DaPC NN Matlab Toolbox: Deep Arbitrary Polynomial Chaos Neural Network
AUTHOR:
Sergey Oladyshkin
AFFILIATION:
Stuttgart Research Centre for Simulation Technology,
Department of Stochastic Simulation and Safety Research for Hydrosystems,
Institute for Modelling Hydraulic and Environmental Systems,
University of Stuttgart, Pfaffenwaldring 5a, 70569 Stuttgart
CONTACT INFORMATION:
E-mail: Sergey.Oladyshkin@iws.uni-stuttgart.de
Phone: +49-711-685-60116
Fax: +49-711-685-51073
Website: http://www.iws.uni-stuttgart.de
SCIENTIFIC LITERATURE:
Oladyshkin S., Praditia T., Kroeker I., Mohammadi F., Nowak W., Otte S., The Deep Arbitrary Polynomial Chaos Neural Network or how Deep Artificial Neural Networks could benefit from Data-Driven Homogeneous Chaos Theory. Neural Networks. Elsevier, 2023. DOI: 10.1016/j.neunet.2023.06.036.
Oladyshkin S. and Nowak W. Data-driven uncertainty quantification using the arbitrary polynomial chaos expansion. Reliability Engineering & System Safety, Elsevier, V. 106, P. 179190, 2012.
Oladyshkin S. and Nowak W. Incomplete statistical information limits the utility of high-order polynomial chaos expansions. Reliability Engineering & System Safety, 169, 137-148, 2018.
GENERAL INFORMATION:
Artificial Intelligence and Machine learning have been widely used in various fields of mathematical computing, physical modeling, computational science, communication science, and stochastic analysis. Approaches based on Deep Artificial Neural Networks are very popular in our days. Depending on the learning task, the exact form of Deep Artificial Neural Networks is determined via their multi-layer architecture, activation functions and the so-called loss function. However, for a majority of deep learning approaches based on Deep Artificial Neural Networks, the kernel structure of neural signal processing remains the same, where the node response is encoded as a linear superposition of neurons, while the non-linearity is triggered by the activation functions. In the current Matlab Toolbox analyses the neural signal processing in Deep Artificial Neural Networks from the point of view of homogeneous chaos theory as known from polynomial chaos expansion introduced by Norbert Wiener in 1938. It employs the data-driven generalization of polynomial chaos expansion theory known as arbitrary polynomial chaos (aPC: Oladyshkin S. and Nowak W., 2012 and 2018) to construct a corresponding multi-layer representation of a Deep Artificial Neural Network. Doing so, we generalize the conventional structure of DANNs to Deep arbitrary polynomial chaos neural networks (DaPC NN: Oladyshkin et al. 2023).
SPECIFIC INFORMATION:
The MATLAB DaPC NN Toolbox represents a workflow using a Deep Arbitrary Polynomial Chaos Neural Network. The key features of the DaPC NN in Matlab toolbox includes:
Network Architecture: users can specify the number of input parameters and define the structure of hidden layers and number of the corresponding neurons.
Orthonormal Representation: the toolbox supports the representation of neural networks using an orthonormal basis.
Polynomial Degrees: the DaPC NN allows users to specify the degree of nonlinearity for each layer allowing to employ the multivariate polynomial degrees for each layer.
Training: the toolbox provides a training function that utilizes a non-linear solver, such as the Levenberg-Marquardt algorithm, to optimize the network weights. Important remark: the standard 'levenberg-marquardt' algorithm provided by MATLAB does not incorporate any form of regularization.
Prediction: Once trained, the toolbox enables users to make predictions using the trained network. The prediction function takes the data-driven input distribution and prediction input as inputs and returns the prediction output.
The script "MainRun_Deep_aPC_NeuralNetwork.mat" provides an example of how to use the DaPC NN Toolbox. It can be summarized as follows:
1. Loading Input Distributions, Training Data, and Validation Data:
Assumption of the Input distributions and Training/Validation Datasets are loaded from a file.
If the variable 'Input_distributions' exists, the data-driven distribution of input parameters based on available assumptions is assigned to 'DataDrivenInputDistribution'.
If 'Input_distributions' does not exist, the data-driven distribution based on the training input is assigned to 'DataDrivenInputDistribution'.
2. Initialization of Deep aPC Neural Network:
An instance of the DeepArbitraryPolynomialChaos class named 'DaPC' is created.
The property 'OrthonormalRepresentation' is set to 'Yes', indicating that the representation uses an orthonormal basis.
The number of input parameters is identified according to the 'TrainingInput' dataset.
3. Network Architecture and Settings:
The network architecture is defined by specifying the hidden layers and their degrees of nonlinearity.
The network has to be specified via hidden layers and the corresponding neurons.
The degrees of nonlinearity for each layers has to be specified as well.
The training algorithm must be specified. In this script, an example is provided on how to use the standard 'levenberg-marquardt' algorithm, which is readily available in MATLAB.
It is important to note that the standard 'levenberg-marquardt' algorithm provided by MATLAB does not incorporate any form of regularization. It is strongly recommended to use regularization during the training process.
The option 'SpecifyObjectiveGradient' is enabled, indicating the use of analytical Jacobian for the objective function.
The 'Jacobian' property is set to 'on', indicating that the Jacobian information will be used.
Other settings include the maximum number of epochs, parallel computing option, display of epoch iterations, initial value of the Levenberg-Marquardt damping factor "lambda", and problem scaling.
4. Structure Initialization of Deep aPC Neural Network:
The structure of the Deep aPC Neural Network is initialized using the 'StuctureInitialization' function, passing the 'DaPC' object.
5. Training of Deep aPC Neural Network:
The Deep aPC Neural Network is trained using the 'train' method of the 'DaPC' object.
The training is performed with the 'DataDrivenInputDistribution', 'TrainingInput', and 'TrainingOutput' datasets.
6. Prediction of Deep aPC Neural Network:
The trained Deep aPC Neural Network is used to predict the output values for the 'DataDrivenInputDistribution' and 'TrainingInput' datasets.
The predicted output values are stored in the 'PredictionOutput' variable.
7. Validation of Deep aPC Neural Network:
The trained Deep aPC Neural Network is used to predict the output values for the 'DataDrivenInputDistribution' and 'ValidationInput' datasets.
The predicted output values are stored in the 'ValidationPredictionOutput' variable.
8. The code shows three metrics to assess the performance of the model on the validation data
Mean Squared Error (MSE): It measures the average squared difference between the predicted outputs and the actual outputs.
Relative error of Mean Value (Mean): It compares the difference between the mean of the predicted outputs and the mean of the actual outputs relative to the mean of the actual outputs.
Relative error of Standard Deviation (Std): It compares the difference between the standard deviation of the predicted outputs and the standard deviation of the actual outputs relative to the standard deviation of the actual outputs.
The code creates a figure with two subplots to visualize the performance of the model: the predicted responses against the training responses (subplot 1) and the predicted responses against the validation references (subplot 2).
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深度任意多项式混沌神经网络DaPC NN Matlab 工具箱.zip
共11个文件
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txt:2个
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深度任意多项式混沌神经网络DaPC NN Matlab 工具箱.zip (11个子文件)
深度任意多项式混沌神经网络DaPC NN Matlab 工具箱
说明.txt 2KB
DaPC NN Matlab Toolbox
DeepArbitraryPolynomialChaos.m 7KB
Deep_aPC_NeuralNetwork.m 10KB
aPC_PsiPolynomialMatrix.mexw64 118KB
1.png 25KB
DaPC-NN.mat 114KB
aPC_MultivariatePolynomialDegrees.m 3KB
Readme.txt 8KB
MainRun_Deep_aPC_NeuralNetwork.m 5KB
aPC_OrthonormalBasis.mexw64 91KB
Data.mat 74KB
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