Neural General Relativity: Re-deriving the Einstein-Hilbert Action from Geometric Invariants

General Relativity describes gravity as the curvature of spacetime, governed by the Einstein-Hilbert action. This paper presents a variational framework for the data-driven discovery of gravitational laws using neural manifold representations. By parameterizing the metric tensor as a neural network (MetricNet) and utilizing automatic differentiation to compute Riemann curvature invariants analytically, we train a GRLagrangianNet to identify the functional form of the gravitational action density from geometric data alone. We demonstrate that by minimizing the Einstein field equation residuals across geometrically diverse spacetimes, the neural network successfully re-derives the Einstein-Hilbert scaling L proportional to R*sqrt(-g). The ratio L/(R*sqrt(-g)) converges to a constant across varied geometries, providing a proof-of-concept for AI-driven discovery in curved spacetime physics and establishing a foundation for probing modified gravity theories beyond General Relativity.