# A Derivative-Free Method for Solving Elliptic Partial Differential Equations with Deep Neural Networks

arXiv: https://arxiv.org/abs/2001.06145 · HTML: https://mlsanity.com/paper/2001.06145/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Jan 2020.

## Built on by (most cited first, 13 of 13)

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- [Hierarchical Learning to Solve Partial Differential Equations Using Physics-Informed Neural Networks](https://mlsanity.com/paper/2112.01254/) (2021)
- [A Neural Network Approach for Homogenization of Multiscale Problems](https://mlsanity.com/paper/2206.02032/) (2022)
- [A brief review of the Deep BSDE method for solving high-dimensional partial differential equations](https://mlsanity.com/paper/2505.17032/) (2025)
- [An analysis of the derivative-free loss method for solving PDEs](https://mlsanity.com/paper/2309.16829/) (2023)
- [Stochastic approach for elliptic problems in perforated domains](https://mlsanity.com/paper/2403.11385/) (2024)
- [Extremization to Fine Tune Physics Informed Neural Networks for Solving Boundary Value Problems](https://mlsanity.com/paper/2406.05290/) (2024)
- [Walk-on-Interfaces: A Monte Carlo Estimator for an Elliptic Interface Problem with Nonhomogeneous Flux Jump Conditions and a Neumann Boundary Condition](https://mlsanity.com/paper/2508.16767/) (2025)
- [Feature Learning for the High Dimensional Stationary Sch"odinger Equation with Deep Ritz Method](https://mlsanity.com/paper/2607.06369/) (2026)

## Abstract (arXiv)

We introduce a deep neural network based method for solving a class of elliptic partial differential equations. We approximate the solution of the PDE with a deep neural network which is trained under the guidance of a probabilistic representation of the PDE in the spirit of the Feynman-Kac formula. The solution is given by an expectation of a martingale process driven by a Brownian motion. As Brownian walkers explore the domain, the deep neural network is iteratively trained using a form of reinforcement learning. Our method is a 'Derivative-Free Loss Method' since it does not require the explicit calculation of the derivatives of the neural network with respect to the input neurons in order to compute the training loss. The advantages of our method are showcased in a series of test problems: a corner singularity problem, an interface problem, and an application to a chemotaxis population model.