Neural Network Representation of Time Integrators

TL;DR

This paper introduces neural network architectures that exactly replicate explicit Runge-Kutta time integrators, enabling error analysis and eliminating training by fixing weights and biases.

Contribution

It presents neural network designs that are mathematically equivalent to classical Runge-Kutta schemes, providing a new approach for physics-based time integration without training.

Findings

Networks exactly match Runge-Kutta schemes

Error analysis is clarified through network design

Example with mass-damper-stiffness system included

Abstract

Deep neural network (DNN) architectures are constructed that are the exact equivalent of explicit Runge-Kutta schemes for numerical time integration. The network weights and biases are given, i.e., no training is needed. In this way, the only task left for physics-based integrators is the DNN approximation of the right-hand side. This allows to clearly delineate the approximation estimates for right-hand side errors and time integration errors. The architecture required for the integration of a simple mass-damper-stiffness case is included as an example.