Neural networks improve classical schemes for solving one-dimensional equations
Neural Network-Assisted Refinement of Traditional Schemes for One-Dimensional Scalar Conservation Laws
Machine Learning
Summary
Numerical methods help computers solve equations that track things like waves or traffic flow. The authors show how small neural networks can enhance these traditional methods to make them more accurate and stable. Their first network rediscovered an old reliable method, and the second mimicked a technique to improve precision. Combining these networks produces better solutions without making the process more complicated.
What this means in practice
- •For computational fluid dynamics engineers: Design numerical solvers with neural network enhancements to improve accuracy of simulations for one-dimensional flow problems.
- •For software developers in scientific computing: Integrate small neural networks into classical numerical schemes to achieve better stability and precision without increasing code complexity.
Authors
Imre Fekete, Ferenc Izsák, Vendel P. Kupás
Abstract
A clear link is established between conventional numerical methods and neural network approximations for solving one-dimensional scalar conservation laws. The focus is on the construction of an appropriate flux term in the case of convex flux functions for improving the classical schemes. The first neural network developed here is able to rediscover Godunov's method, while the second one emulates the behavior of a second-order slope-limiter function. In this way, by merging them, second-order reconstruction-based schemes can be developed. The networks presented here employ a minimal number of parameters, significantly reducing the complexity compared to previous approaches. These networks can also be linked consecutively to get a deep one corresponding to multiple time steps. Training them with an appropriate loss leads to stable schemes, improving even the classical methods without increasing their complexity.