Neural Networks for Dissipative Physics Using Morse-Feshbach Lagrangian

There is a growing attention given to utilizing Lagrangian and Hamiltonian mechanics with network training in order to incorporate physics into the network. Most commonly, conservative systems with zero frictional losses are modeled, which do not accurately represent physical reality. This work addresses systems with dissipation using a novel neural network formulation of the Morse--Feshbach Lagrangian. The Morse-Feshbach Lagrangian models dissipative dynamics by doubling the number of dimensions of the system in order to create a ‘mirror’ latent representation that would counterbalance the dissipation of the observable system, making it a conservative system. We start with their formal approach by redefining a new Dissipative Lagrangian, such that the unknown matrices in the Euler-Lagrange's equations arise as partial derivatives of the Lagrangian with respect to only the observables. We then train a network from simulated training data for dissipative systems. As a model system, we choose a mechanical system with frictional dissipation and show that the approach is able to accurately capture dissipative dynamics. The approach is quite general and can be used to represent other dissipative phenomena such as Fickian diffusion.