Machine learning gravitational waves from binary black hole mergers

We apply machine learning methods to build a time-domain model for gravitational waveforms from binary black hole mergers, called mlgw. The dimensionality of the problem is handled by representing the waveform’s amplitude and phase using a principal component analysis. We train mlgw on about $\mathcal{O}({10}^3)$ teobresums and seobnrv4 effective-one-body waveforms with mass ratios $q\in [1,20]$ and aligned dimensionless spins $s\in [-0.80,0.95]$. The resulting models are faithful to the training sets at the $\sim {10}^{-3}$ level (averaged on the parameter space). The speed up for a single waveform generation is a factor of 10–50 (depending on the binary mass and initial frequency) for teobresums and approximately an order of magnitude more for seobnrv4. Furthermore, mlgw provides a closed form expression for the waveform and its gradient with respect to the orbital parameters; such an information might be useful for future improvements in gravitational wave (GW) data analysis. As demonstration of the capabilities of mlgw to perform a full parameter estimation, we reanalyze the public data from the first GW transient catalog (GWTC-1). We find broadly consistent results with previous analyses at a fraction of the cost, although the analysis with spin-aligned waveforms gives systematic larger values of the effective spins with respect to previous analyses with precessing waveforms. Since the generation time does not depend on the length of the signal, our model is particularly suitable for the analysis of the long signals that are expected to be detected by third-generation detectors. Future applications include the analysis of waveform systematics and model selection in parameter estimation.

Figure 1

Average mismatch between waves ${\mathbf{f}}_{N_{\text{grid}},\alpha }$, as saved in the training dataset, and raw waves from EOB model, as a function of time grid size $N_{\text{grid}}$. Each series refers to a different values of $\alpha$. Clearly, $N_{\text{grid}}\simeq 3\times {10}^3$ and $\alpha \in (0.3,0.5)$ is a good choice for the dataset hyperparameters.

Figure 2

Validation results for a fit of the MoE model. Each point corresponds to an MoE regression for the amplitude (left) and phase (right), with a different value of expert number $N_{\exp }$ and order of polynomial basis function. The amplitude and phase are represented with five and four PCs, respectively. In the color bar, we represent the mismatch on test waves: It is obtained by reconstructing test waves with fitted amplitude (phase) and test phase (amplitude). A model with four experts and with a fourth-order polynomial provides good balance between simplicity and accuracy.

Figure 3

Test mismatch as a function of the number of PCs used in the low-dimensional representation. Label “PCA” refers to waves reconstructed with PCA only; points with label “MoE” are reconstructed after a MoE regression. Data refers to the amplitude (left panel) and phase (right panel). MoE model is chosen to be the optimal one, with four experts and a fourth-order polynomial.

Figure 4

Train and test error for MoE fit of four PCs of phase, as a function of the number of training points. We report train and test reconstruction mismatch (top) and MSE for the first three PCs (below). MoE model employs four experts and a fourth-order polynomial for a basis function expansion. Test mismatch are obtained using test amplitude to reconstruct the waveform; this is not a great limitation, as any error in phase reconstruction dominates the overall mismatch.

Figure 5

Logarithm of mismatch between teobresums and mlgw-teobresums, computed on $N=4000$ test waveforms. Each WF is generated with random masses and spins and with a starting frequency of 10 Hz. The median value $q_{50\%}$ and the positions $q_{5\%}$ and $q_{95\%}$ of the fifth and 95th percentile are reported.

Figure 6

To compare teobresums and mlgw-teobresums, we report test mismatch (left column) and MSE (right column) for the first PC of the phase, as a function of masses and spins. The histograms hold 145061 waveforms, with randomly drawn parameters. Each WF starts 8 s before merger. Apart from poor performances for $q\simeq 1$ and for high (positive) values of $s_1+s_2$, the model performance does not depend much on the input parameters.

Figure 7

Speed up given by mlgw-teobresums (dark-red online) and mlgw-seobnrv4 (blue online), as compared with their respective native implementation. Because of the computational cost, we use $N=500$ test waveforms for seobnrv4 and $N=2000$ waveforms for teobresums. Each WF is generated with random physical parameters and has a minimum frequency of 5 (top panel) and 20 Hz (bottom panel). We set a constant total mass $M=100M_{\odot }$ and the sampling rate $f_{\mathrm{sam}}=2048\mathrm{Hz}$. Median values $q_{50\%}^{\mathrm{TEOB}}$ and $q_{50\%}^{\mathrm{SEOB}}$ for the two models are also reported.

Figure 8

Speed up of mlgw with respect to seobnrv4_rom, computed on $N=2000$ test waveforms. Each WF is generated with random physical parameters and has a minimum frequency of 5 (top panel) and 20 Hz (bottom panel). We set a constant total mass $M=100M_{\odot }$ and the sampling rate $f_{\mathrm{sam}}=2048\mathrm{Hz}$. The median value $q_{50\%}$ and the positions $q_{5\%}$ and $q_{95\%}$ of the fifth and 95th percentile are reported.

Figure 9

Posterior probability densities of the component masses and final masses and spins of all the BBH system in GWTC-1 obtained using mlgw trained with the teobresums [8] spin-aligned waveform model. The contours enclose the 90% credible regions. Left panel: Source-frame component masses $m_1$ and $m_2$. We use the convention $m_1\ge m_2$ which produces the sharp cut in the two-dimensional $(m1,m2)$ distribution (shaded region). Lines of constant mass ratio $q\equiv m_1/m_2$ are shown for $q=\{2,4,8\}$. Right panel: The mass $M_f$ and dimensionless spin magnitude $a_f$ of the final black holes. The figure is consistent with, though different from, Fig. 4 of Ref. [42].

Figure 10

Posterior probability densities of the component masses and final masses and spins of all the BBH system in GWTC-1 obtained using mlgw trained with the seobnrv4 [23] waveform model. The contours enclose the 90% credible regions. Left panel: Source-frame component masses $m_1$ and $m_2$. We use the convention $m_1\ge m_2$ which produces the sharp cut in the two-dimensional $(m_1,m_2)$ distribution (shaded region). Lines of constant mass ratio $q\equiv m_1/m_2$ are shown for $q=\{2,4,8\}$. Right panel: The mass $M_f$ and dimensionless spin magnitude $a_f$ of the final black holes. The differences with Fig. 9 above are practically negligible.

Figure 11

Posterior probability densities of the component, dimensionless, spins for all the BBH systems in GWTC-1 obtained using mlgw-teobresums [8] or mlgw-seobnrv4 [23]. The contours enclose the 90% credible regions.