CBC and MLA groups, for their feedback and for use of computational resources. We thank Vlad Kindratenko for granting us access to GPUs and HPC resources in the Innovative Systems Lab at NCSA. We are grateful to Wolfram Research for providing technical assistance and licenses and to NVIDIA for donating several Tesla P100 GPUs, which we used for our analysis. This document has LIGO number P1700379.
References
- The LIGO Scientific Collaboration et al. (2015) The LIGO Scientific Collaboration, J. Aasi, et al., Classical and Quantum Gravity 32, 074001 (2015), arXiv:1411.4547 [gr-qc] .
- Abbott et al. (2016a) B. P. Abbott, R. Abbott, T. D. Abbott, M. R. Abernathy, F. Acernese, K. Ackley, C. Adams, T. Adams, P. Addesso, R. X. Adhikari, and et al., Physical Review Letters 116, 061102 (2016a), arXiv:1602.03837 [gr-qc] .
- Abbott et al. (2016b) B. P. Abbott, R. Abbott, T. D. Abbott, M. R. Abernathy, F. Acernese, K. Ackley, C. Adams, T. Adams, P. Addesso, R. X. Adhikari, and et al., Physical Review Letters 116, 241103 (2016b), arXiv:1606.04855 [gr-qc] .
- Abbott et al. (2016c) B. P. Abbott, R. Abbott, T. D. Abbott, M. R. Abernathy, F. Acernese, K. Ackley, C. Adams, T. Adams, P. Addesso, R. X. Adhikari, and et al., Physical Review X 6, 041015 (2016c), arXiv:1606.04856 [gr-qc] .
- Abbott et al. (2017) B. P. Abbott, R. Abbott, T. D. Abbott, M. R. Abernathy, F. Acernese, K. Ackley, C. Adams, T. Adams, P. Addesso, R. X. Adhikari, et al., Physical Review Letters 118, 221101 (2017).
- The LIGO Scientific Collaboration et al. (2017) The LIGO Scientific Collaboration, the Virgo Collaboration, B. P. Abbott, R. Abbott, T. D. Abbott, F. Acernese, K. Ackley, C. Adams, T. Adams, P. Addesso, and et al., ArXiv e-prints (2017), arXiv:1709.09660 [gr-qc], arXiv:1709.09660 [gr-qc] .
- Einstein (1915) A. Einstein, Königlich Preussische Akademie der Wissenschaften Zu Berlin, Sitzungberichte 1915, 844 (1915).
- Abbott et al. (2016d) B. P. Abbott, R. Abbott, T. D. Abbott, M. R. Abernathy, F. Acernese, K. Ackley, C. Adams, T. Adams, P. Addesso, R. X. Adhikari, and et al., Physical Review D 94, 064035 (2016d), arXiv:1606.01262 [gr-qc] .
- Usman et al. (2016) S. A. Usman, A. H. Nitz, I. W. Harry, C. M. Biwer, D. A. Brown, M. Cabero, C. D. Capano, T. Dal Canton, T. Dent, S. Fairhurst, M. S. Kehl, D. Keppel, B. Krishnan, A. Lenon, A. Lundgren, A. B. Nielsen, L. P. Pekowsky, H. P. Pfeiffer, P. R. Saulson, M. West, and J. L. Willis, Classical and Quantum Gravity 33, 215004 (2016), arXiv:1508.02357 [gr-qc] .
- Huerta et al. (2017a) E. A. Huerta, R. Haas, E. Fajardo, D. S. Katz, S. Anderson, P. Couvares, J. Willis, T. Bouvet, J. Enos, W. T. C. Kramer, H. W. Leong, and D. Wheeler, ArXiv e-prints (2017a), arXiv:1709.08767 [cs.DC] .
- Weitzel et al. (2017) D. Weitzel, B. Bockelman, D. A. Brown, P. Couvares, F. Würthwein, and E. Fajardo Hernandez, ArXiv e-prints (2017), arXiv:1705.06202 [cs.DC] .
- The LIGO Scientific Collaboration and The Virgo Collaboration (2017) The LIGO Scientific Collaboration and The Virgo Collaboration, ArXiv e-prints (2017), arXiv:1710.05832 [gr-qc] .
- Eichler et al. (1989) D. Eichler, M. Livio, T. Piran, and D. N. Schramm, Nature 340, 126 (1989).
- Paczynski (1986) B. Paczynski, Astrophysical Journal Letters 308, L43 (1986).
- Narayan et al. (1992) R. Narayan, B. Paczynski, and T. Piran, Astrophysical Journal Letters 395, L83 (1992), astro-ph/9204001 .
- Kochanek and Piran (1993) C. S. Kochanek and T. Piran, Astrophysical Journal 417, L17 (1993), arXiv:astro-ph/9305015 [astro-ph] .
- Ott (2009) C. D. Ott, Classical and Quantum Gravity 26, 063001 (2009), arXiv:0809.0695 .
- Phinney (2009) E. S. Phinney, in The Astronomy and Astrophysics Decadal Survey, Astronomy, Vol. 2010 (2009) arXiv:0903.0098 [astro-ph.CO] .
- Gerosa et al. (2013) D. Gerosa, M. Kesden, E. Berti, R. O’Shaughnessy, and U. Sperhake, Physical Review D 87, 104028 (2013), arXiv:1302.4442 [gr-qc] .
- Rodriguez et al. (2016) C. L. Rodriguez, S. Chatterjee, and F. A. Rasio, Physical Review D 93, 084029 (2016), arXiv:1602.02444 [astro-ph.HE] .
- Rodriguez et al. (2015) C. L. Rodriguez, M. Morscher, B. Pattabiraman, S. Chatterjee, C.-J. Haster, and F. A. Rasio, Physical Review Letters 115, 051101 (2015), arXiv:1505.00792 [astro-ph.HE] .
- Antonini et al. (2016) F. Antonini, S. Chatterjee, C. L. Rodriguez, M. Morscher, B. Pattabiraman, V. Kalogera, and F. A. Rasio, Astrophysical Journal 816, 65 (2016), arXiv:1509.05080 .
- Naoz et al. (2013) S. Naoz, B. Kocsis, A. Loeb, and N. Yunes, Astrophysical Journal 773, 187 (2013), arXiv:1206.4316 [astro-ph.SR] .
- Samsing et al. (2014) J. Samsing, M. MacLeod, and E. Ramirez-Ruiz, Astrophysical Journal 784, 71 (2014), arXiv:1308.2964 [astro-ph.HE] .
- Huerta et al. (2017b) E. A. Huerta, P. Kumar, B. Agarwal, D. George, H.-Y. Schive, H. P. Pfeiffer, R. Haas, W. Ren, T. Chu, M. Boyle, D. A. Hemberger, L. E. Kidder, M. A. Scheel, and B. Szilagyi, Physical Review D 95, 024038 (2017b).
- Lehner and Pretorius (2014) L. Lehner and F. Pretorius, ARA&A 52, 661 (2014), arXiv:1405.4840 [astro-ph.HE] .
- Klimenko et al. (2016) S. Klimenko, G. Vedovato, M. Drago, F. Salemi, V. Tiwari, G. A. Prodi, C. Lazzaro, K. Ackley, S. Tiwari, C. F. Da Silva, and G. Mitselmakher, Physical Review D 93, 042004 (2016), arXiv:1511.05999 [gr-qc] .
- Huerta et al. (2014) E. A. Huerta, P. Kumar, S. T. McWilliams, R. O’Shaughnessy, and N. Yunes, Physical Review D 90, 084016 (2014), arXiv:1408.3406 [gr-qc] .
- Huerta and Brown (2013) E. A. Huerta and D. A. Brown, Physical Review D 87, 127501 (2013), arXiv:1301.1895 [gr-qc] .
- Harry et al. (2016) I. Harry, S. Privitera, A. Bohé, and A. Buonanno, Physical Review D 94, 024012 (2016), arXiv:1603.02444 [gr-qc] .
- Lecun et al. (2015) Y. Lecun, Y. Bengio, and G. Hinton, Nature 521, 436 (2015).
- Goodfellow et al. (2016) I. Goodfellow, Y. Bengio, and A. Courville, Deep Learning (MIT Press, 2016).
- Schmidhuber (2015) J. Schmidhuber, Neural Networks 61, 85 (2015).
- Esteva et al. (2017) A. Esteva, B. Kuprel, R. A. Novoa, J. Ko, S. M. Swetter, H. M. Blau, and S. Thrun, Nature 542, 115 (2017).
- Silver et al. (2017) D. Silver, J. Schrittwieser, K. Simonyan, I. Antonoglou, A. Huang, A. Guez, T. Hubert, L. Baker, M. Lai, A. Bolton, et al., Nature 550, 354 (2017).
- Moravčík et al. (2017) M. Moravčík, M. Schmid, N. Burch, V. Lisý, D. Morrill, N. Bard, T. Davis, K. Waugh, M. Johanson, and M. Bowling, Science (2017), 10.1126/science.aam6960.
- van den Oord et al. (2016) A. van den Oord, S. Dieleman, H. Zen, K. Simonyan, O. Vinyals, A. Graves, N. Kalchbrenner, A. Senior, and K. Kavukcuoglu, ArXiv e-prints (2016), arXiv:1609.03499 [cs.SD] .
- O’Shea et al. (2016) T. J. O’Shea, J. Corgan, and T. C. Clancy, “Convolutional radio modulation recognition networks,” in Engineering Applications of Neural Networks: 17th International Conference, EANN 2016, Aberdeen, UK, September 2-5, 2016, Proceedings, edited by C. Jayne and L. Iliadis (Springer International Publishing, Cham, 2016) pp. 213–226.
- George and Huerta (2017) D. George and E. A. Huerta, ArXiv e-prints (2017), arXiv:1701.00008 [astro-ph.IM] .
- LeCun et al. (1998) Y. LeCun, L. Bottou, Y. Bengio, and P. Haffner, Proceedings of the IEEE 86, 2278 (1998).
- Belczynski et al. (2016) K. Belczynski, D. E. Holz, T. Bulik, and R. O’Shaughnessy, Nature 534, 512 (2016), arXiv:1602.04531 [astro-ph.HE] .
- Taracchini et al. (2014) A. Taracchini, A. Buonanno, Y. Pan, T. Hinderer, M. Boyle, D. A. Hemberger, L. E. Kidder, G. Lovelace, A. H. Mroué, H. P. Pfeiffer, M. A. Scheel, B. Szilágyi, N. W. Taylor, and A. Zenginoglu, Physical Review D 89, 061502 (2014), arXiv:1311.2544 [gr-qc] .
- Owen and Sathyaprakash (1999) B. J. Owen and B. S. Sathyaprakash, Physical Review D 60, 022002 (1999).
- Yu and Koltun (2016) F. Yu and V. Koltun, in ICLR (2016).
- Chen et al. (2015) T. Chen, M. Li, Y. Li, M. Lin, N. Wang, M. Wang, T. Xiao, B. Xu, C. Zhang, and Z. Zhang, CoRR abs/1512.01274 (2015).
- Chetlur et al. (2014) S. Chetlur, C. Woolley, P. Vandermersch, J. Cohen, J. Tran, B. Catanzaro, and E. Shelhamer, CoRR abs/1410.0759 (2014).
- Kingma and Ba (2014) D. P. Kingma and J. Ba, CoRR abs/1412.6980 (2014).
- Zevin et al. (2016) M. Zevin, S. Coughlin, S. Bahaadini, E. Besler, N. Rohani, S. Allen, M. Cabero, K. Crowston, A. Katsaggelos, S. Larson, T. K. Lee, C. Lintott, T. Littenberg, A. Lundgren, C. Oesterlund, J. Smith, L. Trouille, and V. Kalogera, ArXiv e-prints (2016), arXiv:1611.04596 [gr-qc] .
- George et al. (2017) D. George, H. Shen, and E. A. Huerta, ArXiv e-prints (2017), arXiv:1706.07446 [gr-qc] .
- Mukund et al. (2017) N. Mukund, S. Abraham, S. Kandhasamy, S. Mitra, and N. S. Philip, Phys. Rev. D 95, 104059 (2017).
- Powell et al. (2017) J. Powell et al., Classical and Quantum Gravity 34, 034002 (2017), arXiv:1609.06262 [astro-ph.IM] .
- Powell et al. (2015) J. Powell, D. Trifirò, E. Cuoco, I. S. Heng, and M. Cavaglià, Classical and Quantum Gravity 32, 215012 (2015), arXiv:1505.01299 [astro-ph.IM] .
- Löffler et al. (2012) F. Löffler, J. Faber, E. Bentivegna, T. Bode, P. Diener, R. Haas, I. Hinder, B. C. Mundim, C. D. Ott, E. Schnetter, G. Allen, M. Campanelli, and P. Laguna, Classical and Quantum Gravity 29, 115001 (2012), arXiv:1111.3344 [gr-qc] .
- Chu et al. (2016) T. Chu, H. Fong, P. Kumar, H. P. Pfeiffer, M. Boyle, D. A. Hemberger, L. E. Kidder, M. A. Scheel, and B. Szilagyi, Classical and Quantum Gravity 33, 165001 (2016), arXiv:1512.06800 [gr-qc] .
Comments
There are no comments yet.