Q-NET: A Formula for Numerical Integration of a Shallow Feed-forward Neural Network

Numerical integration is a computational procedure that is widely encountered across disciplines when reasoning about data. We derive a formula in closed form to calculate the multidimensional integral of functions fw that are representable using a shallow feed-forward neural network with weights w and a sigmoid activation function. We demonstrate its applicability in estimating numerical integration of arbitrary functions f over hyper-rectangular domains in the absence of a prior. To achieve this, we first train the network to learn $fw \approx f$ using point-samples of the integrand. We then use our formula to calculate the exact integral of the learned function fw. Our formula operates on the weights w of the trained approximator network. We show that this formula can itself be expressed as a shallow feed-forward network, which we call a Q-NET, with w as its inputs. Although the Q-NET does not have any learnable parameters, we use this abstraction to derive a family of elegant parametric formulae that represent the marginal distributions of the input function over arbitrary subsets of input dimensions in functional form. We perform empirical evaluations of Q-NETs for integrating smooth functions as well as functions with discontinuities.