# Walk-on-Spheres Monte Carlo and deep neural network approximations of elliptic PDEs with drift and killing

arXiv: https://arxiv.org/abs/2608.09494 · HTML: https://mlsanity.com/paper/2608.09494/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 33 references that are in the map.
- First posted on arXiv 10 Aug 2026.

## Builds on (most cited first, 15 of 15)

- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [An overview on deep learning-based approximation methods for partial differential equations](https://mlsanity.com/paper/2012.12348/) (2020)
- [A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations](https://mlsanity.com/paper/1809.02362/) (2018)
- [Analysis of the Generalization Error: Empirical Risk Minimization over Deep Artificial Neural Networks Overcomes the Curse of Dimensionality in the Numerical Approximation of Black-Scholes Partial Differential Equations](https://mlsanity.com/paper/1809.03062/) (2018)
- [A proof that deep artificial neural networks overcome the curse of dimensionality in the numerical approximation of Kolmogorov partial differential equations with constant diffusion and nonlinear drift coefficients](https://mlsanity.com/paper/1809.07321/) (2018)
- [Deep neural network approximation for high-dimensional elliptic PDEs with boundary conditions](https://mlsanity.com/paper/2007.05384/) (2020)
- [Mathematical Introduction to Deep Learning: Methods, Implementations, and Theory](https://mlsanity.com/paper/2310.20360/) (2023)
- [Rectified deep neural networks overcome the curse of dimensionality for nonsmooth value functions in zero-sum games of nonlinear stiff systems](https://mlsanity.com/paper/1903.06652/) (2019)
- [Grid-Free Monte Carlo for PDEs with Spatially Varying Coefficients](https://mlsanity.com/paper/2201.13240/) (2022)
- [Space-time error estimates for deep neural network approximations for differential equations](https://mlsanity.com/paper/1908.03833/) (2019)
- [Deep neural networks with ReLU, leaky ReLU, and softplus activation provably overcome the curse of dimensionality for space-time solutions of semilinear partial differential equations](https://mlsanity.com/paper/2406.10876/) (2024)
- [Nonlinear Monte Carlo methods with polynomial runtime for Bellman equations of discrete time high-dimensional stochastic optimal control problems](https://mlsanity.com/paper/2303.03390/) (2023)
- [From Monte Carlo to neural networks approximations of boundary value problems](https://mlsanity.com/paper/2209.01432/) (2022)
- [Multilevel Picard approximations and deep neural networks with ReLU, leaky ReLU, and softplus activation overcome the curse of dimensionality when approximating semilinear parabolic partial differential equations in Lp-sense](https://mlsanity.com/paper/2409.20431/) (2024)
- [Deep neural networks can provably solve Bellman equations for Markov decision processes without the curse of dimensionality](https://mlsanity.com/paper/2506.22851/) (2025)

## Abstract (arXiv)

In this paper we provide Monte Carlo and deep neural network approximations for stochastic representations of solutions to linear elliptic partial differential equations with constant diffusion, drift and killing. Building on the modified Walk-on-Spheres algorithm of Beznea et al. (arXiv:2209.01432), we introduce Monte Carlo estimators that explicitly incorporate sampled random times arising in the analyzed stochastic representations. We establish uniform error bounds for these estimators and show that, under suitable assumptions, a prescribed approximation accuracy is achieved with sample complexities growing at most polynomially in both the inverse accuracy and the problem dimension. Furthermore, we prove a deep neural network approximation result for the stochastic representations. Assuming suitable neural network representations of the boundary data and the distance function to the boundary, we use the constructed Monte Carlo to design deep neural networks that approximate the representation uniformly with a number of parameters growing at most polynomially in the inverse accuracy and the problem dimension. These results extend previous complexity analyses to a broader class of elliptic equations involving drift and killing.