  Neural networks with PDEs embedded in their loss functions (physics-informed
neural networks) are employed as a function approximators to find solutions to
the Ricci flow (a curvature based evolution) of Riemannian metrics. A general
method is developed and applied to the real torus. The validity of the solution
is verified by comparing the time evolution of scalar curvature with that found
using a standard PDE solver, which decreases to a constant value of 0 on the
whole manifold. We also consider certain solitonic solutions to the Ricci flow
equation in two real dimensions. We create visualisations of the flow by
utilising an embedding into $\mathbb{R}^3$. Snapshots of highly accurate
numerical evolution of the toroidal metric over time are reported. We provide
guidelines on applications of this methodology to the problem of determining
Ricci flat Calabi--Yau metrics in the context of String theory, a long standing
problem in complex geometry.
