In this project we investigate operator learning based on energy minimization. The aim of operator learning is to approximate solution operators to partial differential equations (PDEs) using tools from machine learning. The input is typically a coefficient or a parameter of the PDE and the output is the corresponding solution. Access to the (approximate) solution operator is key towards achieving fast numerical approximations for a family of parameters. Important applications are problems where a PDE typically needs to be solved many times for different parameter values such as, non-linear problems, inverse problems, optimal control, etc. We focus on so called data-free methods, meaning that there is no labeled data for the training. When learning solution operators, training data would have to be generated by solving the PDE many times which is costly. Instead, we use a neural network with a loss function that is based on the corresponding energy formulation of the PDE. From classical PDE theory, it is well known that minimizing the energy is equivalent to solving the (variational or weak form of the) PDE. This implies that the network will be penalized to find the correct solution. During training, values of the parameter or coefficient are randomly drawn from a given probability measure. This results in a network that approximates the solution operator to the PDE for a wide range of input parameters. We emphasize that the solution must be represented on a set of points, e.g. the nodes in a finite element mesh. The output of the neural network is thus the nodal values at these points. These nodal values are used to calculate the energy which is typically a set of integrals that can be computed locally on each triangle in the finite element mesh. Thus, the computation of the loss function can be performed efficiently. The main advantage of this proposed operator learning method though, is that once the network is trained the solution (at all the nodes) for a given parameter value is directly accessible by one simple feed forward through the network.