A Discussion on Solving Partial Differential Equations using Neural Networks
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.
Code (1)
Tasks
Ensemble LearningSimilar Papers 제목 키워드 기반
Hyena Neural Operator for Partial Differential Equations
Numerically solving partial differential equations typically requires fine discretization to resolve necessary spatiotemporal scales, which can be computationally expensive. Recent advances in deep learning have provided…
Forward-Backward Stochastic Neural Networks: Deep Learning of High-dimensional Partial Differential Equations
Classical numerical methods for solving partial differential equations suffer from the curse dimensionality mainly due to their reliance on meticulously generated spatio-temporal grids. Inspired by modern deep learning b…
Deep Neural networks for solving high-dimensional parabolic partial differential equations
The numerical solution of high dimensional partial differential equations (PDEs) is severely constrained by the curse of dimensionality (CoD), rendering classical grid--based methods impractical beyond a few dimensions. …
Solving Nonlinear and High-Dimensional Partial Differential Equations via Deep Learning
In this work we apply the Deep Galerkin Method (DGM) described in Sirignano and Spiliopoulos (2018) to solve a number of partial differential equations that arise in quantitative finance applications including option pri…
Vocal Bursts Intensity PredictionOne-Shot Transfer Learning of Physics-Informed Neural Networks
Solving differential equations efficiently and accurately sits at the heart of progress in many areas of scientific research, from classical dynamical systems to quantum mechanics. There is a surge of interest in using P…
Transfer Learning