Neural Quadrature Rule and Autoregressive Adaptive Sampling

Monte Carlo integration is widely used in computer graphics, especially in rendering, but we identify two key limitations. First, for each sample, we can often obtain rich auxiliary information, such as sample position, or geometric information. However, a classical Monte Carlo estimator cannot effectively use this information and only averages the function values. Second, the Monte Carlo formulation makes it difficult to adapt the sampling distribution toward truly informative regions, which can be critical for reconstructing the signal. To address these limitations, we argue that a sampler and integrator beyond the standard Monte Carlo methods is needed. We therefore propose an end-to-end sampling-integration approach that jointly learns both a sampler and an integrator using neural networks, enabling samples to be drawn and used in a more coupled and principled manner. By training on a dataset of integrands, the estimator can further use learned priors over integrand structure and specialize to a family of problems. We evaluate our method on diverse applications in lighting, transmittance, generalized winding number, and walk-on-spheres, spanning both linear and nonlinear cases, and a broad range of low- and high-dimensional settings. Even though the networks add computational overheads, in the equal-sample setting, our method achieves substantial improvements, providing a powerful alternative to traditional quadrature rules and sampling methods.