We introduce the concept of geometry-informed neural networks (GINNs), which encompasses (i) learning under geometric constraints, (ii) neural fields as a suitable representation, and (iii) generating diverse solutions to under-determined systems as often found in geometric tasks. Notably, GINNs are formulated for scenarios where no training data is required, and as such can be considered as generative modeling driven purely by constraints. We enforce sample diversity as a remedy to mode collapse. We formulate GINNs on various problems by considering several differentiable losses. In particular, we use Morse theory to optimize for discrete requirements such as connectedness of components. Experimentally, we demonstrate the efficacy of the GINN learning paradigm across a range of two and three-dimensional scenarios, exhibiting varying levels of complexity.