Informational entropy : a failure tolerance and reliability surrogate for water distribution networks

Evolutionary algorithms are used widely in optimization studies on water distribution networks. The optimization algorithms use simulation models that analyse the networks under various operating conditions. The solution process typically involves cost minimization along with reliability constraints that ensure reasonably satisfactory performance under abnormal operating conditions also. Flow entropy has been employed previously as a surrogate reliability measure. While a body of work exists for a single operating condition under steady state conditions, the effectiveness of flow entropy for systems with multiple operating conditions has received very little attention. This paper describes a multi-objective genetic algorithm that maximizes the flow entropy under multiple operating conditions for any given network. The new methodology proposed is consistent with the maximum entropy formalism that requires active consideration of all the relevant information. Furthermore, an alternative but equivalent flow entropy model that emphasizes the relative uniformity of the nodal demands is described. The flow entropy of water distribution networks under multiple operating conditions is discussed with reference to the joint entropy of multiple probability spaces, which provides the theoretical foundation for the optimization methodology proposed. Besides the rationale, results are included that show that the most robust or failure-tolerant solutions are achieved by maximizing the sum of the entropies

Analysis and Decentralised Optimal Flow Control of Heterogeneous Computer Communication Network Models

General closed queueing networks are used to model the local flow control in multiclass computer communication networks with single and multiple transmission links. The problem of analysing multiclass general closed queueing network models with single server and multiserver is presented followed by the problem of decentralised optimal local flow control of multiclass general computer communication networks with single and multiple transmission links. The generalised exponential (GE) distributional model with known first two moments has been used to represent general interarrival and transmission time distributions as various users have various traffic characteristics. A new method of general model reduction using the Norton' s theorem for general queueing networks in conjunction with the universal maximum entropy algorithm is proposed for the analysis of large general closed queueing networks. This extension to Norton's theorem has an advantage over the direct application of the universal maximum entropy approach whereby the study of a subset of queueing centres of interest can be done without repeatedly solving the entire network. The principle of maximum entropy is used to derive new approximate solutions for the joint queue length distributions of multiclass general queueing network models with single server and multiserver and favourable comparisons with other methods are made. The decentralised optimal local flow control of the multiclass computer communication networks with single and multiple transmission links is shown to be a state dependent window type mechanism that has been traditionally used in practice

Maximum Entropy Analysis of Flow Networks: Theoretical Foundation and Applications

The concept of a “flow network”—a set of nodes and links which carries one or more flows—unites many different disciplines, including pipe flow, fluid flow, electrical, chemical reaction, ecological, epidemiological, neurological, communications, transportation, financial, economic and human social networks. This Feature Paper presents a generalized maximum entropy framework to infer the state of a flow network, including its flow rates and other properties, in probabilistic form. In this method, the network uncertainty is represented by a joint probability function over its unknowns, subject to all that is known. This gives a relative entropy function which is maximized, subject to the constraints, to determine the most probable or most representative state of the network. The constraints can include “observable” constraints on various parameters, “physical” constraints such as conservation laws and frictional properties, and “graphical” constraints arising from uncertainty in the network structure itself. Since the method is probabilistic, it enables the prediction of network properties when there is insufficient information to obtain a deterministic solution. The derived framework can incorporate nonlinear constraints or nonlinear interdependencies between variables, at the cost of requiring numerical solution. The theoretical foundations of the method are first presented, followed by its application to a variety of flow networks

Advances in Deep Generative Modeling With Applications to Image Generation and Neuroscience

Deep generative modeling is an increasingly popular area of machine learning that takes advantage of recent developments in neural networks in order to estimate the distribution of observed data. In this dissertation we introduce three advances in this area. The first one, Maximum Entropy Flow Networks, allows to do maximum entropy modeling by combining normalizing flows with the augmented Lagrangian optimization method. The second one is the continuous Bernoulli, a new [0,1]-supported distribution which we introduce with the motivation of fixing the pervasive error in variational autoencoders of using a Bernoulli likelihood for non-binary data. The last one, Deep Random Splines, is a novel distribution over functions, where samples are obtained by sampling Gaussian noise and transforming it through a neural network to obtain the parameters of a spline. We apply these to model texture images, natural images and neural population data, respectively; and observe significant improvements over current state of the art alternatives

Ecological flow analysis of network collapse II: Indicators of ecosystem level vulnerability

Using donor-controlled, bottom-up equations to describe network collapse we systematically investigate the impact each species has on the survival or extinction of other species. Short of extinction, one can determine the integrated losses experienced by the ecosystem. These losses are aggregated into system level indicators, such as entropy, average gain/loss, average time to extinction, etc. The methodology is applied to 18 ecological flow networks available in the literature. We calculate the correlations between various indicators and determine high positive correlation between: number of nodes & maximal trophic level; connectedness & average entropy losses; number of nodes & average number of extinct nodes; and, maximum trophic level & evenness of links. A high negative correlation was found between: number of nodes & connectedness; connectedness & maximal trophic level; maximum tropic level & average entropy loss; and, connectedness & evenness of flows. Lastly, a low correlation was found between: average number of extinct compartments & evenness of flows; number of nodes & evenness of stocks; and, evenness of flows & evenness of stocks

Entropy maximizing evolutionary design optimization of water distribution networks under multiple operating conditions

The informational entropy model for flow networks was formulated over 30 years ago by Tanyimboh and Templeman (University of Liverpool, UK) for a single discrete operating condition that typically comprises the maximum daily demands and was undefined for water distribution networks (WDNs) under multiple operating conditions. Its extension to include multiple independent discrete operating conditions was investigated experimentally herein considering the relationships between flow entropy and hydraulic capacity reliability and redundancy. A novel penalty-free multi-objective genetic algorithm was developed to minimize the initial construction cost and maximize the flow entropy subject to the design constraints. Furthermore, optimized designs derived from the maximum daily demands as a single discrete operating condition were compared to those derived from a combination of discrete operating conditions. Optimized designs from a combination of discrete operating conditions outperformed those from a single operating condition in terms of performance and initial construction cost. The best results overall were achieved by maximizing the sum of the flow entropies of the discrete operating conditions. The logical inference from the results is that the flow entropy of multiple discrete operating conditions is the sum of their respective entropies. In addition, a crucial property of the resulting flow entropy model is that it is bias free with respect to the individual operating conditions; hitherto a fundamental weakness concerning the practical application of the flow entropy model to WDNs is thus addressed

Reliability assessment of water distribution systems with statistical entropy and other surrogate measures

There is ever increasing commercial and regulatory pressure to minimise the cost of water distribution networks even as the demand for them keeps on growing. But cost minimizing is only one of the demands placed on network design. Satisfactory networks are required to operate above a minimum level even if they experience failure of components. Reliable hydraulic performance can be achieved if sufficient redundancy is built in the network. This has given rise to various water distribution system optimization methods including genetic algorithms and other evolutionary computing methods. Evolutionary computing approaches frequently assess the suitability of enormous numbers of potential solutions for which the calculation of accurate reliability measures could be computationally prohibitive. Therefore, surrogate reliability measures are frequently used to ease the computational burden. The aim of this paper is to assess the correlation of surrogate reliability measures in relation to more accurate measures. The surrogate measures studied are statistical entropy, network resilience, resilience index and modified resilience index. The networks were simulated with the prototype software PRAAWDS that produces more realistic results for pressure-deficient water distribution systems. Statistical entropy outperformed resilience index in this study. The results also demonstrate there is a strong correlation between entropy and failure tolerance