Differential Walk on Spheres

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.

18 pages, includes demo video of results. Project page https://imaging.cs.cmu.edu/differential_walk_on_spheres

References in corpus (12)

• Characterizing possible failure modes in physics-informed neural networks
• Deep Marching Tetrahedra: a Hybrid Representation for High-Resolution 3D Shape Synthesis
• Flexible Isosurface Extraction for Gradient-Based Mesh Optimization
• Walk on Stars: A Grid-Free Monte Carlo Method for PDEs with Neumann Boundary Conditions
• A Practical Walk-on-Boundary Method for Boundary Value Problems
• Boundary Value Caching for Walk on Spheres
• An unbiased ray-marching transmittance estimator
• MeshSDF: Differentiable Iso-Surface Extraction
• Sum-of-Squares Geometry Processing
• Fluidic Topology Optimization with an Anisotropic Mixture Model
• Physics-Based Inverse Rendering using Combined Implicit and Explicit Geometries
• Spelunking the Deep: Guaranteed Queries on General Neural Implicit Surfaces via Range Analysis