  Partial differential equations have a wide range of applications in modeling
multiple physical, biological, or social phenomena. Therefore, we need to
approximate the solutions of these equations in computationally feasible terms.
Nowadays, among the most popular numerical methods for solving partial
differential equations in engineering, we encounter the finite difference and
finite element methods. An alternative numerical method that has recently
gained popularity for numerically solving partial differential equations is the
use of artificial neural networks.
  Artificial neural networks, or neural networks for short, are mathematical
structures with universal approximation properties. In addition, thanks to the
extraordinary computational development of the last decade, neural networks
have become accessible and powerful numerical methods for engineers and
researchers. For example, imaging and language processing are applications of
neural networks today that show sublime performance inconceivable years ago.
  This dissertation contributes to the numerical solution of partial
differential equations using neural networks with the following two-fold
objective: investigate the behavior of neural networks as approximators of
solutions of partial differential equations and propose neural-network-based
methods for frameworks that are hardly addressable via traditional numerical
methods.
  As novel neural-network-based proposals, we first present a method inspired
by the finite element method when applying mesh refinements to solve parametric
problems. Secondly, we propose a general residual minimization scheme based on
a generalized version of the Ritz method. Finally, we develop a memory-based
strategy to overcome a usual numerical integration limitation when using neural
networks to solve partial differential equations.
