# ARDO: A Weak Formulation Deep Neural Network Method for Elliptic and Parabolic PDEs Based on Random Differences of Test Functions

arXiv: https://arxiv.org/abs/2509.03757 · HTML: https://mlsanity.com/paper/2509.03757/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 2 of its 7 references that are in the map.
- First posted on arXiv 3 Sep 2025.

## Builds on (most cited first, 2 of 2)

- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Automatic Differentiation is Essential in Training Neural Networks for Solving Differential Equations](https://mlsanity.com/paper/2405.14099/) (2024)

## Abstract (arXiv)

We propose ARDO method for solving PDEs and PDE-related problems with deep learning techniques. This method uses a weak adversarial formulation but transfers the random difference operator onto the test function. The main advantage of this framework is that it is fully derivative-free with respect to the solution neural network. This framework is particularly suitable for Fokker-Planck type second-order elliptic and parabolic PDEs.