Constructing general partial differential equations using polynomial and neural networks

Zjavka, Ladislav; Pedrycz, Witold

doi:10.1016/j.neunet.2015.10.001

Cited by 27 publications

(22 citation statements)

References 31 publications

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“…The general linear PDE (1) can describe an unknown separable u function of n-inputs, possible to be expressed in convergent sum series (2) of partial u k function solutions of two-variable sub-PDEs (3) [24] a…”

Section: Novel Polynomial Pde Substitution Based On Ocmentioning

confidence: 99%

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PV power intra‐day predictions using PDE models of polynomial networks based on operational calculus

Zjavka¹

2020

IET Renewable Power Generation

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Precise daily forecasts of photo-voltaic (PV) power production are necessary for its planning, utilisation and integration into the electrical grid. PV power is conditioned by the current amount of specific solar radiation components. Numerical weather prediction systems are usually run every 6 h and provide only rough local prognoses of cloudiness with a delay. Statistical methods can predict PV power, considering a specific plant situation. Their intra-day models are usually more precise if rely only on the latest data observations and power measurements. Differential polynomial neural network (D-PNN) is a novel neuro-computing technique based on analogies with brain pulse signal processing. D-PNN decomposes the general partial differential equation (PDE), being able to describe the local atmospheric dynamics, into specific sub-PDEs in its nodes. These are converted using adapted procedures of operational calculus to obtain the Laplace images of unknown node functions, which are inverse L-transformed to obtain the originals. D-PNN can select from dozens of input variables to produce applicable sum PDE components which can extend, one by one, its composite models towards the optima. The PDE models are developed with historical spatial data from the estimated optimal numbers of the last days for each 1-9-h inputs-output timeshift to predict clear sky index in the trained time-horizon.

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Section: Novel Polynomial Pde Substitution Based On Ocmentioning

confidence: 99%

“…1) is calculated using the GMDH polynomials (5). The designed sigmoidal transformation of squared variables (10) improves the polynomial ability to approximate periodic functions [24]…”

Section: Novel Polynomial Pde Substitution Based On Ocmentioning

confidence: 99%

PV power intra‐day predictions using PDE models of polynomial networks based on operational calculus

Zjavka¹

2020

IET Renewable Power Generation

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“…Each block contains a single output GMDH polynomial Equation 2, without a derivative part. The neurons do not affect the block output but each of them can be directly included in the total network output sum of the PDE model calculation (Zjavka and Pedrycz, 2016).…”

Section: Appendix Bmentioning

confidence: 99%

“…Sub-solutions of the second-order PDE Equation B4 model unknown two-variable partial functions u k . This type of PDE is used by preference for modelling physical or natural systems nonlinearities (Zjavka and Pedrycz, 2016). y 1 = w 1 (a 0 + a 1 x 1 + a 2 x 2 + a 3 x 1 x 2 + a 4 x 2 1 + a 5 x 2 2 )…”

Section: Appendix Bmentioning

confidence: 99%

“…The blocks of the preceding layers form internal functions B7 which substitute for the next hidden layer input variables of neuron and block polynomials Equation B8. Composite terms (CT), which substitute for the derivatives with respect to input variables of the previous layer blocks, are formed according to the composite function partial derivation rules Equations B9 and B10 (Zjavka and Pedrycz, 2016).…”

Section: Appendix Bmentioning

confidence: 99%

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Local improvements in numerical forecasts of relative humidity using polynomial solutions of general differential equations

Zjavka

Sokol

2018

Quart J Royal Meteoro Soc

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Large‐scale forecast models are based on the numerical integration of differential equation systems, which can describe atmospheric processes in light of global meteorological observations. Mesoscale forecast systems need to define the initial and lateral boundary conditions, which may be carried out with robust global numerical models. Their overall solutions are able to describe the dynamic weather system on the Earth scale using a large number of complete globe 3D matrix variables in several atmospheric layers. Post‐processing methods using local measurements were developed in order to clarify surface weather details and adapt numerical weather prediction model outputs for local conditions. Differential polynomial network is a new type of neural network that can model local weather using spatial data observations to process forecasts of the input variables and revise the target 24 h prognosis. It defines and solves general partial differential equations, being able to describe unknown dynamic systems. The proposed forecast correction method uses a differential network to estimate the optimal numbers of training days and form derivative prediction models. It can improve final numerical forecasts, processed with additional data analysis and statistical techniques, in the majority of cases. The two‐stage procedure presented is analogous to the perfect‐prognosis method using real observations to derive a model, which is applied to the forecasts of the predictors to calculate output predictions.

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Photovoltaic power intra‐ and day‐ahead predictions with differential learning producing PDE‐modular models based on the node L‐transform derivatives

Zjavka

2022

Env Prog and Sustain Energy

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Predictions of photovoltaic (PV) energy supply, based on data statistics of weather and PV records, are required in short-intra and day-ahead planning of PV plant operations. Numerical weather prediction (NWP), based on physical consideration, can simulate the progress of local cloudiness, although their prognoses are usually delayed by a few hours and are not provided with the quality desired by PV operators. Differential polynomial neural network (D-PNN) is an unconventional hybrid regression technique, based on a novel learning strategy and able to resolve some problems in modeling patterns in high-dynamical and chaotic non-deterministic systems. D-PNN splits the general partial differential equation (PDE) of the k th order into a summation form of two-variable PDEs of a predetermined low order. PDE derivatives are transformed and separated into the Laplace counterparts, from which the unknown node originals are restored using the inverse L-operation. D-PNN searches for the best 2-input couples, inserting node by node into its changeable tree structure, to expand the modular sum model by usable PDE components. The PDE-modularization enhances the D-PNN capability in modeling high uncertainty in weather patterns in the ground-layer atmosphere, analogous to NWP. The designed PV intra-and dayahead prediction schemes, using hourly increasing and fixed-time prediction horizons, were compared with different computing approaches based on differential, probabilistic and statistical learning. Machine learning (ML) models calculate the output PV power in each single hour separately or in an all-day processing series with trained input delay. The results are analyzed and interpreted for each day and predictionhour. The quantitative results for both prediction procedures are comparable, showing the average intraday 11.6% and all-day 9.6% (or rather 12.3% respectively) inaccuracy with respect to the daily peak power. Training data samples with spatial overlap in the regional scope and ML initialized similarity intervals are related to the fixed-day or hourly increased input-output delay. This approach can compensate for unpredictable midterm variances in local weather patterns (caused by terrain or convection anomalies) and possible errors in model initialization. PV power was converted to

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Constructing general partial differential equations using polynomial and neural networks

Cited by 27 publications

References 31 publications

PV power intra‐day predictions using PDE models of polynomial networks based on operational calculus

PV power intra‐day predictions using PDE models of polynomial networks based on operational calculus

Local improvements in numerical forecasts of relative humidity using polynomial solutions of general differential equations

Photovoltaic power intra‐ and day‐ahead predictions with differential learning producing PDE‐modular models based on the node L‐transform derivatives

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