Q-NET: A Network for Low-Dimensional Integrals of Neural Proxies

TL;DR

This paper introduces Q-NET, a neural network-based method that efficiently computes exact integrals of low-dimensional functions by leveraging neural proxies and input transformations, benefiting applications like rendering and simulation.

Contribution

The paper proposes Q-NET, a novel neural network architecture that calculates exact integrals over subsets of dimensions for neural proxies, enabling efficient integral estimation without resampling.

Findings

Q-NET accurately computes integrals in low-dimensional spaces.

The method reduces computational costs in applications like rendering.

Q-NET is effective with sparse and adaptive sampling strategies.

Abstract

Many applications require the calculation of integrals of multidimensional functions. A general and popular procedure is to estimate integrals by averaging multiple evaluations of the function. Often, each evaluation of the function entails costly computations. The use of a \emph{proxy} or surrogate for the true function is useful if repeated evaluations are necessary. The proxy is even more useful if its integral is known analytically and can be calculated practically. We propose the use of a versatile yet simple class of artificial neural networks -- sigmoidal universal approximators -- as a proxy for functions whose integrals need to be estimated. We design a family of fixed networks, which we call Q-NETs, that operate on parameters of a trained proxy to calculate exact integrals over \emph{any subset of dimensions} of the input domain. We identify transformations to the input space for which integrals may be recalculated without resampling the integrand or retraining the proxy. We highlight the benefits of this scheme for a few applications such as inverse rendering, generation of procedural noise, visualization and simulation. The proposed proxy is appealing in the following contexts: the dimensionality is low ($<10$D); the estimation of integrals needs to be decoupled from the sampling strategy; sparse, adaptive sampling is used; marginal functions need to be known in functional form; or when powerful Single Instruction Multiple Data/Thread (SIMD/SIMT) pipelines are available for computation.