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We propose a partial differential-integral equation (PDE) framework for deep neural networks (DNNs) and their associated learning problem by taking the continuum limits of both network width and depth. The proposed model captures the complex interactions among hidden nodes, overcoming limitations of traditional discrete and ordinary differential equation (ODE)-based models. We explore the well-posedness of the forward propagation problem, analyze the existence and properties of minimizers for the learning task, and provide a detailed examination of necessary and sufficient conditions for the existence of critical points. Controllability and optimality conditions for the learning task with its associated PDE forward problem are established using variational calculus, the Pontryagin Maximum Principle, and the Hamilton-Jacobi-Bellman equation, framing the deep learning process as a PDE-constrained optimization problem. In this context, we prove the existence of viscosity solutions for the latter and we establish optimal feedback controls based on the value functional. This approach facilitates the development of new network architectures and numerical methods that improve upon traditional layer-by-layer gradient descent techniques by introducing forward-backward PDE discretization. The paper provides a mathematical foundation for connecting neural networks, PDE theory, variational analysis, and optimal control, partly building on and extending the results of \cite{liu2020selection}, where the main focus was the analysis of the forward evolution. By integrating these fields, we offer a robust framework that enhances deep learning models' stability, efficiency, and interpretability.

In recent years, the study of biological transportation networks has attracted significant interest, focusing on their self-regulating, demand-driven nature. This paper examines a mathematical model for these networks, featuring nonlinear elliptic equations for pressure and an auxiliary variable, and a reaction-diffusion parabolic equation for the conductivity tensor, introduced in \cite{portaro2022emergence}. The model, based on an energy functional with diffusive and metabolic terms, allows for various entropy generating functions, facilitating its application to different biological scenarios. We proved a local well-posedness result for the problem in H\"older spaces employing Schauder and semigroup theory. Then, after a suitable parameter reduction through scaling, we computed the numerical solution for the proposed system using a recently developed ghost nodal finite element method \cite{astuto2024nodal}. An interesting aspect emerges when the solution is very articulated and the branches occupy a wide region of the domain.

We propose a mesoscopic modeling framework for optimal transportation networks with biological applications. The network is described in terms of a joint probability measure on the phase space of tensor-valued conductivity and position in physical space. The energy expenditure of the network is given by a functional consisting of a pumping (kinetic) and metabolic power-law term, constrained by a Poisson equation accounting for local mass conservation. We establish convexity and lower semicontinuity of the functional on approriate sets. We then derive its gradient flow with respect to the 2-Wasserstein topology on the space of probability measures, which leads to a transport equation, coupled to the Poisson equation. To lessen the mathematical complexity of the problem, we derive a reduced Wasserstein gradient flow, taken with respect to the tensor-valued conductivity variable only. We then construct equilibrium measures of the resulting PDE system. Finally, we derive the gradient flow of the constrained energy functional with respect to the Fisher-Rao (or Hellinger-Kakutani) metric, which gives a reaction-type PDE. We calculate its equilibrium states, represented by measures concentrated on a hypersurface in the phase space.