Super-Resolution without High-Resolution Labels for Black Hole Simulations

Abstract

Generating high-resolution simulations is key for advancing our understanding of one of the universe's most violent events: Black Hole mergers. However, generating Black Hole simulations is limited by prohibitive computational costs and scalability issues, reducing the simulation's fidelity and resolution achievable within reasonable time frames and resources. In this work, we introduce a novel method that circumvents these limitations by applying a super-resolution technique without directly needing high-resolution labels, leveraging the Hamiltonian and momentum constraints-fundamental equations in general relativity that govern the dynamics of spacetime. We demonstrate that our method achieves a reduction in constraint violation by one to two orders of magnitude and generalizes effectively to out-of-distribution simulations.

Figures (7)

• Figure 1: Schematic representation of our framework: (1) We first apply a commonly used interpolation method to up-sample our simulation, then (2) a network takes the up-sampled simulation and produces a correction $\delta x$. This correction is then (3) multiplied by the scaling factor s and added to the up-sampled simulation. The corrected simulation results in reduced constraint violations, leading to an improved simulation.
• Figure 2: Visualization of the solution surface with $\mathcal{H} = 0$ and $\mathcal{M}_i = 0$. The blue line represents the ideal solution when following Einstein's equations, while the red line illustrate a numerical solution. The dots on both the red and purple line represent the discretization of time. Our method (purple) projects the solution back to the surface where constraint are fulfilled to produce a numerical result closer to the ideal solution.
• Figure 3: Comparing the performance of models trained with ${L_1}$ loss (which requires high-resolution labels) and $\mathcal{L}_{\rm GR}$ loss (which does not require high-resolution labels). As shown in the left figure, both ${L_1}$ and $\mathcal{L}_{\rm GR}$ losses converge to similarly small values of $\mathcal{L}_{\rm GR}$. However, in the right figure, we observe that both $\mathcal{L}_{\rm GR}$ and $L_1$ converge to different values of $L_1$. This suggests that the two losses lead to different solutions, likely due to the under-specified nature of the constraint-based $\mathcal{L}_{\rm GR}$ loss. The dark straight line is the interpolation used in GRTL code.
• Figure 4: Our framework (purple and yellow) outperforms the baseline (dotted black) by two orders of magnitude. In NR simulations, the mass of a Black Hole is a parameter for defining the simulation. We evaluate the loss of the validation set in the in-distribution scenario (purple). However, we aimed to stress-test our framework by varying the Black Hole's mass, enabling us to evaluate its ability to generalize to out-of-distribution scenarios (yellow). Remarkably, even with a 41% variation in the Black Hole's mass, our framework still outperforms the baseline. A more complete overview can be found in Tabel \ref{['tab:Money_table']}.
• Figure 5: Multiple solutions: Depending on starting point, we can control what state our simulations falls into when using the $\mathcal{L}_{\rm GR}$ loss. When we start with a pre-trained network (marked as Warmed-up initialization), we can converge towards the ground truth.