Structure and asymptotic preserving deep neural surrogates for uncertainty quantification in multiscale kinetic equations

Abstract: The high dimensionality of kinetic equations with stochastic parameters poses major computational challenges for uncertainty quantification (UQ). Traditional Monte Carlo (MC) sampling methods, while widely used, suffer from slow convergence and high variance, which become increasingly severe as the dimensionality of the parameter space grows. To accelerate MC sampling, we adopt a multiscale control variates strategy that leverages low-fidelity solutions from simplified kinetic models to reduce variance. To further improve sampling efficiency and preserve the underlying physics, we introduce surrogate models based on structure and asymptotic preserving neural networks (SAPNNs). These deep neural networks are specifically designed to satisfy key physical properties, including positivity, conservation laws, entropy dissipation, and asymptotic limits. By training the SAPNNs on low-fidelity models and enriching them with selected high-fidelity samples from the full Boltzmann equation, our method achieves significant variance reduction while maintaining physical consistency and asymptotic accuracy. The proposed methodology enables efficient large-scale prediction in kinetic UQ and is validated across both homogeneous and nonhomogeneous multiscale regimes. Numerical results demonstrate improved accuracy and computational efficiency compared to standard MC techniques.