Forward and Inverse Mantle Convection with Neural Operators

TL;DR

This study demonstrates that neural operators can dramatically accelerate both forward and inverse mantle convection calculations. A physics-informed Stokes operator $S_0_5$ learns instantaneous velocity fields from temperature without training data, while data-driven forward and reverse convection operators $F^{+n0t}_5$ and $F^{-n0t}_5$ map between thermal states across long time intervals, enabling long-range predictions and operator discovery. In mantle-state reconstructions, four methods are compared; backward buoyancy and reverse convection suffer from noise sensitivity, whereas terminal-state inversion benefits from additional chronological data, with joint inversion using terminal state and surface-velocity observations providing the most robust and accurate reconstructions—extending recoverable histories to about 1 transit time in ideal cases and to $-0.86t_{tr}$ under realistic noise. The results suggest a practical pathway to large-scale mantle-state inversions informed by seismic tomography and plate reconstructions, with neural-operator workflows offering substantial speedups and favorable scaling, especially for time-dependent inversions. The work lays a foundation for extending these techniques to 3-D, spherical geometries and variable rheologies, potentially transforming geodynamic modeling and tectonic reconstructions.

Abstract

Thermal state reconstruction -- reversing convection to recover the thermal structure of the mantle at an earlier geologic time -- is an important tool to understand the evolution of mantle convection and its relation to seismic tomographic images and observations at the surface. Thermal state reconstructions are computationally expensive. Here we transformed the basic computational element, numerical solvers, into neural operators, a class of machine learning models for learning mappings between function spaces. Focusing on a specific architecture, Fourier Neural Operators, we demonstrate that they can represent not only a surrogate model like the Stokes system of equations using a purely physics informed approach, but also discover operators without explicit mathematical formulations or even ill-posedness from data, including the direct mapping between two convecting thermal states separated by a long time interval much larger than the Courant Fredrich Lewy condition and its reversal. These neural operators significantly accelerate forward and inverse convection modeling by transforming forward physical processes into surrogate models with lower complexity while utilizing auto-differentiation to calculate gradients. With this framework, we demonstrate the strength and weaknesses of four methods for thermal state reconstructions: Reverse buoyancy, reverse convection operator, an inversion with only the terminal thermal state, and a joint inversion with the terminal thermal state and surface velocity evolution. The reverse convection operator is shown to perform poorly in the presence of observational noise, but the joint inversion overcomes this limitation. The joint technique could probably become a solution to large-scale thermal state inversion problems using seismic tomography and plate tectonic reconstructions.

Figures (11)

• Figure 1: An example of forward convection computed with Underworld, which is used as an evaluation data sequence in this study. The computation is initiated from a Gaussian random initial thermal field that satisfies the prescribed boundary conditions. The total integration time is around 34 transit times, during which the convection pattern evolves from a chaotic one to a relatively steady state, as shown by the tracked Nusselt number. The temperature field is displayed at eight instances to depict the thermal evolution.
• Figure 2: The architecture of three neural operators described in this study. Detailed MLP and FB parameters are listed in Table \ref{['table:no_par']}.
• Figure 3: Comparisons between forward computations using $\mathbf{F}_{\phi_7}^{+n}$ and Underworld, $\text{Ra}=10^7$. Row 1 to 4 shows the forward evolution snapshots of the systems. Column 1: temperature snapshots and velocity streamlines computed by $\mathbf{F}_{\phi_7}^{+1}$, system integrated by $\mathbf{F}_{\phi_7}^{+1}$; Column 2: temperature and velocity computed by Underworld, system integrated by Underworld; Column 3: velocity computed by $\mathcal{S}_\phi$ based on thermal fields integrated by Underworld as inputs. Row 5 and 6: $\text{Nu}$ and $\mathbf{u}_{x0}$ tracked in systems integrated with different methods. Black short bars indicate the instants where the snapshots within row 1 to 4 are chosen.
• Figure 4: Correlation coefficient of reconstructed thermal fields with ground-truth fields versus backwards time. Colored lines denote different reconstruction methods. (a) Reconstruction with synthesized observations (no noise); (b) Reconstruction with synthesized observations polluted with $5\%$ pink noise.
• Figure S1: The evaluation method of PDE losses for the training of $\mathbf{S}_\phi$. The gray shaded area shows the physical domain $\Omega$. The orange dots show the primary nodes on which the physical quantities, $T$, $\mathbf{u}$, and $p$ are discretized, and the momentum loss $L_M$ is evaluated. The blue dots show the sub-nodes on which the continuity loss $L_C$ is evaluated. Two dots circled with heavy boundaries show examples how the physical information from the neighbor nodes are utilized during the evaluation. Since two side boundaries are periodic, an extra column of sub-nodes are attached to the side of physical domain, in order to evaluate the continuity across the side boundaries.