Forecasting high waters at Venice Lagoon using chaotic time series analisys and nonlinear neural netwoks

Time series analysis using nonlinear dynamics systems theory and multilayer neural networks models have been applied to the time sequence of water level data recorded every hour at 'Punta della Salute' from Venice Lagoon during the years 1980-1994. The first method is based on the reconstruction of the state space attractor using time delay embedding vectors and on the characterisation of invariant properties which define its dynamics. The results suggest the existence of a low dimensional chaotic attractor with a Lyapunov dimension, DL, of around 6.6 and a predictability between 8 and 13 hours ahead. Furthermore, once the attractor has been reconstructed it is possible to make predictions by mapping local-neighbourhood to local-neighbourhood in the reconstructed phase space. To compare the prediction results with another nonlinear method, two nonlinear autoregressive models (NAR) based on multilayer feedforward neural networks have been developed. From the study, it can be observed that nonlinear forecasting produces adequate results for the 'normal' dynamic behaviour of the water level of Venice Lagoon, outperforming linear algorithms, however, both methods fail to forecast the 'high water' phenomenon more than 2-3 hours ahead.Publicad

Short-term power demand forecasting using the differential polynomial neural network

Power demand forecasting is important for economically efficient operation and effective control of power systems and enables to plan the load of generating unit. The purpose of the short-term electricity demand forecasting is to forecast in advance the system load, represented by the sum of all consumers load at the same time. A precise load forecasting is required to avoid high generation cost and the spinning reserve capacity. Under-prediction of the demands leads to an insufficient reserve capacity preparation and can threaten the system stability, on the other hand, over-prediction leads to an unnecessarily large reserve that leads to a high cost preparations. Differential polynomial neural network is a new neural network type, which forms and resolves an unknown general partial differential equation of an approximation of a searched function, described by data observations. It generates convergent sum series of relative polynomial derivative terms which can substitute for the ordinary differential equation, describing 1-parametric function time-series. A new method of the short-term power demand forecasting, based on similarity relations of several subsequent day progress cycles at the same time points is presented and tested on 2 datasets. Comparisons were done with the artificial neural network using the same prediction method.Web of Science8230629

Inverse Problems and Dynamical Systems in Tomography and Optics

The dissertation concerns inverse problems and dynamical systems. Inverse problems, as a subfield of mathematics, studies the mathematical theory of indirect measurements. It is an active area of research with extensive mathematical theory and numerous applications. Many inverse problems are concerned with physical systems that evolve in time. A mathematical model that describes how a quantity evolves in time is called a dynamical system. X-ray computed tomography is a technique where the inner structure of an object is computed from a number of its X-ray images. In the first publication of the dissertation we consider X-ray computed tomography in a setting where the orientations in which the object was imaged are unknown. This problem is called tomography with unknown view angles; such a problem arises e.g. in cryogenic electron microscopy of viral particles. We show that under general assumptions it is possible to reconstruct the structure of the object in tomography with unknown view angles. Diffusion is the flow of a substance from areas of high concentration to areas of low concentration. In the second publication we consider an inverse problem for the space–time fractional diffusion equation. This equation models diffusion and anomalous diffusion processes, such as those sometimes observed in fractured geological formations. We show that the geometry of the underlying space can be determined by observing the evolution of a solution of the equation in a subset of the space. In the third and fourth publications we consider a dynamical system that models injection locking in a laser. Injection locking is a technique where light from one laser is injected into another laser's cavity (the part of a laser where the emitted light is created) with the intention of altering the laser's properties. The idea is to consider injection locking as a process that can provide the basis for optical computing devices. In the third publication we derive an approximation for the nonlinear relationship between the injected light and the injection-locked emitted light, and we show that it is possible to construct an optical logic gate based on this relationship. In the fourth publication we do a detailed analysis of the dynamical system that models injection locking in lasers, and based on this analysis, we propose a design for an optical neural network.Väitöskirja käsittelee käänteisten ongelmien ja dynaamisten järjestelmien matemaattista teoriaa. Käänteisissä ongelmissa pyritään ymmärtämään ilmiön syy sen seurauksia. Esimerkki käänteisestä ongelmasta on röntgenkuvaus, jossa kappaleen läpäisseiden säteiden vaimenemisesta (seuraus) halutaan päätellä vaimenemisen syy (kappaleen sisäinen rakenne). Usein käänteisen ongelman liittyvä tarkasteltava suure on dynaaminen, eli se muuttuu ajan mukana. Tällaisen ilmiön matemaattista mallia kutsutaan dynaamiseksi järjestelmäksi. Väitöskirjan ensimmäisessä julkaisussa tutkitaan tietokonekerroskuvauksen matemaattista teoriaa. Tietokonekerroskuvauksessa tarkasteltavasta kohteesta kuvataan röntgenkuvia useista eri suunnista, ja näistä kuvista pyritään laskemaan kohteen sisäinen rakenne. Usein suunnat joista kuvat on otettu tunnetaan, mutta tietyissä tilanteissa nämä kuvaussuunnat ovat tuntemattomia. Tällainen tilanne on esimerkiksi virusten kuvantamisessa käytetyssä kryogeenisessa elektronimikroskopiassa. Julkaisussa osoitetaan, että kohteen rakenteen määrittäminen on yleensä mahdollista vaikka kuvaussuunnat olisivatkin tuntemattomia. Lämmön johtumisen matemaattista mallia kutsutaan lämpöyhtälöksi. Lämmön johtuminen aineen sisällä riippuu sekä aineen kyvystä johtaa lämpöä (ns. lämmönjohtavuuskertoimesta), että tarkasteltavan kappaleen muodosta. Väitöskirjan toisessa julkaisussa tutkitaan käänteistä ongelmaa lämpöyhtälölle. Julkaisussa osoitetaan, että tarkastelemalla kappaleen lämpötilan käyttäytymistä osassa kappaletta voidaan päätellä sekä koko kappaleen muoto, että sen lämmönjohtavuuskerroin. Kolmannessa ja neljännessä julkaisussa tutkitaan optiseen laskentaan liittyvän dynaamisen järjestelmän matemaattista teoriaa. Nykyisten puolijohdekomponentteihin perustuvien tietokoneiden toiminta perustuu sähköön, optisessa laskennassa tavoite on rakentaa tietokone jonka toiminta perustuu sähkön sijaan valoon. Julkaisuissa tutkitaan lasereiden synkronoitumiseen liittyvän ilmiön (ns. injektiolukituksen) matemaattisia ominaisuuksia. Julkaisuissa osoitetaan että ominaisuuksiensa puolesta injektiolukitusta voi olla mahdollista hyödyntää optisten transistoreiden tai optisten neuroverkkojen rakentamisessa

Design and Analysis of Monte Carlo Experiments

monte carlo experiments;simulation models;mathematical analysis;sensitivity analysis;experimental design

On quadrature rules for solving Partial Differential Equations using Neural Networks

Neural Networks have been widely used to solve Partial Differential Equations. These methods require to approximate definite integrals using quadrature rules. Here, we illustrate via 1D numerical examples the quadrature problems that may arise in these applications and propose several alternatives to overcome them, namely: Monte Carlo methods, adaptive integration, polynomial approximations of the Neural Network output, and the inclusion of regularization terms in the loss. We also discuss the advantages and limitations of each proposed numerical integration scheme. We advocate the use of Monte Carlo methods for high dimensions (above 3 or 4), and adaptive integration or polynomial approximations for low dimensions (3 or below). The use of regularization terms is a mathematically elegant alternative that is valid for any spatial dimension; however, it requires certain regularity assumptions on the solution and complex mathematical analysis when dealing with sophisticated Neural Networks