@article{scharzenberger2021deep_roa,
title={{Learning To Estimate Regions Of Attraction Of Autonomous Dynamical Systems Using Physics-Informed Neural Networks}},
author={Scharzenberger, Cody and Hays, Joe},
journal={https://arxiv.org/abs/2111.09930},
year={2021}
}
This repository contains the code necessary to train a non-spiking deep safety network using Pytorch to predict the boundary of the region of attraction (ROA) associated with a stable equilibrium point of a given autonomous dynamical system. Our approach is best understood as an application of a modified physics-informed neural network (PINN) framework to a specific partial differential equation (PDE), namely the Yuan-Li PDE, whose solution yields an implicit representation of the ROA boundary as time approaches infinity.
We refer to our methodology as using a modified PINN framework due to the fact that we incorporate the standard PINN loss terms, such as the initial, boundary, and residual losses, as well as additional loss terms, such as the variational loss and monotonicity loss. In the examples shown here, we provide the system dynamics of interest to the Yuan-Li PDE directly in the form of differential equations, but this is not strictly necessary.
The safety network produced by this code is a fully connected feed-forward non-spiking network with a number of inputs equal to the dimension of the state space of the dynamical system being analyzed plus one for the single temporal variable, a number of hidden layers specified by the user, and a single scalar output representing the stability of the input states.
When training is successful, the network identifies stable states as those whose output is negative and unstable states as those whose output is positive.
This repository contains one example directory called "Closed_ROA" which itself contains the main script "main.py" as well as three subfolders: (1) Utilities, (2) Save, and (3) Load.
The Utilities folder contains the various classes required to implement this methodology. These various classes implement the framework for training safety networks discussed above and do not themselves need to be edited.
The deep roa class is the highest level Python wrapper that utilities all of the other lower level classes to build, train, and evaluate a safety network using the specified hyperparameters. It encapsulates all of the functionality of the other classes and is the primary class that the user interacts with in their main script.
The project options class contains all of the numerous parameters and settings that the user can select when building, training, and evaluating a safety network. All of the parameters that directly impact the the safety network structure or training are contained in the hyperparameter class (see below), while the other non-essential options are left as attributes of the project options class.
The hyperparameters class contains all of the hyperparameters for building and training a safety network. Changing any of these parameters will impact the quality of the resulting safety network ROA estimation.
The problem specifications class contains all of the parameters required to define the underlying dynamical system that is being analyzed, including the domain of analysis, the flow of the dynamical system, and the relevant initial and boundary conditions.
The network class contains all of the code necessary to implement out modified physics-informed network framework. Specially, this class stores the network properties, such as its weights and biases, and implements the methods necessary to train and evaluate the network, such as the various loss functions used during training.
The PDE class contains information related to the underlying dynamical system whose stable equilibrium we are analyzing. This includes the explicit differential equations as well as methods for evaluating the flow of the system.
The initial-boundary condition class contains information related to the creation and application of the initial and boundary conditions that are relevant to the Yuan-Li PDE.
The IBC, residual, and variational data classes contain information pertaining to the initial-boundary condition, residual, and variational training / testing data, respectively.
The finite element class contains the code necessary to implement the various finite element integration techniques that we use to evaluate the variational loss. This includes such things as constructing finite elements and Gauss-Legendre integration.
The domain class simply contains information related to the domain over which we are analyzing the dynamical system. This includes such things as the limits of the domain as well as the spatiotemporal discretization.
In addition to the aforementioned classes, there are numerous classes whose names include the ``utilities'' keyword. These classes do not themselves contain many properties but are instead organizational tools that collect many relevant, low level functions into a single utilities module.
The utilities classes are leverage by many of the previous classes to perform common basic functions. Importantly, the model utilities class stores all of the possible dynamical systems of interest. To study a new dynamical system, the associated system of first order ODEs would first need to be added to the model utilities class.
The save directory contains all of the projection options, hyperparameters, and problem specifications associated with the most recent network training session. Depending on user specifications during training, the save directory also includes saved network states during and after training, as well as plots that evaluate network performance.
The load directory contains all of the project option, hyperparameters, problem specifications, and network save states that can be loaded before initiating a new training session if specified by the user.