Physics-Informed Classical Lagrange / Hamilton Neural Networks in Deep Learning

Abstract

The principles of constructing deep machine learning systems based on taking into account information about the physical properties of the studied control object, such as an autonomous robot, are considered. The platform for the development of intelligent tools is deep machine learning models applying physics-informed neural networks. Most of the methods under development for constructing system identification models are either "black box" models (i.e., general models based on training data) or so-called "white box" models (e.g., state-space/control models, which can be explicitly expressed mathematically). Thus, the direction of development is to study the gray box model in the state space. A gray box model means a model that is trained on data while being guided by information about some physical properties or laws that apply. Such models can be further applied for adaptive сontrol and self-organization. Therefore, the use of state-space models is considered. A feature of physically informed neural networks is that they initially take into account the underlying description of the physical interpretation of partial or ordinary differential equations, that is, the physics of the problem, instead of trying to derive a solution based solely on data, that is, by approximating a set of pairs by a neural network "state-value". Lagrange and Newton neural networks are considered as such a class of learning systems. Specific examples show the advantages and features of using the discussed types of physics-informed neural networks.

Keywords

• physics-informed neural networks
• lagrange neural networks
• hamilton neural networks
• neural networks with physical constraints