Data-driven intrinsic localized mode detection and classification in one-dimensional crystal lattice model

In this work we propose Support Vector Machine classification algorithms to classify one-dimensional crystal lattice waves from locally sampled data. Three different learning datasets of particle displacements, momenta and energy density values are considered. Efficiency of the classification algorithms are further improved by two dimensionality reduction techniques: Principal Component Analysis and Locally Linear Embedding. Robustness of classifiers are investigated and demonstrated. Developed algorithms are successfully applied to detect localized intrinsic modes in three numerical simulations considering a case of two localized stationary breather solutions, a single stationary breather solution in noisy background and two mobile breather collision.

Locally-symplectic neural networks for learning volume-preserving dynamics

We propose locally-symplectic neural networks LocSympNets for learning volume-preserving dynamics. The construction of LocSympNets stems from the theorem of local Hamiltonian description of the vector field of a volume-preserving dynamical system and the splitting methods based on symplectic integrators. Modified gradient modules of recently proposed symplecticity-preserving neural networks SympNets are used to construct locally-symplectic modules, which composition results in volume-preserving neural networks. LocSympNets are studied numerically considering linear and nonlinear dynamics, i.e., semi-discretized advection equation and Euler equations of the motion of a free rigid body, respectively. LocSympNets are able to learn linear and nonlinear dynamics to high degree of accuracy. When learning a single trajectory of the rigid body dynamics LocSympNets are able to learn both invariants of the system with absolute relative errors below 1% in long-time predictions and produce qualitatively good short-time predictions, when the learning of the whole system from randomly sampled data is considered.