Fast Prediction and Evaluation of Gravitational Waveforms Using Surrogate Models

We propose a solution to the problem of quickly and accurately predicting gravitational waveforms within any given physical model. The method is relevant for both real-time applications and more traditional scenarios where the generation of waveforms using standard methods can be prohibitively expensive. Our approach is based on three offline steps resulting in an accurate reduced order model in both parameter and physical dimensions that can be used as a surrogate for the true or fiducial waveform family. First, a set of $m$ parameter values is determined using a greedy algorithm from which a reduced basis representation is constructed. Second, these $m$ parameters induce the selection of $m$ time values for interpolating a waveform time series using an empirical interpolant that is built for the fiducial waveform family. Third, a fit in the parameter dimension is performed for the waveform’s value at each of these $m$ times. The cost of predicting $L$ waveform time samples for a generic parameter choice is of order $\mathcal{O}(mL+m{c}_{\text{fit}})$ online operations, where $c_{\text{fit}}$ denotes the fitting function operation count and, typically, $m\ll L$. The result is a compact, computationally efficient, and accurate surrogate model that retains the original physics of the fiducial waveform family while also being fast to evaluate. We generate accurate surrogate models for effective-one-body waveforms of nonspinning binary black hole coalescences with durations as long as $10^{5}M$, mass ratios from 1 to 10, and for multiple spherical harmonic modes. We find that these surrogates are more than 3 orders of magnitude faster to evaluate as compared to the cost of generating effective-one-body waveforms in standard ways. Surrogate model building for other waveform families and models follows the same steps and has the same low computational online scaling cost. For expensive numerical simulations of binary black hole coalescences, we thus anticipate extremely large speedups in generating new waveforms with a surrogate. As waveform generation is one of the dominant costs in parameter estimation algorithms and parameter space exploration, surrogate models offer a new and practical way to dramatically accelerate such studies without impacting accuracy. Surrogates built in this paper, as well as others, are available from GWSurrogate, a publicly available python package.

Popular Summary

Data-driven modeling studies are often characterized by rapid, repeated evaluation of solutions to parametrized, time-dependent differential equations. Such studies, which allow scientists to infer system parameters or identify best model fits, are crucial for extracting a wealth of scientific information from models and experiments. Gravitational-wave science is a modeling-dependent field that requires the repeated evaluation of physically important yet prohibitively expensive source models. A single production-quality numerical relativity simulation of binary black hole mergers typically requires weeks or months of large-scale supercomputing time. We describe a new approach to gravitational-wave modeling, reduced-order surrogate modeling, that can be used to evaluate a high-accuracy approximation to a parametrized waveform family in a fraction of the time required for a direct solution of the model.

The high computational costs associated with modeling binary black hole mergers make it intractable to comprehensively explore the physical parameter space. We build surrogate models by evaluating the underlying expensive models at a representative set of a few parameter values and tying together their samples in a judicious way. This offline building stage needs to be performed only once. Our approach is nonintrusive to existing solvers, which can be complicated or publicly unavailable. The surrogate model may be systematically improved as more simulations become available, without having to discard previous simulations. The models themselves are portable, compact to store, and easy to use. Furthermore, our methodology is generic and may, in principle, be applied to a wide class of underlying gravitational waveform models. Parameter estimation studies performed with this semianalytical model, which previously took months to carry out, can return statistically indistinguishable results in less than a day.

Gravitational-wave detections will allow scientists to test the predictions of general relativity in strong gravitational fields generated by binary black hole mergers, provide clues about the progenitors of final black hole formation, and help to determine the event rate of compact object mergers. Fast parameter estimation studies would ensure that as much scientific information as possible is derived from each gravitational-wave detection.

Figure 1

A schematic of the method for building and evaluating the surrogate model. The red dots show the greedy selection of parameter points for building the reduced basis (step 1, offline), the blue dots (step 2, offline) show the associated empirical nodes in time from which a waveform can be reconstructed by interpolation with high accuracy, and the blue lines (step 3, offline) indicate a fit for the waveform’s parametric dependence at each empirical time. The yellow dot shows a generic parameter, which is predicted at the yellow diamonds and filled in between for arbitrary times using the empirical interpolant, represented as a dotted black line (step 4, online).

Figure 2

Time series of a normalized (2,2) mode of an EOB waveform for an equal-mass, nonspinning black hole binary coalescence. This waveform, corresponding to about 70 gravitational wave cycles, is representative of the structure encountered when building a surrogate model.

Figure 3

Greedy error, as defined by Eq. (9), over 501 EOB training set waveforms with mass ratios between 1 and 2. Labels at the dots indicate the selected mass ratios at each step in the greedy algorithm.

Figure 4

Histogram of parameters selected by the greedy algorithm for the reduced basis of Fig. 3.

Figure 5

Location of the empirical nodes for the fiducial family of EOB waveforms with mass ratio $q\in [1,2]$. Knowing the waveform in this parameter range at these specific times is sufficient to reconstruct the former with very high accuracy at any other time using the empirical interpolant in Eq. (18).

Figure 6

A comparison of errors for the example family of EOB waveforms. The dashed line shows the greedy error ${\sigma }_m$ in Eq. (9). The solid line shows the maximum empirical interpolant error (21) taken over 1,000 randomly selected waveforms (i.e., not taken from the training set) for $q\in [1,2]$. The dash-dotted line shows the error bound provided by the right side of Eq. (20) and is based solely on the greedy error and ${\mathrm{\Lambda }}_m$. All three errors display similar decay rates.

Figure 7

Top panel: Amplitude (solid line) and phase (dashed line) of the fiducial EOB training space waveform at the 15th selected empirical time as a function of $q$ along with the greedy data (circles). The empirical time is $T_{15}=28.5M$ after merger and corresponds to the largest pointwise relative error for the least squares fit to the amplitude as quantified by Eq. (26). Bottom panel: The pointwise least squares errors for the amplitude (red) and phase (blue) at $T_{15}$ evaluated for 1,000 randomly selected waveforms. The dashed lines correspond to the maximum pointwise error for the second empirical node $T_2=-2,367M$, which has the smallest maximum error of all the nodes.

Figure 8

The relative amplitude differences and phase differences of the least squares fits, as defined by Eqs. (26) and (27), maximized over the greedy mass ratios at each empirical time for our EOB example. The top panel shows these errors when using a polynomial least squares fit and the bottom panel when using a fitting function inspired by the post-Newtonian amplitude and phase. Both types of fits exhibit very low errors at all of the empirical times.

Figure 9

Top panel: Surrogate model error defined by Eq. (31), which is related to the overlap error through Eq. (3), for 1,000 randomly selected mass ratios. The mass ratio yielding the largest surrogate model error is $q=1.068$. Middle panel: The fiducial EOB waveform and its surrogate prediction for $q=1.068$. There is visual agreement throughout the entire duration of $\approx 12,000M$. Bottom panel: The fractional errors (32) in the amplitude and the phase difference between the fiducial EOB waveform and its surrogate model prediction for $q=1.068$. The differences are smaller than the errors intrinsic to the EOB model itself, as well as those of state-of-the-art numerical relativity simulations.

Figure 10

Top panel: Average time to generate a single fiducial EOB waveform from a standard EOB code (circles) and through evaluation of its surrogate (crosses). Here, we show results for the nominal example when using polynomial least squares fits for the amplitudes and phases. Bottom panel: The speedup, defined as the ratio of waveform generation times for EOB-LAL code to the surrogate model.

Figure 11

Top panel: The greedy error in Eq. (9) computed for 1,000 randomly selected waveforms (dashed line) and the error (31) of the resulting surrogate model (solid line) as a function of the number of basis waveforms $m$. Because of the fitting errors (see Sec. 3c), the surrogate error is roughly constant after $m=7$, implying little practical gain in using more than seven basis waveforms. The dash-dotted line shows an averaged error bound provided by the right side of Eq. (33). Bottom panel: Average time to evaluate a surrogate waveform (at a sampling rate of 2,048 Hz) as a function of $m$. As expected, there is only mild growth with $m$.

Figure 12

Top panel: The greedy error in Eq. (9) computed for waveforms of fixed duration and mass ratios in $[1,q_{\text{max}}]$ with $q_{\text{max}}=2$, 4, 6, 8, and 10. The curves correspond to cases 1, 3, 4, 5, and 6 from Table 1. Bottom panel: Greedy error for mass ratios in [1,2] with different durations in time and thus numbers of cycles. The curves correspond to cases 1, 7, and 8 from Table 1.

Figure 13

The optimal degree of each polynomial from a least squares fit at each empirical time. Out of a possible maximum of $n_{\mathrm{LS}}=19$, polynomial degrees between 9 and 14 are most often selected during the inspiral phase. The degree of the first fit for the phase is zero because the initial phases are chosen to vanish for all mass ratios.

Figure 14

Top panel: EOB waveforms for $q=1$, 2 starting at the same initial frequency but not aligned at the peak amplitudes. Bottom panel: Not aligning the waveforms results in needing more reduced basis elements to accurately span the space of waveforms. Here, we see that nearly all 501 points in the training space are selected, whereas only 19 points are required if the waveforms are aligned at the peak amplitude (compare with Fig. 3).

Figure 15

The dependence of the error (31) when using the surrogate to model an EOB waveform (with $q=1.068$ from Fig. 9) as a function of the resolution (i.e., time steps) of the peak amplitude. The trend is linear in $\mathrm{\Delta }t/M$. The resolution leads to an uncertainty in estimating the peak amplitudes and thus in aligning the waveforms. This is the dominant source of error in the surrogate model that translates directly into errors in the fits of the last offline step for building the surrogate.

Figure 16

The curves depict a variety of projection coefficients $c_i(q)$ along with the greedy data as a function of mass ratio for our EOB example case introduced in Sec. 3. Only a representative few curves are shown. The top panel shows the kind of structure that the coefficients have, thereby preventing accurate global (polynomial) fits without additional data points. The bottom panel shows the transition in the behavior of these functions with mass ratio from smooth to noisy.

Figure 17

The curves depict the values of the real parts of the waveforms along with the greedy data as a function of the mass ratio for our EOB example case. Only curves at a few representative empirical times are shown. While there is less structure here than in the coefficients shown in Fig. 16, the majority of functions still require additional sampling to be accurately resolved by global polynomial fits.