On ab initio-based, free and closed-form expressions for gravitational waves

Abstract

We introduce a new approach for finding high accuracy, free and closed-form expressions for the gravitational waves emitted by binary black hole collisions from ab initio models. More precisely, our expressions are built from numerical surrogate models based on supercomputer simulations of the Einstein equations, which have been shown to be essentially indistinguishable from each other. Distinct aspects of our approach are that: (i) representations of the gravitational waves can be explicitly written in a few lines, (ii) these representations are free-form yet still fast to search for and validate and (iii) there are no underlying physical approximations in the underlying model. The key strategy is combining techniques from Artificial Intelligence and Reduced Order Modeling for parameterized systems. Namely, symbolic regression through genetic programming combined with sparse representations in parameter space and the time domain using Reduced Basis and the Empirical Interpolation Method enabling fast free-form symbolic searches and large-scale a posteriori validations. As a proof of concept we present our results for the collision of two black holes, initially without spin, and with an initial separation corresponding to 25–31 gravitational wave cycles before merger. The minimum overlap, compared to ground truth solutions, is 99%. That is, 1% difference between our closed-form expressions and supercomputer simulations; this is considered for gravitational (GW) science more than the minimum required due to experimental numerical errors which otherwise dominate. This paper aims to contribute to the field of GWs in particular and Artificial Intelligence in general.

Gravitational wave surrogates through automated machine learning

“…In Eureqa, the best models are ranked following a compromise between accuracy and complexity summarized in a Pareto front curve. Early applications of these methods in gravitational wave modeling can be found in [35]. Support Vector Machines.-SVM algorithms [36,37] approximate functions by fitting expansions of the form…”

Deep Residual Error and Bag-of-Tricks Learning for Gravitational Wave Surrogate Modeling

“…proach) were used to obtain closed-form expressions for the GW emission of BBH collision, modeling the entire coalescence at once, i.e. without the need to distinguish between the different regimes of inspiral, merger and ringdown [103].…”

Reduced Order and Surrogate Models for Gravitational Waves

“…Figure 11: Representation of amplitude A(t, λ = q) and phase φ(t, λ = q) surfaces for the waveform mode (2, 2) corresponding to a symbolic model [137] for the surrogate plotted in Fig. 10 above.…”

“…One could also ask for a symbolic model representing the numerical surrogate one. In this line, in [137] the authors constructed the first ab initio free-form symbolic model (that is, analytical expressions in terms of elementary functions) for gravitational waves using symbolic regression (SR) through genetic programming. The fiducial model corresponds to the principal (2, 2)-mode of the surrogate SpEC q1 10 NoSpin described in [25], which is taken as ground truth solution of the Einstein field equations, since it is practically indistinguishable (meaning, within the numerical relativity errors themselves) from supercomputer numerical simulations.…”