# Geometric Integration for Neural Control Variates

arXiv: https://arxiv.org/abs/2509.15538 · HTML: https://mlsanity.com/paper/2509.15538/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 17 of its 31 references that are in the map.
- First posted on arXiv 19 Sep 2025.

## Builds on (most cited first, 17 of 17)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Instant Neural Graphics Primitives with a Multiresolution Hash Encoding](https://mlsanity.com/paper/2201.05989/) (2022)
- [Neural Geometric Level of Detail: Real-time Rendering with Implicit 3D Shapes](https://mlsanity.com/paper/2101.10994/) (2021)
- [AutoInt: Automatic Integration for Fast Neural Volume Rendering](https://mlsanity.com/paper/2012.01714/) (2020)
- [Neural Importance Sampling](https://mlsanity.com/paper/1808.03856/) (2018)
- [Real-time Neural Radiance Caching for Path Tracing](https://mlsanity.com/paper/2106.12372/) (2021)
- [Neural Control Variates](https://mlsanity.com/paper/2006.01524/) (2020)
- [Neural Radiosity](https://mlsanity.com/paper/2105.12319/) (2021)
- [SplineCam: Exact Visualization and Characterization of Deep Network Geometry and Decision Boundaries](https://mlsanity.com/paper/2302.12828/) (2023)
- [Neural Control Variates with Automatic Integration](https://mlsanity.com/paper/2409.15394/) (2024)
- [Primary-Space Adaptive Control Variates using Piecewise-Polynomial Approximations](https://mlsanity.com/paper/2008.06722/) (2020)
- [Importance Sampling BRDF Derivatives](https://mlsanity.com/paper/2304.04088/) (2023)
- [A Practical Walk-on-Boundary Method for Boundary Value Problems](https://mlsanity.com/paper/2305.04403/) (2023)
- [Neural Intersection Function](https://mlsanity.com/paper/2306.07191/) (2023)
- [Regression-based Monte Carlo Integration](https://mlsanity.com/paper/2211.07422/) (2022)
- [Local Positional Encoding for Multi-Layer Perceptrons](https://mlsanity.com/paper/2309.15101/) (2023)
- [Residual path integrals for re-rendering](https://mlsanity.com/paper/2406.16302/) (2024)

## Abstract (arXiv)

Control variates are a variance-reduction technique for Monte Carlo integration. The principle involves approximating the integrand by a function that can be analytically integrated, and integrating using the Monte Carlo method only the residual difference between the integrand and the approximation, to obtain an unbiased estimate. Neural networks are universal approximators that could potentially be used as a control variate. However, the challenge lies in the analytic integration, which is not possible in general. In this manuscript, we study one of the simplest neural network models, the multilayered perceptron (MLP) with continuous piecewise linear activation functions, and its possible analytic integration. We propose an integration method based on integration domain subdivision, employing techniques from computational geometry to solve this problem in 2D. We demonstrate that an MLP can be used as a control variate in combination with our integration method, showing applications in the light transport simulation.