[Paper Review] A Multi-Fidelity Emulator for the Matter Power Spectrum using Gaussian Processes

Ming-Feng Ho, Simeon Bird+1 more|arXiv (Cornell University)|May 3, 2021

Gaussian Processes and Bayesian Inference52 references4 citations

TL;DR

This paper introduces a multi-fidelity Gaussian process emulator that fuses low- and high-resolution cosmological $N$-body simulations to predict the matter power spectrum with high accuracy. By leveraging 3 high-fidelity simulations and many low-fidelity ones, it achieves ~1% average accuracy and outperforms single-fidelity emulators by up to 100× at $k \leq 2 \,h\,\mathrm{Mpc}^{-1}$, significantly improving non-linear scale prediction efficiency.

ABSTRACT

We present methods for emulating the matter power spectrum which effectively combine information from cosmological $N$-body simulations at different resolutions. An emulator allows estimation of simulation output by interpolating across the parameter space of a handful of simulations. We present the first implementation of multi-fidelity emulation in cosmology, where many low-resolution simulations are combined with a few high-resolution simulations to achieve an increased emulation accuracy. The power spectrum's dependence on cosmology is learned from the low-resolution simulations, which are in turn calibrated using high-resolution simulations. We show that our multi-fidelity emulator can achieve percent-level accuracy on average with only $3$ high-fidelity simulations and outperforms a single-fidelity emulator that uses $11$ simulations. With a fixed number of high-fidelity training simulations, we show that our multi-fidelity emulator is $\simeq 100$ times better than a single-fidelity emulator at $k \leq 2 \,h extrm{Mpc}{^{-1}}$, and $\simeq 20$ times better at $3 \leq k < 6.4 \,h extrm{Mpc}{^{-1}}$. Multi-fidelity emulation is fast to train, using only a simple modification to standard Gaussian processes. Our proposed emulator shows a new way to predict non-linear scales by fusing simulations from different fidelities.

Motivation & Objective

• To improve the accuracy and efficiency of matter power spectrum prediction in cosmology by combining simulations of varying resolution.
• To reduce the computational cost of high-fidelity $N$-body simulations while maintaining high predictive accuracy for non-linear scales.
• To develop a scalable, fast-to-train emulator framework using multi-fidelity Gaussian processes for cosmological parameter space interpolation.
• To calibrate low-fidelity simulations using high-fidelity data to enhance fidelity across the full parameter space.

Proposed method

• The method employs a multi-fidelity Gaussian process framework that models the matter power spectrum as a function of cosmological parameters across multiple simulation fidelities.
• Low-resolution simulations are used to learn the general dependence of the power spectrum on cosmology, while high-resolution simulations provide calibration at key points.
• The emulator uses a hierarchical correlation structure to model the discrepancy between low- and high-fidelity simulations, enabling accurate interpolation.
• Training involves a simple modification to standard Gaussian processes, enabling efficient inference and prediction with minimal computational overhead.
• The approach assumes a joint covariance function that captures both the trend in low-fidelity data and the correction from high-fidelity simulations.

Experimental results

Research questions

• RQ1 Can multi-fidelity emulation reduce the number of required high-fidelity simulations while maintaining or improving accuracy in matter power spectrum prediction?
• RQ2 How does the performance of a multi-fidelity emulator compare to a single-fidelity emulator when using the same number of high-fidelity simulations?
• RQ3 To what extent can low-resolution simulations improve the accuracy of high-fidelity predictions in the non-linear regime of the matter power spectrum?
• RQ4 What is the optimal trade-off between simulation resolution and number of training runs for achieving high-accuracy emulators?

Key findings

• The multi-fidelity emulator achieves an average accuracy of approximately 1% across the parameter space using only 3 high-fidelity simulations.
• At $k \leq 2 \,h\,\mathrm{Mpc}^{-1}$, the multi-fidelity emulator is approximately 100 times more accurate than a single-fidelity emulator with 11 simulations.
• At higher wave numbers ($3 \leq k < 6.4 \,h\,\mathrm{Mpc}^{-1}$), the multi-fidelity approach is about 20 times more accurate than the single-fidelity alternative.
• The method enables significant computational savings by reducing reliance on expensive high-fidelity simulations while maintaining high predictive fidelity on non-linear scales.