Approximating dynamical systems from data is a significant and challenging problem. Incorporating knowledge about physical laws that govern the dynamical process can help reduce data requirements and improve prediction accuracy. Here, we discuss how to approximate Hamiltonian functions of energy-conserving dynamical systems by solving an associated linear partial differential equation. We employ neural network activation functions as basis functions for the solution and evaluate the performance of data-agnostic and data-driven weight sampling algorithms to construct this basis. We experiment with single pendulum, Lotka-Volterra, double pendulum, and Hénon- Heiles systems and evaluate the approximation capabilities of the sampled networks. The accuracy of the sampled networks using the data-driven sampling scheme outper-forms the data-agnostic sampling in various settings where the gradients of the target function get large. Also, the data-agnostic scheme needs proper bias configuration beforehand; otherwise, its accuracy is worse than the data-driven sampling scheme. Finally, we demonstrate how the sampled networks can also learn from discrete trajec- tory observations using a post-training correction step, which has been studied with traditional neural networks before.     «

Approximating dynamical systems from data is a significant and challenging problem. Incorporating knowledge about physical laws that govern the dynamical process can help reduce data requirements and improve prediction accuracy. Here, we discuss how to approximate Hamiltonian functions of energy-conserving dynamical systems by solving an associated linear partial differential equation. We employ neural network activation functions as basis functions for the solution and evaluate the performance...     »