Machine Learning for Initial Value Problems of Parameter-Dependent Dynamical Systems

TL;DR

This paper explores using neural networks to efficiently approximate the relationship between parameters and trajectories in nonlinear dynamical systems with physical parameters, reducing computational effort.

Contribution

It introduces a machine learning approach with neural networks to approximate solutions of parameter-dependent initial value problems, offering a faster alternative to traditional methods.

Findings

Neural networks accurately approximate trajectories for the test example.

The approach reduces computational cost compared to solving initial value problems directly.

Numerical results demonstrate the method's effectiveness.

Abstract

We consider initial value problems of nonlinear dynamical systems, which include physical parameters. A quantity of interest depending on the solution is observed. A discretisation yields the trajectories of the quantity of interest in many time points. We examine the mapping from the set of parameters to the discrete values of the trajectories. An evaluation of this mapping requires to solve an initial value problem. Alternatively, we determine an approximation, where the evaluation requires low computation work, using a concept of machine learning. We employ feedforward neural networks, which are fitted to data from samples of the trajectories. Results of numerical computations are presented for a test example modelling an electric circuit.