title: ÐÑÑÐ»ÐµÐ´Ð¾Ð²Ð°Ð½Ð¸Ðµ Ð¸ Ð°Ð½Ð°Ð»Ð¸Ð· Ð´Ð¸Ð½Ð°Ð¼Ð¸ÐºÐ¸ Ð¿ÑÐ¾ÑÐµÑÑÐ¾Ð² Ð¾Ð±ÑÑÐµÐ½Ð¸Ñ Ð¼Ð½Ð¾Ð³Ð¾ÑÐ»Ð¾Ð¹Ð½Ð¾Ð¹ Ð½ÐµÐ¹ÑÐ¾Ð½Ð½Ð¾Ð¹ ÑÐµÑÐ¸
title: Ð²ÑÐ¿ÑÑÐºÐ½Ð°Ñ ÐºÐ²Ð°Ð»Ð¸ÑÐ¸ÐºÐ°ÑÐ¸Ð¾Ð½Ð½Ð°Ñ ÑÐ°Ð±Ð¾ÑÐ° Ð¼Ð°Ð³Ð¸ÑÑÑÐ°
titleType: Subtitle
title: Investigation and Analysis of the Dynamics of Training Processes in Multilayer Neural Networks
Creators: name: Ð¢ÐºÐ°ÑÐµÐ½ÐºÐ¾, ÐÐ¸ÑÐ°Ð¸Ð»
Publisher: name: Ð¡Ð°Ð½ÐºÑ-ÐÐµÑÐµÑÐ±ÑÑÐ³ÑÐºÐ¸Ð¹ Ð¿Ð¾Ð»Ð¸ÑÐµÑÐ½Ð¸ÑÐµÑÐºÐ¸Ð¹ ÑÐ½Ð¸Ð²ÐµÑÑÐ¸ÑÐµÑ ÐÐµÑÑÐ° ÐÐµÐ»Ð¸ÐºÐ¾Ð³Ð¾
Year: 2023
Subjects: Ð¼ÐµÐ¶ÑÐ»Ð¾Ð¹Ð½ÑÐµ Ð¿ÑÐ¾ÑÐµÑÑÑ
ÑÑÐ°ÑÐ¸ÑÑÐ¸ÑÐµÑÐºÐ¸Ðµ Ð¸Ð½Ð²Ð°ÑÐ¸Ð°Ð½ÑÑ
ÑÐ¾Ð¿Ð¾Ð»Ð¾Ð³Ð¸ÑÐµÑÐºÐ¸Ðµ Ð¸Ð½Ð²Ð°ÑÐ¸Ð°Ð½ÑÑ
ÑÐ²ÑÑÑÐ¾ÑÐ½Ð°Ñ Ð½ÐµÐ¹ÑÐ¾Ð½Ð½Ð°Ñ ÑÐµÑÑ
Ð¾Ð±ÑÑÐµÐ½Ð¸Ðµ Ð½ÐµÐ¹ÑÐ¾Ð½Ð½Ð¾Ð¹ ÑÐµÑÐ¸
Ð¿ÐµÑÐµÐ¾Ð±ÑÑÐµÐ½Ð¸Ðµ
Ð¿Ð»Ð°ÑÐ¾Ð² Ð² Ð¾Ð±ÑÑÐµÐ½Ð¸Ð¸
Ð°ÐºÑÐ¸Ð²Ð°ÑÐ¸Ð¸
Ð³ÑÐ°Ð´Ð¸ÐµÐ½ÑÑ
Ð²ÐµÑÐ°
ÑÐ¿Ð¾Ñ Ð°
ÑÐ°Ð³ Ð¾Ð±ÑÑÐµÐ½Ð¸Ñ
TDA
ÑÐ¾ÑÐ½Ð¾ÑÑÑ ÐºÐ»Ð°ÑÑÐ¸ÑÐ¸ÐºÐ°ÑÐ¸Ð¸
ÑÑÐ½ÐºÑÐ¸Ñ Ð¿Ð¾ÑÐµÑÑ
Ð¾Ð±ÑÑÐµÐ½Ð½Ð¾ÑÑÑ ÑÐµÑÐ¸
Ð°ÑÑ Ð¸ÑÐµÐºÑÑÑÐ° ÑÐµÑÐ¸
interlayer processes
statistical invariants
topological invariants
convolutional neural network
neural network training
overfitting
learning curves
activations
gradients
weights
epoch
learning rate
classification accuracy
loss function
network performance
network architecture

description: Ð¦ÐµÐ»Ñ Ð´Ð°Ð½Ð½Ð¾Ð¹ ÑÐ°Ð±Ð¾ÑÑ Ð·Ð°ÐºÐ»ÑÑÐ°ÐµÑÑÑ Ð² Ð¸Ð·ÑÑÐµÐ½Ð¸Ð¸ Ð¸ Ð°Ð½Ð°Ð»Ð¸Ð·Ðµ Ð¿ÑÐ¾ÑÐµÑÑÐ¾Ð², ÐºÐ¾ÑÐ¾ÑÑÐµ Ð¿ÑÐ¾Ð¸ÑÑÐ¾Ð´ÑÑ Ð¼ÐµÐ¶Ð´Ñ ÑÐ»Ð¾ÑÐ¼Ð¸ Ð² Ð¼Ð½Ð¾Ð³Ð¾ÑÐ»Ð¾Ð¹Ð½ÑÑ ÑÐ²ÐµÑÑÐ¾ÑÐ½ÑÑ Ð½ÐµÐ¹ÑÐ¾Ð½Ð½ÑÑ ÑÐµÑÑÑ. Ð ÑÐ¾Ð´Ðµ Ð¸ÑÑÐ»ÐµÐ´Ð¾Ð²Ð°Ð½Ð¸Ñ Ð±ÑÐ»Ð¸ ÑÐ¾Ð±ÑÐ°Ð½Ñ Ð´Ð°Ð½Ð½ÑÐµ Ð¾Ð± Ð°ÐºÑÐ¸Ð²Ð°ÑÐ¸ÑÑ, Ð²ÐµÑÐ°Ñ Ð¸ Ð³ÑÐ°Ð´Ð¸ÐµÐ½ÑÐ°Ñ ÑÐµÑÐ¸. ÐÐ»Ñ ÑÑÐ°ÑÐ¸ÑÑÐ¸ÑÐµÑÐºÐ¾Ð³Ð¾ Ð°Ð½Ð°Ð»Ð¸Ð·Ð° ÑÑÐ¸ Ð´Ð°Ð½Ð½ÑÐµ Ð±ÑÐ»Ð¸ Ð²Ð¸Ð·ÑÐ°Ð»Ð¸Ð·Ð¸ÑÐ¾Ð²Ð°Ð½Ñ Ð² Ð²Ð¸Ð´Ðµ Ð³ÑÐ°ÑÐ¸ÐºÐ¾Ð² Ð¸ Ð´Ð¸Ð°Ð³ÑÐ°Ð¼Ð¼ ÑÐ°ÑÑÐµÑÐ½Ð¸Ñ, Ð° Ð´Ð»Ñ ÑÐ¾Ð¿Ð¾Ð»Ð¾Ð³Ð¸ÑÐµÑÐºÐ¾Ð³Ð¾ Ð°Ð½Ð°Ð»Ð¸Ð·Ð° Ð±ÑÐ»Ð¸ Ð¸ÑÐ¿Ð¾Ð»ÑÐ·Ð¾Ð²Ð°Ð½Ñ Ð¿ÐµÑÑÐ¸ÑÑÐµÐ½ÑÐ½ÑÐµ Ð´Ð¸Ð°Ð³ÑÐ°Ð¼Ð¼Ñ. 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ÐÐ¾ Ð¼ÐµÑÐµ Ð¾Ð±ÑÑÐµÐ½Ð¸Ñ ÑÑÐ¸ ÐºÐ¾Ð»ÐµÐ±Ð°Ð½Ð¸Ñ Ð´Ð¾Ð»Ð¶Ð½Ñ ÑÐ¼ÐµÐ½ÑÑÐ°ÑÑÑÑ, ÑÑÐ¾ ÑÐ²Ð¸Ð´ÐµÑÐµÐ»ÑÑÑÐ²ÑÐµÑ Ð¾Ð± ÑÑÐ¿ÐµÑÐ½Ð¾Ð¼ Ð¾Ð±ÑÑÐµÐ½Ð¸Ð¸ ÑÐµÑÐ¸. Ð ÑÐ°Ð±Ð¾ÑÐµ Ð¸ÑÐ¿Ð¾Ð»ÑÐ·Ð¾Ð²Ð°Ð»Ð¸ÑÑ ÑÐ»ÐµÐ´ÑÑÑÐ¸Ðµ Ð´Ð°ÑÐ°ÑÐµÑÑ Ð´Ð»Ñ Ð¾Ð±ÑÑÐµÐ½Ð¸Ñ ÑÐ²ÐµÑÑÐ¾ÑÐ½ÑÑ Ð½ÐµÐ¹ÑÐ¾Ð½Ð½ÑÑ ÑÐµÑÐµÐ¹: ï­ MNIST: Ð½Ð°Ð±Ð¾Ñ Ð´Ð°Ð½Ð½ÑÑ Ð´Ð»Ñ ÑÐ°ÑÐ¿Ð¾Ð·Ð½Ð°Ð²Ð°Ð½Ð¸Ñ ÑÑÐºÐ¾Ð¿Ð¸ÑÐ½ÑÑ ÑÐ¸ÑÑ, ÑÐ¾ÑÑÐ¾ÑÑÐ¸Ð¹ Ð¸Ð· 60 000 Ð¾Ð±ÑÑÐ°ÑÑÐ¸Ñ Ð¸ 10 000 ÑÐµÑÑÐ¾Ð²ÑÑ Ð¸Ð·Ð¾Ð±ÑÐ°Ð¶ÐµÐ½Ð¸Ð¹ ÑÐ°Ð·Ð¼ÐµÑÐ¾Ð¼ 28x28 Ð¿Ð¸ÐºÑÐµÐ»ÐµÐ¹. ï­ Cats vs Dogs (Kaggle): Ð½Ð°Ð±Ð¾Ñ Ð´Ð°Ð½Ð½ÑÑ Ñ Ð¸Ð·Ð¾Ð±ÑÐ°Ð¶ÐµÐ½Ð¸ÑÐ¼Ð¸ ÐºÐ¾ÑÐµÐº Ð¸ ÑÐ¾Ð±Ð°Ðº Ð´Ð»Ñ Ð·Ð°Ð´Ð°ÑÐ¸ ÐºÐ»Ð°ÑÑÐ¸ÑÐ¸ÐºÐ°ÑÐ¸Ð¸. ÐÑÐ½Ð¾Ð²Ð½ÑÐµ Ð·Ð°Ð´Ð°ÑÐ¸, ÑÐµÑÐ°ÐµÐ¼ÑÐµ Ð² ÑÐ°Ð±Ð¾ÑÐµ, Ð²ÐºÐ»ÑÑÐ°Ð»Ð¸: 1. 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Ð Ð°Ð·ÑÐ°Ð±Ð¾ÑÐºÐ° Ð¿ÑÐ¾Ð³ÑÐ°Ð¼Ð¼Ð½Ð¾Ð³Ð¾ Ð¿ÑÐ¾Ð´ÑÐºÑÐ° Ð½Ð° ÑÐ·ÑÐºÐµ Python, ÐºÐ¾ÑÐ¾ÑÑÐ¹ Ð²ÑÐ¿Ð¾Ð»Ð½ÑÐ» ÑÑÐ½ÐºÑÐ¸Ð¸ Ð¿Ð¾Ð´Ð³Ð¾ÑÐ¾Ð²ÐºÐ¸ Ð´Ð°Ð½Ð½ÑÑ Ð´Ð»Ñ Ð¾Ð±ÑÑÐµÐ½Ð¸Ñ Ð½ÐµÐ¹ÑÐ¾Ð½Ð½ÑÑ ÑÐµÑÐµÐ¹, Ð¸Ð·Ð²Ð»ÐµÑÐµÐ½Ð¸Ñ ÑÑÐ°ÑÐ¸ÑÑÐ¸ÑÐµÑÐºÐ¸Ñ Ð´Ð°Ð½Ð½ÑÑ Ð² Ð¿ÑÐ¾ÑÐµÑÑÐµ Ð¾Ð±ÑÑÐµÐ½Ð¸Ñ Ð´Ð»Ñ Ð°Ð½Ð°Ð»Ð¸Ð·Ð° Ð²Ð·Ð°Ð¸Ð¼Ð¾Ð´ÐµÐ¹ÑÑÐ²Ð¸Ñ Ð¼ÐµÐ¶Ð´Ñ ÑÐ»Ð¾ÑÐ¼Ð¸ ÑÐµÑÐ¸, Ð° ÑÐ°ÐºÐ¶Ðµ Ð¿Ð¾ÑÑÑÐ¾ÐµÐ½Ð¸Ñ Ð³ÑÐ°ÑÐ¸ÐºÐ¾Ð², Ð´Ð¸Ð°Ð³ÑÐ°Ð¼Ð¼ ÑÐ°ÑÑÐµÑÐ½Ð¸Ñ Ð¸ Ð´Ð¸Ð°Ð³ÑÐ°Ð¼Ð¼ Ð¿ÐµÑÑÐ¸ÑÑÐµÐ½ÑÐ½Ð¾ÑÑÐ¸ Ð´Ð»Ñ Ð²Ð¸Ð·ÑÐ°Ð»Ð¸Ð·Ð°ÑÐ¸Ð¸ Ð¿Ð¾Ð»ÑÑÐµÐ½Ð½ÑÑ Ð´Ð°Ð½Ð½ÑÑ. 3. ÐÐ±ÑÑÐµÐ½Ð¸Ðµ ÑÐ²ÐµÑÑÐ¾ÑÐ½ÑÑ Ð½ÐµÐ¹ÑÐ¾Ð½Ð½ÑÑ ÑÐµÑÐµÐ¹ Ð½Ð° Ð¿Ð¾Ð´Ð³Ð¾ÑÐ¾Ð²Ð»ÐµÐ½Ð½ÑÑ Ð½Ð°Ð±Ð¾ÑÐ°Ñ Ð´Ð°Ð½Ð½ÑÑ. 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description: The objective of this research is to study and analyze the processes that occur between layers in multilayer convolutional neural networks. During the investigation, data on activations, weights, and gradients of the network were collected. For statistical analysis, these data were visualized using graphs and scatter plots, while persistent diagrams were used for topological analysis. One of the main objectives of this research was to apply statistical and topological invariants to assess the effectiveness of training and the problem of overfitting in multilayer neural networks. Additionally, the study involved examining other parameters based on the dynamics of interlayer processes. The hypothesis of this work posited that at the beginning of training, the neural network exhibits significant fluctuations in interlayer processes due to substantial changes in weights and activations of the layers. As training progresses, these fluctuations should diminish, indicating successful network learning. The following datasets were used for training the convolutional neural networks: ï­ MNIST: A dataset for handwritten digit recognition, consisting of 60,000 training and 10,000 test images of size 28x28 pixels. ï­ Cats vs Dogs (Kaggle): A dataset with images of cats and dogs for classification tasks. The main tasks addressed in this work included: 1. Investigation of the architecture of convolutional neural networks and the training process for data classification. This task involved analyzing various network architectures and training methods, with a focus on classification tasks. 2. Development of a Python software product that performs data preprocessing for neural network training, extraction of statistical data during training to analyze interlayer interactions, and construction of graphs, scatter plots, and persistent diagrams for data visualization. 3. Training of convolutional neural networks on prepared datasets. This task involved training neural networks using different architectures and training methods on the datasets considered in the work, aiming for image classification. 4. Analysis of the obtained statistical data using topological analysis methods. This task encompassed analyzing the collected statistical data using topological analysis methods to identify statistical and topological invariants, anomalies, and other characteristics. 5. Comparison of the results of interlayer process analysis with commonly accepted indicators of network learning for data classification. This task involved comparing the obtained results of interlayer process analysis with known indicators characterizing the networks learning state for data classification tasks. As a result of this work, features, statistical invariants, and topological invariants were identified that confirm the hypotheses of the research. Additionally, additional features were discovered that can be utilized for further analysis of the training process in convolutional neural networks.