# Neural Control Variates with Automatic Integration

arXiv: https://arxiv.org/abs/2409.15394 · HTML: https://mlsanity.com/paper/2409.15394/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 41 references that are in the map.
- First posted on arXiv 23 Sep 2024.

## Builds on (most cited first, 15 of 15)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Instant Neural Graphics Primitives with a Multiresolution Hash Encoding](https://mlsanity.com/paper/2201.05989/) (2022)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014)
- [AutoInt: Automatic Integration for Fast Neural Volume Rendering](https://mlsanity.com/paper/2012.01714/) (2020)
- [Modulated Periodic Activations for Generalizable Local Functional Representations](https://mlsanity.com/paper/2104.03960/) (2021)
- [Neural Control Variates](https://mlsanity.com/paper/2006.01524/) (2020)
- [Using Large Ensembles of Control Variates for Variational Inference](https://mlsanity.com/paper/1810.12482/) (2018)
- [Grid-Free Monte Carlo for PDEs with Spatially Varying Coefficients](https://mlsanity.com/paper/2201.13240/) (2022)
- [Neural Field Convolutions by Repeated Differentiation](https://mlsanity.com/paper/2304.01834/) (2023)
- [Neural network integral representations with the ReLU activation function](https://mlsanity.com/paper/1910.02743/) (2019)
- [Walk on Stars: A Grid-Free Monte Carlo Method for PDEs with Neumann Boundary Conditions](https://mlsanity.com/paper/2302.11815/) (2023)
- [Boundary Value Caching for Walk on Spheres](https://mlsanity.com/paper/2302.11825/) (2023)
- [Regression-based Monte Carlo Integration](https://mlsanity.com/paper/2211.07422/) (2022)

## Built on by (most cited first, 5 of 5)

- [Operator Learning Using Weak Supervision from Walk-on-Spheres](https://mlsanity.com/paper/2603.01193/) (2026)
- [WoSNN: Stochastic Solver for PDEs with Machine Learning](https://mlsanity.com/paper/2509.00204/) (2025)
- [Geometric Integration for Neural Control Variates](https://mlsanity.com/paper/2509.15538/) (2025)
- [Distance Marching for Generative Modeling](https://mlsanity.com/paper/2602.02928/) (2026)
- [Monte Carlo Steklov Operators for Large-Scale Geometry Processing in the Wild](https://mlsanity.com/paper/2606.05581/) (2026)

## Abstract (arXiv)

This paper presents a method to leverage arbitrary neural network architecture for control variates. Control variates are crucial in reducing the variance of Monte Carlo integration, but they hinge on finding a function that both correlates with the integrand and has a known analytical integral. Traditional approaches rely on heuristics to choose this function, which might not be expressive enough to correlate well with the integrand. Recent research alleviates this issue by modeling the integrands with a learnable parametric model, such as a neural network. However, the challenge remains in creating an expressive parametric model with a known analytical integral. This paper proposes a novel approach to construct learnable parametric control variates functions from arbitrary neural network architectures. Instead of using a network to approximate the integrand directly, we employ the network to approximate the anti-derivative of the integrand. This allows us to use automatic differentiation to create a function whose integration can be constructed by the antiderivative network. We apply our method to solve partial differential equations using the Walk-on-sphere algorithm. Our results indicate that this approach is unbiased and uses various network architectures to achieve lower variance than other control variate methods.