ADEPTS: An auto-differentiable framework for time-dependent nonlinear thermo-chemical mantle convection inversion

Abstract

Time-dependent mantle-dynamics inversion must address the high dimensionality of the initial state, nonlinear rheology, and gradient propagation through long-term thermo-mechanical evolution. We develop ADEPTS, a two-dimensional staggered-grid finite-difference framework for mantle-dynamics inversion based on automatic differentiation. The forward model solves incompressible Stokes flow, temperature advection-diffusion, and compositional advection with temperature- and strain-rate-dependent viscosity and plastic yielding. For the nonlinear Stokes system, we compare two gradient strategies: unrolled differentiation through a fixed number of Picard iterations and implicit differentiation of the converged discrete residual equations. Numerical experiments show that unrolled differentiation remains stable even when the nonlinear solve is not fully converged, whereas implicit differentiation requires sufficiently accurate nonlinear solutions; otherwise gradient consistency and optimization convergence deteriorate. With sufficiently converged solves, implicit differentiation recovers accurate gradients and reconstruction quality comparable to unrolled differentiation. Joint thermo-chemical twin experiments show that ADEPTS can simultaneously recover a high-dimensional initial temperature field and low-dimensional physical parameters, including compositional density, reference viscosity, and stress exponent, while fitting final-time temperature, surface horizontal velocity, and surface normal stress. These results demonstrate the feasibility of differentiable time-dependent mantle-dynamics inversion and clarify the different convergence requirements of unrolled and implicit differentiation.