Mathematical Introduction to Deep Learning: Methods, Implementations, and Theory (2310.20360v1)

Abstract: This book aims to provide an introduction to the topic of deep learning algorithms. We review essential components of deep learning algorithms in full mathematical detail including different artificial neural network (ANN) architectures (such as fully-connected feedforward ANNs, convolutional ANNs, recurrent ANNs, residual ANNs, and ANNs with batch normalization) and different optimization algorithms (such as the basic stochastic gradient descent (SGD) method, accelerated methods, and adaptive methods). We also cover several theoretical aspects of deep learning algorithms such as approximation capacities of ANNs (including a calculus for ANNs), optimization theory (including Kurdyka-{\L}ojasiewicz inequalities), and generalization errors. In the last part of the book some deep learning approximation methods for PDEs are reviewed including physics-informed neural networks (PINNs) and deep Galerkin methods. We hope that this book will be useful for students and scientists who do not yet have any background in deep learning at all and would like to gain a solid foundation as well as for practitioners who would like to obtain a firmer mathematical understanding of the objects and methods considered in deep learning.

• The paper introduces a comprehensive mathematical framework for neural network architectures and activation functions with detailed Python implementations.
• It rigorously analyzes approximation, optimization, and generalization, providing insights into network design and error decomposition.
• The study applies deep learning to PDEs, demonstrating methods like PINNs, DGMs, and deep Kolmogorov techniques for solving complex equations.

Okay, here is a detailed summary of the paper "Mathematical Introduction to Deep Learning: Methods, Implementations, and Theory" (Jentzen et al., 2023 ), focusing on its practical implementation aspects for developers and practitioners.

This paper presents a comprehensive mathematical introduction to deep learning, structured like a textbook. It aims to provide a solid foundation for both newcomers and practitioners seeking a deeper mathematical understanding, bridging the gap between theory and real-world implementation. The content is divided into six main parts, supplemented with Python code examples available on GitHub.

Part 1: Artificial Neural Networks (ANNs)

• Basics (Chapter 1): This chapter meticulously defines various fundamental ANN architectures:
  • Fully-connected feedforward ANNs: Introduces both a vectorized description (where all parameters form a single vector, useful for optimization algorithms) and a structured description (using weight matrices and bias vectors, often closer to implementation libraries). Provides the mathematical formulas for forward propagation (\cref{def:FFNN}, \cref{def:ANNrealization}).
  • Activation Functions (Section 2): Critically reviews numerous popular activation functions (ReLU, Leaky ReLU, ELU, GELU, Swish, Softplus, Sigmoid, Tanh, Softsign, Sine, Heaviside, Softmax) with their mathematical definitions and graphical illustrations (\cref{fig:relu_plot} - \cref{fig:heaviside_plot}). Discusses the concept of applying 1D activations element-wise to higher-dimensional tensors (\cref{def:multidim_version}). Code examples for plotting these functions are provided (\cref{code:relu_plot} - \cref{code:heaviside_plot}).
  • Convolutional Neural Networks (CNNs) (Section 4): Defines discrete convolutions (\cref{def:convolution}) and introduces feedforward CNNs mathematically (\cref{def:CNN}, \cref{def:CNNrealisation}). Includes a PyTorch implementation example (\cref{code:convann}).
  • Residual Neural Networks (ResNets) (Section 5): Defines ResNets with skip connections mathematically, including learnable linear maps for the skip connections (\cref{def:ResNet}, \cref{def:ResNetrealisation}). Provides a PyTorch implementation example (\cref{code:resann}).
  • Recurrent Neural Networks (RNNs) (Section 6): Defines RNNs based on the concept of unrolling a recurrent node function (\cref{def:unrolling}, \cref{def:RNNs}). Details simple fully-connected RNN nodes (\cref{def:RNNNode}) and the resulting RNN (\cref{def:VanillaRNN}). Briefly mentions LSTMs (\cref{sec:LSTM}).
  • Other Architectures (Section 7): Briefly reviews Autoencoders, Transformers/Attention (\cref{sec:attention}), Graph Neural Networks (GNNs), and Neural Operators (FNOs, DeepONets), providing references for further exploration.
• ANN Calculus (Chapter 2): Presents operations like composition (\cref{def:ANNcomposition}) and parallelization (\cref{def:simpleParallelization}, \cref{def:generalParallelization}) of ANNs. This provides a mathematical framework for building complex networks from basic blocks and analyzing their properties (like parameter count and realization functions), relevant for modular design and understanding network expressivity. It includes specific ANN constructions for identity mappings (\cref{def:ReLu:identity}), affine transformations (\cref{def:ANN:affine}), scalar multiplication (\cref{def:ANNscalar}), sums (\cref{def:ANN:sum}), and concatenations (\cref{def:ANN:extension}).

Part 2: Approximation

• Focus: Analyzes the theoretical capacity of ANNs to approximate functions.
• Practical Implication: Justifies why ANNs are suitable for tasks requiring function approximation. While theoretical, understanding these limits helps in choosing appropriate network sizes and architectures.
• Structure: Starts with simpler one-dimensional cases (\cref{sect:onedApprox}) using linear interpolation (\cref{def:lin_interp}) and constructing ReLU ANNs for them (\cref{interpol_ANN_points}). It establishes explicit approximation rates relating network size (parameter count) to approximation error (\cref{interpol_ANN_function2}, \cref{interpol_ANN_function_implicit2}). Then extends to multi-dimensional functions (\cref{sect:multidApprox}), using tools like supremal convolutions (\cref{lem:lipschitz_extension}) and covering numbers (\cref{def:covering_number}) to derive approximation results (\cref{prop:approximation_error_structured_eps}, \cref{approx_error_rate}). It provides ANN constructions for the 1-norm (\cref{def:dnn:l1norm}) and maxima (\cref{def:max_d}).

Part 3: Optimization

• Core Topic: Discusses methods for training ANNs, typically by minimizing a loss function (\cref{sec:intro_training_anns}, \cref{sec:intro_SGD}).
• Gradient Flows (Chapter 3): Introduces Gradient Flow ODEs (\cref{def:gradientflow}) as a continuous-time intuition for gradient descent methods. Analyzes convergence using Lyapunov functions (\cref{subsec:Lyapunov_flow}) and coercivity (\cref{subsec:Lyapunov_coercivity}). Reviews common loss functions like MSE (\cref{def:mseloss}) and Cross-Entropy (\cref{def:crossentropyloss}).
• Deterministic GD (Chapter 4): Covers plain Gradient Descent (\cref{def:GD}) and variants with momentum (Classical \cref{def:determ_momentum}, Nesterov \cref{def:determ_nesterov}), adaptive learning rates (Adagrad \cref{def:determ_adagrad}, RMSprop \cref{def:determ_RMSprop}, Adadelta \cref{def:determ_adadelta}), and Adam (\cref{def:determ_adam}). Includes convergence analysis (\cref{GD_convergence}, \cref{GD_convergence_Nesterov}) and discusses optimal learning rates (\cref{GD_convergence_sharp}).
• Stochastic GD (Chapter 5): Presents the widely used SGD method (\cref{def:SGD}) and its variants (Momentum \cref{def:momentum}, Nesterov \cref{def:nesterov}, Adagrad \cref{def:adagrad}, RMSprop \cref{def:rmsprop}, Adadelta \cref{def:adadelta}, Adam \cref{def:adam}). Discusses the bias-variance trade-off (\cref{L2_distance}) and provides concentration inequalities (Hoeffding \cref{cor:Hoeffding2}) relevant to understanding the stochastic nature. Includes code examples comparing optimizers (\cref{code:mnist_optim}, \cref{fig:mnist_optim}).
• Backpropagation (Chapter 6): Derives the backpropagation algorithm (\cref{backprop_for_ANNs}, \cref{backprop_for_ANNs_minibatch}) for efficiently computing gradients of the loss function with respect to ANN parameters. This is fundamental for practical implementation of training.
• KL Inequalities (Chapter 7): Introduces the Kurdyka–Łojasiewicz (KL) inequality (\cref{def:KLinequality_standard}, \cref{def:KLinequality}) as a tool to analyze convergence in non-convex optimization landscapes typical for ANNs. Shows that analytic functions (\cref{thm:KL}) and thus ANNs with analytic activations (\cref{cor:analyticity_empirical_risk}, \cref{thm:convergence_gradient_flow}) satisfy this property.
• Batch Normalization (Chapter 8): Rigorously defines Batch Normalization (\cref{def:batch_norm}) for training and inference (\cref{def:BNANNrealisation}, \cref{def:BNANNgivenrealisation}), explaining its mechanics (batch mean/variance calculation, normalization operation). BN is a widely used practical technique for improving training stability and speed.
• Random Initializations (Chapter 9): Briefly discusses the practical strategy of using multiple random initializations to find better minima.

Part 4: Generalization

• Focus: Analyzes the generalization error – the difference between a model's performance on the training data and unseen data.
• Content: Reviews probabilistic generalization bounds using tools like covering numbers (\cref{sect:covering_numbers}) and concentration inequalities (\cref{sect:concentration_inequ}), leading to estimates relating model complexity (covering number) and sample size to expected error (\cref{lem:cov6}). Also covers strong $L^p$-type generalization estimates (\cref{sec:generalisation_error}).
• Practical Implication: Helps understand factors influencing overfitting and how to potentially control it (though the bounds are often loose in practice).

Part 5: Composed Error Analysis

• Goal: Combines estimates from the previous parts (Approximation, Optimization, Generalization) into an overall error decomposition (\cref{sect:overall_error_decomp}, \cref{prop:error_decomposition2}).
• Application: Illustrates (\cref{sec:composed_error}, \cref{cor:SGD_simplfied}) how these components contribute to the final performance of an ANN trained with SGD and random initializations, providing a theoretical framework for analyzing the sources of error in a learning system.

Part 6: Deep Learning for PDEs

• Application Focus: Demonstrates applying deep learning to solve Partial Differential Equations (PDEs).
• Methods Covered:
  • PINNs & DGMs (Chapter 11): Explains how Physics-Informed Neural Networks (PINNs) and Deep Galerkin Methods (DGMs) reformulate PDE problems as optimization problems by minimizing residuals of the PDE equations and boundary/initial conditions (\cref{thm:dgm}). Includes implementations (\cref{lst.pinn}, \cref{lst.dgm}) for Allen-Cahn type equations (\cref{fig:pinn}, \cref{fig:dgm}).
  • DKMs (Chapter 12): Presents Deep Kolmogorov Methods, which leverage stochastic representations (Feynman-Kac formulas) of PDE solutions to formulate optimization problems (\cref{prop:heat_min}, \cref{cor:kolmogorovtime}). Includes an implementation for the heat equation (\cref{lst.kolmogorov}, \cref{fig:kolmogorov}).
  • Further Methods (Chapter 13): Briefly reviews other approaches based on strong formulations, weak/variational formulations (like Deep Ritz, VPNNs, WANs), and other stochastic representations (like Deep BSDE methods). Also points to theoretical error analysis literature for these methods (\cref{sec:ANN_approx_PDEs}).

In essence, this paper serves as a rigorous mathematical textbook that covers the foundations of deep learning, approximation capabilities, optimization algorithms (crucially including backpropagation and modern techniques like Adam and BN), generalization theory, and applications to PDEs. Its strength lies in the detailed mathematical treatment combined with concrete definitions, algorithms, and illustrative code examples, making it valuable for practitioners who need both a deep understanding and practical implementation guidance.