# A Discussion on Solving Partial Differential Equations using Neural Networks

arXiv: https://arxiv.org/abs/1904.07200 · HTML: https://mlsanity.com/paper/1904.07200/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Apr 2019.

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

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## Abstract (arXiv)

Can neural networks learn to solve partial differential equations (PDEs)? We investigate this question for two (systems of) PDEs, namely, the Poisson equation and the steady Navier--Stokes equations. The contributions of this paper are five-fold. (1) Numerical experiments show that small neural networks (< 500 learnable parameters) are able to accurately learn complex solutions for systems of partial differential equations. (2) It investigates the influence of random weight initialization on the quality of the neural network approximate solution and demonstrates how one can take advantage of this non-determinism using ensemble learning. (3) It investigates the suitability of the loss function used in this work. (4) It studies the benefits and drawbacks of solving (systems of) PDEs with neural networks compared to classical numerical methods. (5) It proposes an exhaustive list of possible directions of future work.