Abstract
We introduce a Monte Carlo method for computing derivatives of the solution to a partial differential equation (PDE) with respect to problem parameters (such as domain geometry or boundary conditions). Derivatives can be evaluated at arbitrary points, without performing a global solve or constructing a volumetric grid or mesh. The method is hence well suited to inverse problems with complex geometry, such as PDE-constrained shape optimization. Like other walk on spheres (WoS) algorithms, our method is trivial to parallelize, and is agnostic to boundary representation (meshes, splines, implicit surfaces, etc.), supporting large topological changes. We focus in particular on screened Poisson equations, which model diverse problems from scientific and geometric computing. As in differentiable rendering, we jointly estimate derivatives with respect to all parameters---hence, cost does not grow significantly with parameter count. In practice, even noisy derivative estimates exhibit fast, stable convergence for stochastic gradient-based optimization, as we show through examples from thermal design, shape from diffusion, and computer graphics.
Video
Resources
Paper: Our paper is available on the ACM Digital Library,, on arXiv, and locally.
Code: Our code is available on Github.
Citation
@article{Miller:DiffWoS:2024,
author = {Miller, Bailey and Sawhney, Rohan and Crane, Keenan and Gkioulekas, Ioannis},
title = {Differential Walk on Spheres},
year = {2024},
issue_date = {December 2024},
publisher = {Association for Computing Machinery},
address = {New York, NY, USA},
volume = {43},
number = {6},
issn = {0730-0301},
url = {https://doi.org/10.1145/3687913},
doi = {10.1145/3687913},
journal = {ACM Trans. Graph.},
month = nov,
articleno = {174},
numpages = {18},
}
Acknowledgments
This work was supported by National Science Foundation (NSF) awards 2008123 and 2212290; National Institute of Food and Agriculture award 2023-67021-39073; a gift from Adobe Research; NSF Graduate Research Fellowship DGE2140739 and an NVIDIA graduate fellowship for Bailey Miller; a Packard Foundation Fellowship for Keenan Crane; and Alfred P. Sloan Research Fellowship FG202013153 for Ioannis Gkioulekas. Rohan Sawhney thanks Ken Museth for supporting this work and the authors thank Gilles Daviet for helpful discussion.