Learning Parametric Integrals from Integrands

Parametric integration requires repeatedly evaluating families of integrals with varying integrands and integration domains. Learning a neural representation of such integrals can amortize this computation, but conventionally requires integral supervision, typically obtained by numerically integrating pointwise evaluations of the integrand. We introduce a neural representation that instead learns parametric integrals directly from pointwise integrand supervision. Our key idea is to decompose the integrand into a component that is constructed to integrate to zero and a second component that directly represents the integral. We realize the zero-integral component using the divergence of a neural field and handle parameterized integration domains through a change of variables. Once trained, our representation directly evaluates parametric integrals without numerical integration and remains differentiable with respect to its parameters. Across diverse applications such as filtering, rendering, and integral transforms, our approach achieves lower approximation error than supervision on Monte Carlo integral estimates under the same training-time budget, despite using fewer integrand evaluations overall.