Ladislav Zjavka

Constructing ordinary sum differential equations using polynomial networks

Data relations can define general sum partial differential equations of a composite function additive derivative model. Time-series data observations can analogously describe an ordinary sum differential equation with time derivatives, which is possible to be solved using partial derivative term substitutions of time-dependent series. Differential polynomial neural network is a new type of neural network, which constructs and substitutes for an unknown general partial differential equation from data observations, developed by the author. It generates sum series of convergent partial polynomial derivative terms, which can describe an unknown complex function time-series. This type of non-linear regression decomposes a system model, described by the general differential equation, into many partial low order derivative specifications of selected relative sum terms. Common soft-computing techniques in general can apply input variables of only absolute interval values of a specific data range. The character of relative data allows processing a wider range of test interval values than defined by a training set. The characteristics of the composite sum differential equation solutions can facilitate a much greater variety of model forms than is allowed using standard soft computing methods. Recurrent neural network proved to form simple solid time-series models, most of which it is possible to describe using ordinary differential equations, so the comparisons were done.

Constructing general partial differential equations using polynomial and neural networks

Sum fraction terms can approximate multi-variable functions on the basis of discrete observations, replacing a partial differential equation definition with polynomial elementary data relation descriptions. Artificial neural networks commonly transform the weighted sum of inputs to describe overall similarity relationships of trained and new testing input patterns. Differential polynomial neural networks form a new class of neural networks, which construct and solve an unknown general partial differential equation of a function of interest with selected substitution relative terms using non-linear multi-variable composite polynomials. The layers of the network generate simple and composite relative substitution terms whose convergent series combinations can describe partial dependent derivative changes of the input variables. This regression is based on trained generalized partial derivative data relations, decomposed into a multi-layer polynomial network structure. The sigmoidal function, commonly used as a nonlinear activation of artificial neurons, may transform some polynomial items together with the parameters with the aim to improve the polynomial derivative term series ability to approximate complicated periodic functions, as simple low order polynomials are not able to fully make up for the complete cycles. The similarity analysis facilitates substitutions for differential equations or can form dimensional units from data samples to describe real-world problems.

Irradiance prediction using Echo State Queueing Networks and Differential polynomial Neural Networks

This paper investigates the estimation of a real time-series benchmark: the solar irradiance forcasting. The global solar irradiance is an important variable in the production of renewable energy sources. These variable is very unstable and hard to be predicted. For the prediction, we use two new models for time-series modeling: Echo State Queueing Networks and Differential polynomial Neural Networks. Both tools have been proven to be efficient for forecasting and time-series modeling. We compare their performances for this particular data set.

Numerical weather prediction revisions using the locally trained differential polynomial network

Differential polynomial neural network (D-PNN) extends the GMDH network structure.D-PNN constructs and solves the general partial differential equation with sum series.D-PNN is trained with historical time-series for actual local weather data relations.The correction model can apply NWP outputs to revise one target 24-hour forecast.NWP model revisions of the temperature, relative humidity and dew point were done. Meso-scale forecasts result from global numerical weather prediction models, which are based upon the differential equations for atmospheric dynamics that do not perfectly determine weather conditions near the ground. Statistical corrections can combine complex numerical models, based on the physics of the atmosphere to forecast the large-scale weather patterns, and regression in post-processing to clarify surface weather details according to local observations and climatological conditions. Neural networks trained with local relevant weather observations of fluctuant data relations in current conditions, entered by numerical model outcomes of the same data types, may revise its one target short-term prognosis (e.g. relative humidity or temperature) to stand for these methods. Polynomial neural networks can compose general partial differential equations, which allow model more complicated real system functions from discrete time-series observations than using standard soft-computing methods. This new neural network technique generates convergent series of substitution relative derivative terms, which combination sum can define and solve an unknown general partial differential equation, able to describe dynamic processes of the weather system in a local area, analogous to the differential equation systems of numerical models. The trained network model revises hourly-series of numerical prognosis of one target variable in sequence, applying the general differential equation solution of the correction multi-variable function to corresponding output variables of the 24-hour numerical forecast. The experimental results proved this polynomial network type can successfully revise some numerical weather prognoses after this manner.

Short-term Power Demand Forecasting using the Differential Polynomial Neural Network

AbstractPower 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 differ...

Power Output Models of Ordinary Differential Equations by Polynomial and Recurrent Neural Networks

The production of renewable energy sources is unstable, influenced a weather frame. Photovoltaic power plant output is primarily dependent on the solar illuminance of a locality, which is possible to predict according to meteorological forecasts (Aladin). Wind charger power output is induced mainly by a current wind speed, which depends on several weather standings. Presented time-series neural network models can define incomputable functions of power output or quantities, which direct influence it. Differential polynomial neural network is a new neural network type, which makes use of data relations, not only absolute interval values of variables as artificial neural networks do. Its output is formed by a sum of fractional derivative terms, which substitute a general differential equation, defining a system model. In the case of time-series data application an ordinary differential equation is created with time derivatives. Recurrent neural network proved to form simple solid time-series models, which can replace the ordinary differential equation description.

Approximation of multi-parametric functions using the differential polynomial neural network

Unknown data relations can describe a lot of complex systems through a partial differential equation solution of a multi-parametric function approximation. Common artificial neural network techniques of a pattern classification or function approximation in general are based on whole-pattern similarity relations of trained and tested data samples. It applies input variables of only absolute interval values, which may cause problems by far various training and testing data ranges. Differential polynomial neural network is a new type of neural network developed by the author, which constructs and resolves an unknown general partial differential equation, describing a system model of dependent variables. It creates a sum of fractional polynomial terms, defining partial mutual derivative changes of input variables combinations. This type of regression is based on learned generalized data relations. It might improve dynamic system models a standard time-series prediction, as the character of relative data allows to apply a wider range of input interval values than defined by the trained data. Also the characteristics of differential equation solutions facilitate a great variety of model forms.

Failure and power utilization system models of differential equations by polynomial neural networks

Reliability modeling of electronic circuits can be best performed by the stressor - susceptibility interaction model. A circuit or a system is deemed to be failed once the stressor has exceeded the susceptibility limits. Complex manufacturing systems often require a high level of reliability from the incoming electricity supply. Modern industrial time power quality monitoring systems can be used for the pre-fault load alarming. Neural networks can successfully model and predict the failure frame of critical electronic systems and power utilization in power plants described only a few input quantities. Differential polynomial neural network is a new type of neural network, which constructs and substitutes an unknown general sum partial differential equation with a total sum of fractional polynomial terms. The system model describes partial relative derivative dependent changes of some input combinations of variables. This type of non-linear regression is based on trained generalized data relations decomposed by partial low order polynomials of 2-input variables. Experimental results indicate that the proposed method is efficient.

Forecast Models of Partial Differential Equations Using Polynomial Networks

Unknown data relations can describe lots of complex systems through partial differential equation solutions of a multi-parametric function approximation. Common neural network techniques of pattern classification or function approximation problems in general are based on whole-pattern similarity relationships of trained and tested data samples. They apply input variables of only absolute interval values, which may cause problems by far various training and testing data ranges. Differential polynomial neural network is a new type of neural network developed by the author, which constructs and substitutes an unknown general sum partial differential equation, defining a system model of dependent variables. It generates a total sum of fractional polynomial terms defining partial relative derivative dependent changes of some combinations of input variables. This type of regression is based only on trained generalized data relations. The character of relative data allows processing a wider range of test interval values than defined by the training set. The characteristics of differential equation solutions also in general facilitate a greater variety of model forms than allow standard soft computing methods.

Direct Wind Power Forecasting Using a Polynomial Decomposition of the General Differential Equation

The wind power is primarily induced by the local wind speed, whose accurate daily forecasts are important for planning and utilization of the unstable power generation and its integration into the electrical grid. The main problem of the wind speed or direct output power forecasting is its intermittent nature due to the high correlation with chaotic large-scale pattern atmospheric circulation processes, which together with local characteristics and anomalies largely influence its temporal flow. Numerical global weather systems solve sets of differential equations to describe the time change of each three-dimensional grid cell in several atmospheric layers. They can provide as a rule only rough short-term surface wind speed prognoses, which are not entirely adequate to specific local conditions, e.g., the wind farm siting, surrounding terrain, and ground level (hub height). Statistical methods using historical observations can particularize the daily forecasts or provide independent predictions in several hours. Extended polynomial networks can produce fraction substitution sum terms in all the nodes, in consideration of data samples, to decompose and substitute for the general linear partial differential equation, being able to describe the local atmospheric dynamics. The designed method using the inverse Laplace transformation aims at the formation of stand-alone spatial derivative models, which can represent current local weather conditions for a trained input–output time shift to predict the daily wind power up to 12 h ahead. The proposed intraday multistep predictions are more precise than those based on middle-term numerical forecasts or adaptive intelligence techniques using local time series, which are worthless beyond a few hours.