Neural Statistical Functions

2026-05-11

AI Summary

This work addresses the high latency of conventional deep learning approaches in estimating statistical quantities—such as integrals, quantiles, and extrema—in complex decision-making tasks, which typically require repeated model inferences. The authors propose neural statistical functions that unify diverse statistics under a framework of interval-conditional regression by introducing the concept of prefix statistics. They establish an identity relationship between these statistical functions and single-sample predictors, which serves as the learning objective. Leveraging only a pre-trained single-sample model and scattered observational data, the method enables direct inference of arbitrary statistics across continuous operating conditions without explicit sampling. Experiments on tasks including energy accumulation in dynamical systems, aerodynamic response quantiles, and peak stress in collisions demonstrate up to two orders of magnitude reduction in model invocations, substantially improving inference efficiency.

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.

Problem

• statistical estimation
• uncertainty quantification
• extreme-value analysis
• computational latency
• repeated inference

Innovation

• Neural Statistical Functions
• prefix statistics
• interval-conditional modeling
• statistical inference without sampling
• efficient uncertainty quantification