Lagrangian Density Space-Time Deep Neural Network Topology

TL;DR

This paper introduces Lagrangian Density Space-Time Deep Neural Networks (LDDNN), a novel unsupervised neural network topology that models physical phenomena by respecting conservation laws through Lagrangian and Hamiltonian densities.

Contribution

It proposes a new neural network framework that parameterizes the Lagrangian density directly from data, enabling physics-informed learning without explicit solutions.

Findings

Successfully models physical dynamics respecting conservation laws

Enables unsupervised learning of complex PDEs from data

Provides a physics-based interpretation of neural network representations

Abstract

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.