Einstein's Field Equations as Continuous-Time Recurrent Neural Networks

We examine a subset of spatially homogenous and anisotropic solutions to Einstein's field equations: the Bianchi Type A models, and show that they can be written as a continuous-time recurrent neural network (CTRNN). This reformulation of Einstein's equations allows one to write potentially complicated nonlinear equations as a simpler dynamical system consisting of linear combinations of the neural network weights and logistic sigmoid activation functions. The CTRNN itself is trained by using an explicit Runge-Kutta solver to sample a number of solutions of Einstein's equations for the Bianchi Type A models and then using a nonlinear least-squares approach to find the optimal set of weights, time delay constants, and bias parameters that provide the best fit of the CTRNN equations to the Einstein equations. In terms of numerical examples, we specifically provide solutions to Bianchi Type I and II models. We conclude the paper with some comments on optimal parameter probability distributions and ideas for future work.