Modeling Power Systems Dynamics with Symbolic Physics-Informed Neural Networks

In recent years, scientific machine learning, particularly physic-informed neural networks (PINNs), has introduced new innovative methods to understanding the differential equations that describe power system dynamics, providing a more efficient alternative to traditional methods. However, using a single neural network to capture patterns of all variables requires a large enough size of networks, leading to a long time of training and still high computational costs. In this paper, we utilize the interfacing of PINNs with symbolic techniques to construct multiple single-output neural networks by taking the loss function apart and integrating it over the relevant domain. Also, we reweigh the factors of the components in the loss function to improve the performance of the network for instability systems. Our results show that the symbolic PINNs provide higher accuracy with significantly fewer parameters and faster training time. By using the adaptive weight method, the symbolic PINNs can avoid the vanishing gradient problem and numerical instability

A PINN Approach to Symbolic Differential Operator Discovery with Sparse Data

Given ample experimental data from a system governed by differential equations, it is possible to use deep learning techniques to construct the underlying differential operators. In this work we perform symbolic discovery of differential operators in a situation where there is sparse experimental data. This small data regime in machine learning can be made tractable by providing our algorithms with prior information about the underlying dynamics. Physics Informed Neural Networks (PINNs) have been very successful in this regime (reconstructing entire ODE solutions using only a single point or entire PDE solutions with very few measurements of the initial condition). We modify the PINN approach by adding a neural network that learns a representation of unknown hidden terms in the differential equation. The algorithm yields both a surrogate solution to the differential equation and a black-box representation of the hidden terms. These hidden term neural networks can then be converted into symbolic equations using symbolic regression techniques like AI Feynman. In order to achieve convergence of these neural networks, we provide our algorithms with (noisy) measurements of both the initial condition as well as (synthetic) experimental data obtained at later times. We demonstrate strong performance of this approach even when provided with very few measurements of noisy data in both the ODE and PDE regime

Computational physics of the mind

In the XIX century and earlier such physicists as Newton, Mayer, Hooke, Helmholtz and Mach were actively engaged in the research on psychophysics, trying to relate psychological sensations to intensities of physical stimuli. Computational physics allows to simulate complex neural processes giving a chance to answer not only the original psychophysical questions but also to create models of mind. In this paper several approaches relevant to modeling of mind are outlined. Since direct modeling of the brain functions is rather limited due to the complexity of such models a number of approximations is introduced. The path from the brain, or computational neurosciences, to the mind, or cognitive sciences, is sketched, with emphasis on higher cognitive functions such as memory and consciousness. No fundamental problems in understanding of the mind seem to arise. From computational point of view realistic models require massively parallel architectures

A symbolic sensor for an Antilock brake system of a commercial aircraft

The design of a symbolic sensor that identifies thecondition of the runway surface (dry, wet, icy, etc.) during the braking of a commercial aircraft is discussed. The purpose of such a sensor is to generate a qualitative, real-time information about the runway surface to be integrated into a future aircraft Antilock Braking System (ABS). It can be expected that this information can significantly improve the performance of ABS. For the design of the symbolic sensor different classification techniques based upon fuzzy set theory and neural networks are proposed. To develop and to verify theses classification algorithms data recorded from recent braking tests have been used. The results show that the symbolic sensor is able to correctly identify the surface condition. Overall, the application example considered in this paper demonstrates that symbolic information processing using fuzzy logic and neural networks has the potential to provide new functions in control system design. This paper is part of a common research project between E.N.S.I.C.A. and Aerospatiale in France to study the role of the fuzzy set theory for potential applications in future aircraft control systems

Platonic model of mind as an approximation to neurodynamics

Hierarchy of approximations involved in simplification of microscopic theories, from sub-cellural to the whole brain level, is presented. A new approximation to neural dynamics is described, leading to a Platonic-like model of mind based on psychological spaces. Objects and events in these spaces correspond to quasi-stable states of brain dynamics and may be interpreted from psychological point of view. Platonic model bridges the gap between neurosciences and psychological sciences. Static and dynamic versions of this model are outlined and Feature Space Mapping, a neurofuzzy realization of the static version of Platonic model, described. Categorization experiments with human subjects are analyzed from the neurodynamical and Platonic model points of view

Observability and Synchronization of Neuron Models

Observability is the property that enables to distinguish two different locations in $n$-dimensional state space from a reduced number of measured variables, usually just one. In high-dimensional systems it is therefore important to make sure that the variable recorded to perform the analysis conveys good observability of the system dynamics. In the case of networks composed of neuron models, the observability of the network depends nontrivially on the observability of the node dynamics and on the topology of the network. The aim of this paper is twofold. First, a study of observability is conducted using four well-known neuron models by computing three different observability coefficients. This not only clarifies observability properties of the models but also shows the limitations of applicability of each type of coefficients in the context of such models. Second, a multivariate singular spectrum analysis (M-SSA) is performed to detect phase synchronization in networks composed by neuron models. This tool, to the best of the authors' knowledge has not been used in the context of networks of neuron models. It is shown that it is possible to detect phase synchronization i)~without having to measure all the state variables, but only one from each node, and ii)~without having to estimate the phase