用几何分割法实现神经控制变量的精确积分，提升光传输模拟效率。
Geometric Integration for Neural Control Variates
- 将MLP控制变量与计算几何结合，通过区域分割实现解析积分。
- 在2D场景下成功完成神经网络控制变量的精确积分，误差显著降低。
- 适合做渲染、物理仿真等需要高精度积分的科研人员使用。
控制变量是蒙特卡洛积分中的一种方差缩减技术，其核心思想是用一个可解析积分的函数近似被积函数，仅对两者之差进行蒙特卡洛积分，从而获得无偏估计。神经网络作为通用逼近器，可望用于构建控制变量，但其解析积分通常不可行。本文研究最简单的神经网络模型——带连续分段线性激活函数的多层感知机（MLP），探索其解析积分的可能性。提出一种基于积分域分割的方法，结合计算几何技术，在二维情形下成功实现该类MLP的解析积分。实验表明，该方法可有效将MLP作为控制变量应用于光传输模拟，显著降低方差。
原文摘要 · Abstract (English)
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.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。