Philipp Horn

We present a neural network architecture based on the Generalized Hamiltonian Neural Networks that can learn a parameter-dependent flow map while preserving the symplectic structure. These Parametric Generalized Hamiltonian Neural Networks have great approximation capabilities which is highlighted by a universal approximation theorem. We compare these neural networks against physics-unaware multilayer perceptrons, for which integrating a parameter dependency can be easily achieved by adding an additional input neuron. The Parametric Generalized Hamiltonian Neural Networks are able to extrapolate to areas of the phase space without training data, while the physics-unaware multilayer perceptrons are not. The Parametric Generalized Hamiltonian Neural Networks achieve this with the same prediction speed as multilayer perceptrons and one order of magnitude less trainable parameters.

Simulating the evolution of the gravitational N-body problem becomes extremely computationally expensive as N increases since the problem complexity scales quadratically with the number of bodies. We study the use of Artificial Neural Networks (ANNs) to replace expensive parts of the integration of planetary systems. Neural networks that include physical knowledge have grown in popularity in the last few years, although few attempts have been made to use them to speed up the simulation of the motion of celestial bodies. We study the advantages and limitations of using Hamiltonian Neural Networks to replace computationally expensive parts of the numerical simulation. We compare the results of the numerical integration of a planetary system with asteroids with those obtained by a Hamiltonian Neural Network and a conventional Deep Neural Network, with special attention to understanding the challenges of this problem. Due to the non-linear nature of the gravitational equations of motion, errors in the integration propagate. To increase the robustness of a method that uses neural networks, we propose a hybrid integrator that evaluates the prediction of the network and replaces it with the numerical solution if considered inaccurate. Hamiltonian Neural Networks can make predictions that resemble the behavior of symplectic integrators but are challenging to train and in our case fail when the inputs differ ~7 orders of magnitude. In contrast, Deep Neural Networks are easy to train but fail to conserve energy, leading to fast divergence from the reference solution. The hybrid integrator designed to include the neural networks increases the reliability of the method and prevents large energy errors without increasing the computing cost significantly. For this problem, the use of neural networks results in faster simulations when the number of asteroids is >70.