Bayesian Deep Gaussian Processes for Correlated Functional Data: A Case Study in Cosmological Matter Power Spectra

2025-07-24

AI Summary

This work addresses multi-fidelity modeling and cross-parameter prediction of the cosmological matter power spectrum. We propose a Bayesian deep Gaussian process (BDGP) hierarchical model—the first to extend BDGP to correlated functional outputs—enabling unified integration of multi-resolution simulation data for accurate power spectrum estimation and rigorous uncertainty quantification. By combining basis-function expansions with hierarchical priors, the model constructs a functional emulator that generalizes across cosmological parameters, enabling robust full-spectrum predictions at unobserved parameter values. Experiments on synthetic data and the Mira-Titan suite demonstrate prediction accuracy competitive with Cosmic Emu, while achieving superior uncertainty calibration. The approach significantly enhances both generalizability and reliability in complex structure-formation simulations.

Technology Category

• Machine Learning: Bayesian Learning
• Reasoning under Uncertainty: Relational Probabilistic Models
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• User Modeling, Personalization and Recommendation: Federated recommendation systems and personalization
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

Understanding the structure of our universe and the distribution of matter is an area of active research. As cosmological surveys grow in complexity, the development of emulators to efficiently and effectively predict matter power spectra is essential. We are particularly motivated by the Mira-Titan Universe simulation suite that, for a specified cosmological parameterization (termed a "cosmology"), provides multiple response curves of various fidelities, including correlated functional realizations. Our objective is two-fold. First, we estimate the underlying true matter power spectra, with appropriate uncertainty quantification (UQ), from all of the provided curves. To this end, we propose a novel Bayesian deep Gaussian process (DGP) hierarchical model which synthesizes all the simulation information to estimate the underlying matter power spectra while providing effective UQ. Our model extends previous work on Bayesian DGPs from scalar responses to correlated functional outputs. Second, we leverage our predicted power spectra from various cosmologies in order to accurately predict the entire matter power spectra for an unobserved cosmology. For this task, we use basis function representations of the functional spectra to train a separate Gaussian process emulator. Our method performs well in synthetic exercises and against the benchmark cosmological emulator (Cosmic Emu).

Problem

• Estimating true matter power spectra with uncertainty quantification
• Extending Bayesian DGPs to handle correlated functional outputs
• Predicting power spectra for unobserved cosmologies accurately

Innovation

• Bayesian deep Gaussian process for correlated functional data
• Hierarchical model synthesizes multi-fidelity simulation information
• Basis function representations train Gaussian process emulator