Learning Hamiltonian Systems with Mono-Implicit Runge-Kutta Methods
Numerical integrators could be used to form interpolation conditions when training neural networks to approximate the vector field of an ordinary differential equation (ODE) from data. When numerical one-step schemes such as the Runge-Kutta methods are used to approximate the temporal discretization of an ODE with a known vector field, properties such as symmetry and stability are much studied. Here, we show that using mono-implicit Runge-Kutta methods of high order allows for accurate training of Hamiltonian neural networks on small datasets. This is demonstrated by numerical experiments where the Hamiltonian of the chaotic double pendulum in addition to the Fermi-Pasta-Ulam-Tsingou system is learned from data.
Code (1)
Similar Papers 제목 키워드 기반
Learning Dynamical Systems from Noisy Data with Inverse-Explicit Integrators
We introduce the mean inverse integrator (MII), a novel approach to increase the accuracy when training neural networks to approximate vector fields of dynamical systems from noisy data. This method can be used to averag…
Randomized Runge-Kutta-Nyström Methods for Unadjusted Hamiltonian and Kinetic Langevin Monte Carlo
We introduce $5/2$- and $7/2$-order $L^2$-accurate randomized Runge-Kutta-Nystr\"{o}m methods, tailored for approximating Hamiltonian flows within non-reversible Markov chain Monte Carlo samplers, such as unadjusted Hami…
Controlled oscillation modeling using port-Hamiltonian neural networks
Learning dynamical systems through purely data-driven methods is challenging as they do not learn the underlying conservation laws that enable them to correctly generalize. Existing port-Hamiltonian neural network method…
Convolutional Neural Networks combined with Runge-Kutta Methods
A convolutional neural network can be constructed using numerical methods for solving dynamical systems, since the forward pass of the network can be regarded as a trajectory of a dynamical system. However, existing mode…
Image ClassificationError estimation for physics-informed neural networks with implicit Runge-Kutta methods
The ability to accurately approximate trajectories of dynamical systems enables their analysis, prediction, and control. Neural network (NN)-based approximations have attracted significant interest due to fast evaluation…
Prediction