The article presents a detailed, step‑by‑step derivation of the backpropagation algorithm for a fully connected feedforward neural network with sigmoid activations and squared‑error loss. It starts with a single‑weight example, extends to general error terms and matrix equations, and then scales the derivation to multiple training examples and mini‑batch gradient descent. Each equation is annotated with dimensions and linked to a NumPy implementation trained on handwritten Arabic digits, with full code provided.
arXiv:2607. 11289v1 Announce Type: cross Abstract: Backpropagation is the computational engine of deep learning, yet its mathematical structure is typically treated as a procedural traversal of computational graphs.
Backpropagation is the computational engine of deep learning, yet its mathematical structure is typically treated as a procedural traversal of computational graphs. We present a global operator theory of the \emph{F-adjoint} framework, which reformulates the layerwise backward recursion of an $L$-depth feedforward network into a single linear system $(I-\cB)\Xs=\bG$, where $\bG$ is a source vector.
arXiv:2501. 07400v2 Announce Type: replace-cross Abstract: We derive explicit equations governing the cumulative biases and weights in Deep Learning with ReLU activation function, based on gradient descent for the Euclidean loss in the input layer, and under the assumption that the weights are, in a precise sense, adapted to the coordinate system distinguished by the activations.
In this paper, we propose an efficient hybrid least squares/gradient descent (LSGD) method for MIONets to accelerate training. This method generalizes the LSGD method for DeepONets.
arXiv:2607. 07035v1 Announce Type: cross Abstract: The architecture of deep feedforward neural networks is ubiquitous in deep learning, either as a whole system or as a subnetwork of other architectures, and thus its mechanism is a key ingredient of the black box of neural networks.
arXiv:2507. 14177v2 Announce Type: replace-cross Abstract: This paper aims to understand the training solution, which is obtained by the back-propagation algorithm, of two-layer neural networks whose hidden layer is composed of the units with smooth activation functions, including the usual sigmoid type most commonly used before the advent of ReLUs.
The paper introduces a least‑squares method for training quadratic neural networks with regularization, providing a lower bound on the training optimization problem when the regularization coefficient is positive. It delivers closed‑form expressions for both the approximate solution and its sensitivity to data errors, and shows that the solution is optimal when the regularization coefficient is zero. The approach offers computational advantages over iterative methods like backpropagation and is validated on a nonlinear system identification example.
arXiv:2607. 06976v1 Announce Type: cross Abstract: In this paper, we propose an efficient hybrid least squares/gradient descent (LSGD) method for MIONets to accelerate training.
The paper investigates weight perturbation (WP) as a derivative-free alternative to backpropagation for training fully connected MNIST networks. In shallow architectures, large-batch WP attains about 97% test accuracy, close to backpropagation, but under a ten‑epoch budget WP lags behind (90.2% vs. 97.4%). The study also shows diminishing returns beyond a batch size of 4,000 and that learning‑rate reduction and averaging of perturbation updates can stabilize deeper networks, though they do not yet match backpropagation accuracy.
arXiv:2607. 13042v1 Announce Type: new Abstract: This paper traces, with explicit numerical values, how PyTorch's automatic differentiation (AD) engine computes gradients for Physics-Informed Neural Network (PINN) training -- a setting that requires two levels of differentiation: computing the physics derivative $\hat{y}'(t)=d\hat{y}/dt$ through the network, and computing parameter gradients $\nabla_\theta L$ of a loss that itself depends on $\hat{y}'(t)$.
arXiv:2501.18018v2 Announce Type: replace-cross Abstract: The neurons of artificial neural networks were originally invented when much less was known about biological neurons than is known today. Our...
Let's discover how neural networks learn, step by step The post Backpropagation Explained for Beginners (Part 1): Building the Intuition appeared first on Towards Data Science .