Machine Learning for Initial Value Problems of Parameter-Dependent Dynamical Systems
This work addresses computational efficiency for parameter-dependent dynamical systems, but it appears incremental as it applies standard neural networks to a specific test case without broad validation.
The paper tackles the problem of efficiently approximating the mapping from parameters to trajectories in nonlinear dynamical systems with initial value problems, using feedforward neural networks to reduce computational work, and demonstrates results on a test example of an electric circuit.
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.