Walk-on-Spheres Monte Carlo and deep neural network approximations of elliptic PDEs with drift and killing
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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.
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References · 15
- 2016Error bounds for approximations with deep ReLU networkscited by 567
- 2020An overview on deep learning-based approximation methods for partial differential equationscited by 65
- 2018A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equationscited by 55
- 2018Analysis 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 Equationscited by 44
- 2018A 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 coefficientscited by 31
- 2020Deep neural network approximation for high-dimensional elliptic PDEs with boundary conditionscited by 25
- 2023Mathematical Introduction to Deep Learning: Methods, Implementations, and Theorycited by 19
- 2019Rectified deep neural networks overcome the curse of dimensionality for nonsmooth value functions in zero-sum games of nonlinear stiff systemscited by 12
- 2022Grid-Free Monte Carlo for PDEs with Spatially Varying Coefficientscited by 12
- 2019Space-time error estimates for deep neural network approximations for differential equationscited by 6
- 2024Deep neural networks with ReLU, leaky ReLU, and softplus activation provably overcome the curse of dimensionality for space-time solutions of semilinear partial differential equationscited by 6
- 2023Nonlinear Monte Carlo methods with polynomial runtime for Bellman equations of discrete time high-dimensional stochastic optimal control problemscited by 5
3 more
- 2022From Monte Carlo to neural networks approximations of boundary value problemscited by 3
- 2024Multilevel 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-sensecited by 3
- 2025Deep neural networks can provably solve Bellman equations for Markov decision processes without the curse of dimensionalitycited by 1
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