Lagrangian Density Space-Time Deep Neural Network Topology
This work addresses the challenge of interpreting black-box deep learning for physical dynamics, offering a domain-specific approach to infer information from data in physics.
The authors tackled the problem of predicting dynamics of physical systems governed by nonlinear partial differential equations by proposing a Lagrangian Density Space-Time Deep Neural Network (LDDNN) topology, which learns Lagrangian densities from data without requiring hand-crafted solutions, enabling unsupervised training and adherence to conservation laws.
As a network-based functional approximator, we have proposed a "Lagrangian Density Space-Time Deep Neural Networks" (LDDNN) topology. It is qualified for unsupervised training and learning to predict the dynamics of underlying physical science governed phenomena. The prototypical network respects the fundamental conservation laws of nature through the succinctly described Lagrangian and Hamiltonian density of the system by a given data-set of generalized nonlinear partial differential equations. The objective is to parameterize the Lagrangian density over a neural network and directly learn from it through data instead of hand-crafting an exact time-dependent "Action solution" of Lagrangian density for the physical system. With this novel approach, can understand and open up the information inference aspect of the "Black-box deep machine learning representation" for the physical dynamics of nature by constructing custom-tailored network interconnect topologies, activation, and loss/cost functions based on the underlying physical differential operators. This article will discuss statistical physics interpretation of neural networks in the Lagrangian and Hamiltonian domains.