🤖 AI Summary
This work addresses the challenge of estimating statistical moments—such as means and covariances—for continuous-time Markov chains, which typically lack closed-form solutions and incur prohibitive computational costs when using conventional Monte Carlo methods in multi-parameter settings. The authors propose a simulation-based surrogate modeling framework that employs neural networks to learn the mapping from model parameters to statistical moments using Monte Carlo–noisy simulation data, thereby enabling efficient approximation across the entire parameter space. A key contribution lies in uncovering the heterogeneous impact of simulation noise on moment estimation, which informs a tailored allocation of computational resources and a robust learning algorithm. Under a fixed computational budget, the method yields accurate population-level moment estimates and demonstrates significant improvements over baseline approaches in downstream tasks such as whitening.
📝 Abstract
Continuous-time Markov Chains are widely used to model stochastic dynamical systems, but key summary quantities such as means and covariances are often intractable. While Monte Carlo sampling provides asymptotically exact estimates, it becomes computationally prohibitive when moments must be evaluated across many parameter values. We develop a simulation-based surrogate modeling framework that learns parameter-to-moment mappings from Monte Carlo-derived, noise-corrupted training targets, enabling efficient and accurate approximation across the parameter space. We show that Monte Carlo noise affects mean estimation primarily through additive variance, whereas covariance estimation is additionally impacted by bias arising from nonlinear transformations of empirical estimates. Using a stochastic Susceptible-Infected-Recovered model, we demonstrate that neural networks accurately learn both mean and covariance under fixed simulation budgets allocated to constructing the noisy training labels. We further characterize how to allocate computational resources between parameter-space coverage and Monte Carlo replication, showing that covariance estimation requires a balanced allocation to control both variance and bias, while mean estimation benefits more from increased parameter space coverage. Finally, we show that the learned moment mappings produce valid population-level quantities and perform well in downstream tasks such as whitening. These results highlight the importance of accounting for Monte Carlo noise in surrogate modeling and provide practical guidance for simulation-based learning in stochastic systems.