# Algorithms for Solving High Dimensional PDEs: From Nonlinear Monte Carlo to Machine Learning

arXiv: https://arxiv.org/abs/2008.13333 · HTML: https://mlsanity.com/paper/2008.13333/

## Key facts

- Cited by 77 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 31 Aug 2020.

## Built on by (most cited first, 30 of 77)

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- [SVD-PINNs: Transfer Learning of Physics-Informed Neural Networks via Singular Value Decomposition](https://mlsanity.com/paper/2211.08760/) (2022)
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- [DeepHAM: A Global Solution Method for Heterogeneous Agent Models with Aggregate Shocks](https://mlsanity.com/paper/2112.14377/) (2021)
- [Connections between Numerical Algorithms for PDEs and Neural Networks](https://mlsanity.com/paper/2107.14742/) (2021)
- [Achieving High Accuracy with PINNs via Energy Natural Gradients](https://mlsanity.com/paper/2302.13163/) (2023)
- [Neural Q-learning for solving PDEs](https://mlsanity.com/paper/2203.17128/) (2022)
- [Distributional Offline Continuous-Time Reinforcement Learning with Neural Physics-Informed PDEs (SciPhy RL for DOCTR-L)](https://mlsanity.com/paper/2104.01040/) (2021)
- [Imaging Conductivity from Current Density Magnitude using Neural Networks](https://mlsanity.com/paper/2204.02441/) (2022)
- [Towards Size-Independent Generalization Bounds for Deep Operator Nets](https://mlsanity.com/paper/2205.11359/) (2022)
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- [Bayesian Deep Learning with Multilevel Trace-class Neural Networks](https://mlsanity.com/paper/2203.12961/) (2022)
- [A Curriculum-Training-Based Strategy for Distributing Collocation Points during Physics-Informed Neural Network Training](https://mlsanity.com/paper/2211.11396/) (2022)
- [Entropy-dissipation Informed Neural Network for McKean-Vlasov Type PDEs](https://mlsanity.com/paper/2303.11205/) (2023)
- [From continuous-time formulations to discretization schemes: tensor trains and robust regression for BSDEs and parabolic PDEs](https://mlsanity.com/paper/2307.15496/) (2023)
- [Residual Multi-Fidelity Neural Network Computing](https://mlsanity.com/paper/2310.03572/) (2023)
- [A brief review of the Deep BSDE method for solving high-dimensional partial differential equations](https://mlsanity.com/paper/2505.17032/) (2025)
- [Solving Elliptic Problems with Singular Sources using Singularity Splitting Deep Ritz Method](https://mlsanity.com/paper/2209.02931/) (2022)

## Abstract (arXiv)

In recent years, tremendous progress has been made on numerical algorithms for solving partial differential equations (PDEs) in a very high dimension, using ideas from either nonlinear (multilevel) Monte Carlo or deep learning. They are potentially free of the curse of dimensionality for many different applications and have been proven to be so in the case of some nonlinear Monte Carlo methods for nonlinear parabolic PDEs. In this paper, we review these numerical and theoretical advances. In addition to algorithms based on stochastic reformulations of the original problem, such as the multilevel Picard iteration and the Deep BSDE method, we also discuss algorithms based on the more traditional Ritz, Galerkin, and least square formulations. We hope to demonstrate to the reader that studying PDEs as well as control and variational problems in very high dimensions might very well be among the most promising new directions in mathematics and scientific computing in the near future.