A Discussion on Solving Partial Differential Equations using Neural Networks
This work addresses the challenge of PDE solving for computational science and engineering, but it is incremental as it builds on existing neural network approaches without introducing major new methods.
The paper tackles the problem of solving partial differential equations (PDEs) using neural networks, specifically testing on the Poisson equation and steady Navier-Stokes equations, and finds that small neural networks with fewer than 500 parameters can accurately learn complex solutions.
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.