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This paper presents a method that uses neural networks as a caching mechanism to reduce the variance of Monte Carlo Partial Differential Equation solvers, such as the Walk-on-Spheres algorithm [Sawhney and Crane 2020]. While these Monte Carlo PDE solvers have the merits of being unbiased and discretization-free, their high variance often hinders real-time applications. On the other hand, neural networks can approximate the PDE solution, and evaluating these networks at inference time can be very fast. However, neural-network-based solutions may suffer from convergence difficulties and high bias. Our hybrid system aims to combine these two potentially complementary solutions by training a neural field to approximate the PDE solution using supervision from a WoS solver. This neural field is then used as a cache in the WoS solver to reduce variance during inference. We demonstrate that our neural field training procedure is better than the commonly used self-supervised objectives in the literature. We also show that our hybrid solver exhibits lower variance than WoS with the same computational budget: it is significantly better for small compute budgets and provides smaller improvements for larger budgets, reaching the same performance as WoS in the limit. 

Inverse rendering is a powerful approach to modeling objects from photographs, and we extend previous techniques to handle translucent materials that exhibit subsurface scattering. Representing translucency using a heterogeneous bidirectional scattering-surface reflectance distribution function (BSSRDF), we extend the framework of path-space differentiable rendering to accommodate both surface and subsurface reflection. This introduces new types of paths requiring new methods for sampling moving discontinuities in material space that arise from visibility and moving geometry. We use this differentiable rendering method in an end-to-end approach that jointly recovers heterogeneous translucent materials (represented by a BSSRDF) and detailed geometry of an object (represented by a mesh) from a sparse set of measured 2D images in a coarse-to-fine framework incorporating Laplacian preconditioning for the geometry. To efficiently optimize our models in the presence of the Monte Carlo noise introduced by the BSSRDF integral, we introduce a dual-buffer method for evaluating the L2 image loss. This efficiently avoids potential bias in gradient estimation due to the correlation of estimates for image pixels and their derivatives and enables correct convergence of the optimizer even when using low sample counts in the renderer. We validate our derivatives by comparing against finite differences and demonstrate the effectiveness of our technique by comparing inverse-rendering performance with previous methods. We show superior reconstruction quality on a set of synthetic and real-world translucent objects as compared to previous methods that model only surface reflection.