# An overview on deep learning-based approximation methods for partial differential equations

arXiv: https://arxiv.org/abs/2012.12348 · HTML: https://mlsanity.com/paper/2012.12348/

## Key facts

- Cited by 65 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Dec 2020.

## Built on by (most cited first, 30 of 65)

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- [Global Convergence of Deep Galerkin and PINNs Methods for Solving Partial Differential Equations](https://mlsanity.com/paper/2305.06000/) (2023)
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- [From continuous-time formulations to discretization schemes: tensor trains and robust regression for BSDEs and parabolic PDEs](https://mlsanity.com/paper/2307.15496/) (2023)
- [A brief review of the Deep BSDE method for solving high-dimensional partial differential equations](https://mlsanity.com/paper/2505.17032/) (2025)
- [Deep learning approximations for non-local nonlinear PDEs with Neumann boundary conditions](https://mlsanity.com/paper/2205.03672/) (2022)
- [From Monte Carlo to neural networks approximations of boundary value problems](https://mlsanity.com/paper/2209.01432/) (2022)
- [r-Adaptive Deep Learning Method for Solving Partial Differential Equations](https://mlsanity.com/paper/2210.10900/) (2022)
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## Abstract (arXiv)

It is one of the most challenging problems in applied mathematics to approximatively solve high-dimensional partial differential equations (PDEs). Recently, several deep learning-based approximation algorithms for attacking this problem have been proposed and tested numerically on a number of examples of high-dimensional PDEs. This has given rise to a lively field of research in which deep learning-based methods and related Monte Carlo methods are applied to the approximation of high-dimensional PDEs. In this article we offer an introduction to this field of research by revisiting selected mathematical results related to deep learning approximation methods for PDEs and reviewing the main ideas of their proofs. We also provide a short overview of the recent literature in this area of research.