Time-dependent mantle-dynamics inversion must simultaneously address the high dimensionality of the initial state, nonlinear rheology, and the complex propagation of gradients through long-term thermo-mechanical evolution. Here, we develop a geodynamic inversion framework <em>ADEPTS</em> based on a two-dimensional staggered-grid finite-difference discretization and automatic differentiation, with the aim of jointly constraining the initial temperature field and mantle material properties. The forward model solves the incompressible Stokes equations, the temperature advection–diffusion equation, and the compositional advection equation, while accounting for temperature- and strain-rate-dependent nonlinear viscosity and plastic yielding. For the nonlinear Stokes system arising at each time step, we implement two gradient-computation strategies: unrolled differentiation through a fixed number of Picard iterations and implicit differentiation based on the sufficiently converged discrete nonlinear residual equations. Numerical experiments show that unrolled differentiation enables stable inversion even when the nonlinear solve has not fully converged, while the accuracy of implicit differentiation depends on sufficient convergence of the nonlinear Stokes system. When the nonlinear solve is insufficiently accurate, both gradient consistency and optimization convergence deteriorate for implicit differentiation, whereas sufficiently converged nonlinear solves recover the expected gradient accuracy and yield reconstruction results comparable to those obtained with unrolled differentiation. Under idealized noise-free twin-experiment conditions, we further test the joint inversion of the initial temperature field and physical parameters. The joint inversion recovers the initial temperature field together with the density of a compositional anomaly, the reference viscosity, and the stress exponent, while simultaneously fitting observations of the final-time temperature field, surface horizontal velocity, and surface normal stress. The two differentiation strategies impose different requirements on the accuracy of the nonlinear Stokes solve, but both can be used for gradient computation and inversion within a differentiable time-dependent mantle-dynamics framework.