Learning Post-Newtonian Corrections from Numerical Relativity

TL;DR

The paper tackles the challenge of producing accurate gravitational-wave templates across the full inspiral–merger spectrum by bridging post-Newtonian (PN) theory and numerical relativity (NR). It introduces a physics-informed neural network (PINN) that learns residual corrections to PN dynamics and waveforms, anchored to PN physics and constrained by physics-based losses, and trains on a small set of NR surrogate waveforms. The authors demonstrate a two-step validation: first recovering analytically known 3PN corrections from a 2PN evolution, then extending to PN→NR using TaylorT4 up to 4.5PN and NRHybSur3dq8, including a mass-correction branch, achieving mismatches as low as $\sim 10^{-6}$ and showing strong extrapolation to higher mass ratios with sparse additional data. This data-efficient PN→NR bridge improves robustness and generalization beyond existing NR datasets and sets the stage for incorporating spins and eccentricity in future work, potentially enabling more accurate and scalable gravitational-wave modeling.

Abstract

Accurate modeling of gravitational waveforms from compact binary coalescences remains central to gravitational-wave (GW) astronomy. Post-Newtonian (PN) approximations capture the early inspiral dynamics analytically but break down near merger, while numerical relativity (NR) provides the accurate yet computationally expensive waveforms over limited parameter ranges. We develop a physics-informed neural network (PINN) framework that learns corrections mapping PN dynamics and waveforms to their NR counterparts. As a demonstration of the approach, we use the TaylorT4 PN model as the baseline, and train the network on a remarkably small dataset of only eight hybridized NR surrogate waveforms (NRHybSur3dq8) to learn higher-order corrections to the orbital dynamics and waveform modes for nonspinning noneccentric systems. Physically motivated loss terms enforce known limits and symmetries, such as vanishing corrections in the Newtonian limit and suppression of odd-$m$ modes in equal-mass systems, promoting consistent and reliable extrapolation beyond the training region. We simultaneously incorporate corrections that account for the different meaning of mass parameters in PN and NR descriptions. The learned corrections significantly reduce the phase and amplitude error through the inspiral up to about $200M$ before the merger. This approach provides a differentiable and computationally efficient bridge between PN and NR, offering a path toward waveform models that generalize more robustly beyond existing NR datasets.

Figures (9)

• Figure 1: Real parts of $h_{2,2}$ and $h_{2,1}$ for the highest mass ratio validation case of $q=7.93$ for the final NN model for PN to NR correction. The waveform mismatch is reduced from $2\times 10^{-1}$ to $1\times 10^{-6}$. All waveforms are phase-aligned at start. Although the PN waveform agrees well with NR at early inspiral parts, it deviates significantly near merger where both phase and amplitude discrepancies emerge. The NN corrections significantly reduce these discrepancies, allowing the PN waveforms to remain consistent with NR counterparts in the late inspiral regime.
• Figure 2: Neural network architecture for the 2PN$\rightarrow$3PN experiment. The two inputs, $(\nu, v)$, are linearly rescaled to $(\tilde{\nu},\tilde{v})$ to the interval $[-1, 1]$ to improve numerical stability during training. Each network output is multiplied by a tunable scaling coefficient before being applied as a correction to either the orbital dynamics or the waveform modes. The final architecture uses a single hidden layer with eight neurons.
• Figure 3: Real part of the $h_{2,2}$ and $h_{2,1}$ and the orbital velocity $v$, for the worst validation case ($q=7.84$). Including the NN corrections reduces the mismatch from $10^{-1}$ to $4\times10^{-8}$. All the waveforms are phase-aligned at $t=0$. The consistent improvement in the corrected orbital velocity shows that the NN captures genuine orbital corrections rather than simply compensating through the waveform modes corrections.
• Figure 4: Training and validation loss history for the 2PN$\rightarrow$3PN experiment. The validation losses closely follow the training losses, indicating that the neural network generalizes beyond the training set. The sharp increase in loss near epoch, $n=6000$, corresponds to the transition from segmented-waveform training to full-waveform training.
• Figure 5: Comparison of the true and neural-network–learned coefficients for the missing term in the $\dot{v}$ expansion. Solid lines represent the true value while dashed lines the learned coefficients. Different colors corresponding to different values of $v$. The close agreement between the true and learned curves across different values of $v$ demonstrates that the network successfully recovers the missing higher-order orbital corrections despite the degeneracy between orbital and waveform mode corrections.