← Back to all news
arXiv Machine Learning By Konrad Kleinberg, Thomas Kruse

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

Read the original on arXiv Machine Learning →

arXiv:2608. 09494v1 Announce Type: cross Abstract: 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.

Machine-generated by The Flow from the publisher's headline and feed description — not written or checked by a human. The full article lives at arXiv Machine Learning.

Related stories

arXiv Machine Learning
Jun 25

A Zeroth-Order Deep Learning Method for Fully Nonlinear Parabolic Partial Differential Equations with Unknown Coefficients

arXiv:2606. 24999v1 Announce Type: new Abstract: High-dimensional partial differential equations (PDEs) with unknown coefficients arise widely in scientific machine learning, including continuous-time reinforcement learning, yet solving them efficiently in a data-driven way remains challenging.

By Yanwei Jia, Du Ouyang, Huy\^en Pham, Xun Yu Zhou
reinforcement-learningsafety
More like this →
arXiv Machine Learning
Jun 26

Random test functions, $H^{-1}$ norm equivalence, and stochastic variational physics-informed neural networks

arXiv:2605. 03542v2 Announce Type: replace-cross Abstract: The dual norm characterisation of weak solutions of second-order linear elliptic partial differential equations is mathematically natural but computationally intractable: evaluating the $H^{-1}$ norm of the residual requires a supremum over an infinite-dimensional test space.

By Diego Marcondes
More like this →
arXiv Machine Learning
Sep 2

Rigorous Error Certification for Neural PDE Solvers: From Empirical Residuals to Solution Guarantees

The paper presents a theoretical framework for certifying the accuracy of physics‑informed neural networks (PINNs) used to solve partial differential equations. It derives generalization bounds that link the residual loss minimized during training to the actual error in the solution space, showing that if the neural approximation stays within a compact subset, a vanishing residual guarantees convergence to the true solution. Deterministic and probabilistic convergence results are provided, offering explicit error guarantees based on residual, boundary, and initial condition errors.

By Amartya Mukherjee, Maxwell Fitzsimmons, David C. Del Rey Fern\'andez, Jun Liu
More like this →