Signal and Gradient Flows in Deep Neural Networks

Sakino Hisao^*
*Correspondence: Sakino Hisao, Department of Mechanical Engineering, University of Minnesota 111 Church Street SE Minneapolis, MN 55455-0111, USA, Email:

Introduction

The remarkable rise of deep learning has revolutionized the landscape of artificial intelligence, enabling systems to perform complex tasks such as image recognition, natural language understanding, and game playing at superhuman levels. At the heart of deep learning lies the Deep Neural Network (DNN), a layered computational structure inspired by the organization of neurons in the human brain. Despite their wide application and performance success, the inner workings of deep neural networks especially the flow of signals and gradients are often misunderstood or taken for granted by practitioners who rely heavily on pre-built frameworks like TensorFlow or PyTorch. However, to truly master deep learning, one must go beyond abstracted high-level APIs and delve into the mechanics of how information propagates through the network during both the forward pass (signal flow) and backward pass (gradient flow). Understanding these flows is not just an academic exercise itâ??s crucial for designing more efficient architectures, debugging training failures, implementing custom layers or loss functions, and optimizing performance. The forward pass governs how input data transforms into predictions, activating neurons via weighted sums and non-linearities, while the backward pass handles the propagation of gradients used to update parameters during training. These processes are tightly linked by the principle of differentiable computation, enabled by the chain rule of calculus, which allows gradients to be computed efficiently across potentially millions of parameters using back propagation. This tutorial aims to demystify the fundamental algorithms that power these flows, offering a clear, structured, and implementation-focused narrative on how to build and train deep neural networks from scratch, without relying on any deep learning libraries. By understanding the granular details of signal and gradient flows, readers can gain deep insights into the behavior of neural networks and develop the confidence to innovate and optimize beyond the constraints of existing frameworks [1].

Description

To begin constructing a neural network from scratch, one must first understand the architectureâ??s basic building block: the artificial neuron. Each neuron receives inputs, applies a weighted sum, adds a bias, and passes the result through a non-linear activation function like ReLU, Sigmoid, or Tanh. The forward pass starts at the input layer, where raw features enter the network and are propagated layer by layer. At each hidden layer, a matrix-vector multiplication is performed between the layer weights and the incoming signals, followed by the addition of bias and the application of a non-linearity. This transforms the input representation into a higher-dimensional space where patterns become more separable. The output layer then interprets the final transformation into predictions or probabilities, depending on the task such as classification (using softmax) or regression (using linear activation). This flow of data, or signal, determines the prediction output and forms the basis of how the model "understands" input data. At this point, a loss function is used to quantify the difference between the network output and the ground truth labels. Common loss functions include mean squared error for regression, and cross-entropy loss for classification. The second phase of training is the backward pass, which involves computing gradients of the loss with respect to every trainable parameter in the network a process known as back propagation [2].

This is made possible by applying the chain rule through the computational graph of the network. For each layer, we calculate how the error changes with respect to its weights and biases by propagating gradients from the output layer back toward the input layer. This involves storing intermediate values (from the forward pass) like inputs and activation outputs, which are required to compute partial derivatives during the backward pass. Each layer computes its local gradient and multiplies it with the gradient from the layer above to get the total derivative. These gradients are then used by an optimization algorithm such as Stochastic Gradient Descent (SGD), Adam, or RMSProp to update the weights in the direction that minimizes the loss. This iterative process continues for many epochs, gradually refining the networkâ??s parameters to improve its performance. Building such a network from scratch requires defining data structures for weights and biases (typically as matrices and vectors), and implementing forward and backward functions for each layer. Layers must be able to store their parameters, accept inputs, compute outputs, and return gradients. Additionally, each activation function must have a corresponding derivative function, since the back propagation process depends heavily on these gradients. For instance, the derivative of ReLU is simple and efficient for positive inputs and 0 for negative ones while Sigmoidâ??s derivative involves its own output value [3].

A robust implementation also includes gradient clipping to prevent exploding gradients, normalization techniques like batch normalization to stabilize training, and possibly dropout to reduce overfitting. Alongside these, a training loop orchestrates the entire process: shuffling data, batching inputs, computing forward passes, evaluating loss, executing backward passes, and applying updates. When developing a neural network from the ground up, modularity and clarity are essential. By treating each layer or operation as a self-contained component with its own forward and backward functionality, one can construct complex architectures while maintaining clean and debug gable code. For example, convolutional layers for image tasks, recurrent layers for sequence modeling, and fully connected layers for dense representations all follow the same fundamental principles of signal and gradient flow, differing only in how inputs and weights are shaped and multiplied. A deeper understanding of these flows enables developers to create novel architectures such as ResNets with skip connections, Transformers with self-attention, or GANs with adversarial objectives by leveraging the same core machinery. Additionally, debugging becomes more intuitive; if gradients vanish or explode, or if the network fails to converge, one can trace the flow of signals and gradients to identify the root cause. Understanding gradient flow also opens the door to more advanced optimization techniques, such as learning rate scheduling, momentum, or second-order methods like L-BFGS [4].

Another critical concept that emerges from understanding signal and gradient flow is the interpretability and explain ability of neural networks. While DNNs are often seen as black boxes, having insight into how data transforms at each layer helps in visualizing learned features, diagnosing over fitting or under fitting, and designing models that are both powerful and transparent. By tracking the evolution of activations and gradients, one can develop tools like saliency maps, gradient-based attention, and layer-wise relevance propagation to explain predictions to end-users a crucial feature in high-stakes domains like healthcare, finance, or autonomous driving. Furthermore, understanding the flow of signals and gradients provides the foundational intuition behind transfer learning and fine-tuning pre-trained networks, where new layers are stacked on existing models and trained on limited data. In more advanced use cases, signal and gradient flow become essential for implementing emerging paradigms such as meta-learning, reinforcement learning, and differentiable programming. In meta-learning, models must learn how to learn, which involves higher-order gradients and complex gradient accumulation strategies. In reinforcement learning, policies are updated based on reward signals that may be sparse or delayed, requiring careful credit assignment through signal and gradient propagation over time. Differentiable programming, which blurs the line between neural networks and general algorithms, relies on the same back propagation techniques to optimize arbitrary computational graphs. All these methods extend from the same core principle: the efficient, differentiable, and structured flow of information through a computational system [5].

Conclusion

In conclusion, mastering the flow of signals and gradients is fundamental to both understanding and implementing deep neural networks from scratch. The forward pass propagates input data through a sequence of weighted transformations and non-linear activations, culminating in a predictive output. The backward pass, governed by back propagation and the chain rule, computes gradients that guide the learning process by updating network parameters to minimize prediction error. Together, these processes enable neural networks to learn from data, adjust internal representations, and generalize to unseen inputs. By exploring these flows in detail through the lens of math, code, and conceptual clarity developers gain the power to build, customize, and innovate beyond pre-built tools. Such understanding is invaluable for research, real-world applications, and advancing the field itself. Whether one aims to develop efficient networks for edge devices, optimize large-scale training on cloud platforms, or push the frontier with new architectures, a solid grasp of how signals and gradients behave within neural networks provides the critical foundation for success. The journey from raw data to intelligent behavior, from initial weights to learned representations, all hinges on the disciplined yet elegant flow of signals and gradients a dance of mathematics and computation that brings intelligence to code. As we continue to scale models and expand applications, this foundational understanding will remain central to the next generation of breakthroughs in artificial intelligence.

Acknowledgement

None.

Conflict of Interest

None.

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