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

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.