# Grid-Free Monte Carlo for PDEs with Spatially Varying Coefficients

arXiv: https://arxiv.org/abs/2201.13240 · HTML: https://mlsanity.com/paper/2201.13240/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 31 Jan 2022.

## Built on by (most cited first, 12 of 12)

- [Monte Carlo Neural PDE Solver for Learning PDEs via Probabilistic Representation](https://mlsanity.com/paper/2302.05104/) (2023)
- [Solving Poisson Equations using Neural Walk-on-Spheres](https://mlsanity.com/paper/2406.03494/) (2024)
- [Robust Biharmonic Skinning Using Geometric Fields](https://mlsanity.com/paper/2406.00238/) (2024)
- [Neural Control Variates with Automatic Integration](https://mlsanity.com/paper/2409.15394/) (2024)
- [McGrids: Monte Carlo-Driven Adaptive Grids for Iso-Surface Extraction](https://mlsanity.com/paper/2409.06710/) (2024)
- [On the Relationships between Graph Neural Networks for the Simulation of Physical Systems and Classical Numerical Methods](https://mlsanity.com/paper/2304.00146/) (2023)
- [Operator Learning Using Weak Supervision from Walk-on-Spheres](https://mlsanity.com/paper/2603.01193/) (2026)
- [Walk-on-Interfaces: A Monte Carlo Estimator for an Elliptic Interface Problem with Nonhomogeneous Flux Jump Conditions and a Neumann Boundary Condition](https://mlsanity.com/paper/2508.16767/) (2025)
- [WoSNN: Stochastic Solver for PDEs with Machine Learning](https://mlsanity.com/paper/2509.00204/) (2025)
- [MC2: Monte Carlo Correction for Fast Elliptic PDE Solving](https://mlsanity.com/paper/2605.09288/) (2026)
- [Walk-on-Spheres Monte Carlo and deep neural network approximations of elliptic PDEs with drift and killing](https://mlsanity.com/paper/2608.09494/) (2026)
- [Neural Harmonic Measure Operator](https://mlsanity.com/paper/2609.35752/) (2026)

## Abstract (arXiv)

Partial differential equations (PDEs) with spatially-varying coefficients arise throughout science and engineering, modeling rich heterogeneous material behavior. Yet conventional PDE solvers struggle with the immense complexity found in nature, since they must first discretize the problem -- leading to spatial aliasing, and global meshing/sampling that is costly and error-prone. We describe a method that approximates neither the domain geometry, the problem data, nor the solution space, providing the exact solution (in expectation) even for problems with extremely detailed geometry and intricate coefficients. Our main contribution is to extend the walk on spheres (WoS) algorithm from constant- to variable-coefficient problems, by drawing on techniques from volumetric rendering. In particular, an approach inspired by null-scattering yields unbiased Monte Carlo estimators for a large class of 2nd-order elliptic PDEs, which share many attractive features with Monte Carlo rendering: no meshing, trivial parallelism, and the ability to evaluate the solution at any point without solving a global system of equations.