# An analysis of the derivative-free loss method for solving PDEs

arXiv: https://arxiv.org/abs/2309.16829 · HTML: https://mlsanity.com/paper/2309.16829/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 21 references that are in the map.
- First posted on arXiv 28 Sep 2023.

## Builds on (most cited first, 7 of 7)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [A Derivative-Free Method for Solving Elliptic Partial Differential Equations with Deep Neural Networks](https://mlsanity.com/paper/2001.06145/) (2020)
- [A Neural Network Approach for Homogenization of Multiscale Problems](https://mlsanity.com/paper/2206.02032/) (2022)

## Built on by (most cited first, 1 of 1)

- [Stochastic approach for elliptic problems in perforated domains](https://mlsanity.com/paper/2403.11385/) (2024)

## Abstract (arXiv)

This study analyzes the derivative-free loss method to solve a certain class of elliptic PDEs and fluid problems using neural networks. The approach leverages the Feynman-Kac formulation, incorporating stochastic walkers and their averaged values. We investigate how the time interval associated with the Feynman-Kac representation and the walker size influence computational efficiency, trainability, and sampling errors. Our analysis shows that the training loss bias scales proportionally with the time interval and the spatial gradient of the neural network, while being inversely proportional to the walker size. Moreover, we demonstrate that the time interval must be sufficiently long to enable effective training. These results indicate that the walker size can be chosen as small as possible, provided it satisfies the optimal lower bound determined by the time interval. Finally, we present numerical experiments that support our theoretical findings.