Neural Statistical Functions

TL;DR

Neural statistical functions enable direct, efficient estimation of complex statistical quantities over continuous ranges from pre-trained models, significantly reducing inference latency in decision-making tasks.

Contribution

This work introduces neural statistical functions and prefix statistics, unifying diverse statistical tasks into an interval-conditional framework for improved efficiency.

Findings

Achieved up to 100× reduction in model evaluations.

Accurately estimated energy, quantiles, and maximum stress in physical systems.

Unified diverse statistical functions into a single framework.

Abstract

Classical deep learning typically operates on individual cases. Despite its success, real-world usage often requires repeated inference to estimate statistical quantities for complex decision-making tasks involving uncertainty or extreme-value analysis, resulting in substantial latency. We introduce neural statistical functions, a new family of models learned from pre-trained single-sample predictors and scattered data samples, which can directly infer statistics over continuous operating condition ranges without explicit sampling. By introducing the notion of prefix statistics, we transform and unify diverse statistical functions (e.g., integrals, quantiles, and maxima) into an interval-conditional framework, in which a principled identity between the prefix statistics and the individual-case regression serves as the learning objective. Neural statistical functions achieve strong performance in estimating essential statistics of complex physical processes, including accumulated energy in dynamical systems, quantiles of aerodynamic responses, and maximum stress in crash processes, while achieving up to a 100$\times$ reduction in model evaluations.