Fourier Neural Operators Show Faster Thermal State Reconstruction of Mantle Convection

Reconstructing the thermal history of Earth’s mantle is crucial for understanding the planet’s dynamic evolution and interpreting seismic images, yet remains a computationally intensive challenge. Chenxi Kong, Michael Gurnis, and Zachary E. Ross, all from the Seismological Laboratory at the California Institute of Technology, have now demonstrated a significant acceleration of both forward and inverse mantle convection modelling by integrating neural operators, a novel machine learning approach, into traditional numerical solvers. Their work showcases how these operators can learn complex relationships between thermal states, even across vast timescales, and effectively bypass limitations inherent in conventional methods. By comparing four reconstruction techniques, the researchers reveal the strengths of a joint inversion method, which promises a viable pathway towards large-scale thermal state reconstruction using seismic data and plate tectonic history.

Reconstructing Earth’s mantle thermal history with learned neural operator surrogates offers new insights into planetary evolution

Scientists have developed a groundbreaking new approach to reconstructing the thermal history of Earth’s mantle using neural operators, a form of machine learning. This research addresses the computationally intensive problem of thermal state reconstruction, essentially reversing mantle convection to understand past thermal structures.
The team transformed the fundamental computational elements of numerical solvers into these neural operators, enabling them to learn mappings between different functional spaces relevant to mantle dynamics. Specifically, they focused on Fourier Neural Operators to create surrogate models of complex physical processes, like the Stokes system of equations, using a physics-informed approach.

This breakthrough extends beyond simply creating faster simulations; the neural operators can also discover operators directly from data, even in scenarios where traditional mathematical formulations are lacking or the problem is ill-posed. Experiments demonstrate the ability to map between two convecting thermal states separated by vast geological timescales, exceeding the limitations of conventional numerical methods.

By converting forward physical processes into lower-complexity surrogate models and utilising auto-differentiation for gradient calculations, the research significantly accelerates both forward and inverse convection modelling. The study rigorously tested four methods for thermal state reconstruction, including reverse buoyancy, a reverse convection operator, inversion using only the final thermal state, and a joint inversion incorporating both the final thermal state and the evolution of surface velocity.

Results show the reverse convection operator struggles with observational noise, but the joint inversion technique effectively overcomes this limitation. This joint approach holds considerable promise as a solution for large-scale thermal state inversion problems, potentially integrating seismic tomography and plate tectonic reconstructions to reveal deeper insights into Earth’s thermal evolution.

Researchers demonstrated that these neural operators can not only represent established physical models but also learn directly from data, bypassing the need for explicit mathematical definitions. The framework allows for the direct mapping between two thermal states separated by extended periods, a feat previously hindered by computational constraints and the Courant, Friedrichs, Lewy condition.

This innovation significantly reduces the computational burden associated with both forward and inverse modelling, paving the way for more detailed and accurate reconstructions of mantle convection over geological timescales. The work opens new avenues for understanding the interplay between mantle dynamics, seismic observations, and plate tectonics.

Fourier Neural Operators accelerate forward and inverse mantle convection modelling significantly

Scientists engineered a novel approach to thermal state reconstruction by integrating neural operators into mantle convection modeling. The research team transformed the basic computational element of traditional methods, numerical solvers, into Fourier Neural Operators, a machine learning model designed to learn mappings between function spaces.

This technique facilitates the direct mapping between two convecting thermal states separated by time intervals exceeding the Courant, Friedrich, Lewy (CFL) condition, and even reverses this process. Experiments employed these neural operators to significantly accelerate both forward and inverse convection modeling by converting complex forward physical processes into lower-complexity surrogate models.

Auto-differentiation was harnessed to efficiently calculate gradients, streamlining the iterative optimization process inherent in inverse problems. The study pioneered a framework to assess four distinct methods for reconstructing past thermal states: reverse buoyancy, a reverse convection operator, inversion using only the terminal thermal state, and a joint inversion incorporating both the terminal thermal state and surface velocity evolution.

Researchers demonstrated that the reverse convection operator exhibited poor performance when subjected to observational noise. However, the joint inversion technique effectively mitigated this limitation, offering improved accuracy and robustness. This joint approach leverages both the current mantle state and the evolution of surface velocity to constrain the reconstruction.

The system delivers a potential solution for large-scale thermal state inversion problems, particularly those utilizing seismic tomography and plate tectonic reconstructions, by reducing computational demands and improving solution fidelity. This method achieves a breakthrough in handling ill-posed problems by incorporating comprehensive data constraints and efficient surrogate modeling.

Mapping mantle convection states using physics-informed Fourier Neural Operators is a promising approach for geophysical modeling

Scientists achieved a breakthrough in modeling mantle convection by integrating numerical solvers into Fourier Neural Operators, a type of machine learning model. The team demonstrated that these operators can represent complex systems, including the Stokes system of equations, using a physics-informed approach, and even discover operators directly from data without explicit mathematical formulations.

Experiments revealed the capability to map between two convecting thermal states separated by time intervals exceeding the Courant-Friedrich-Lewy condition, and to reverse this process. Results demonstrate a significant acceleration in both forward and inverse convection modeling by transforming forward physical processes into lower-complexity surrogate models, utilising auto-differentiation for accurate gradient calculations.

The physics-informed Stokes neural operator, Sφ, was trained with a loss function, LS, and successfully predicted super-resolution flow even with high-frequency plume structures, achieving an L2 relative error of approximately 5% compared to Underworld simulations. Tests using thermal structures with Rayleigh numbers of 105, 106, and 107, at resolutions of 65, 129, and 257, consistently yielded L2 relative errors around 5% (Table 1).

Measurements confirm that point-wise errors generally align with velocity amplitudes, with the largest errors occurring at the edges of convection cells and diminishing in thermally homogeneous regions. The Stokes neural operator accurately predicted velocities within the thermal cores of upwellings and downwellings, despite generally predicting less vigorous convection patterns.

Data shows that the training of Sφ required no pre-training computational resource on solving the Stokes equations, starting with random source term inputs and evaluating PDE losses from the output end. Furthermore, the team tested forward convection neural operators, F+n∆t φ, across a time window spanning approximately 14 transit times with a Rayleigh number of 107.

The integration time steps for these operators were found to be about 100times larger than those used in traditional solvers, with the largest time step contrast reaching approximately 300times (Table 2, S3). Diagnostic variables, including the Nusselt number, Nu, and maximum horizontal velocity, ux0, were tracked, revealing that the neural operators produced results tracking closely with Underworld simulations for the first transit time. The relative error of the one-step prediction was as low as 0.1% (Table 2), although divergence occurred after 3-4 transit times due to the system’s non-linearity.

Physics-informed neural operators reconstruct thermal convection dynamics efficiently and accurately

Scientists have demonstrated the utility of neural operators for two-dimensional, bottom-heated, Rayleigh-Bénard thermal convection modelling. These machine learning models represent a computational element, numerical solvers, enabling both forward and backward calculations with reduced complexity. The research focused on Fourier Neural Operators, showing their ability to model the Stokes system of equations using a physics-informed approach, and to discover operators directly from data without explicit mathematical formulations.

This allows for the mapping between convecting thermal states separated by time intervals exceeding conventional computational limits. The findings establish a framework for accelerating forward and inverse convection modelling through surrogate models and auto-differentiation for gradient calculations.

Four methods for thermal state reconstruction were compared, revealing that a joint inversion, incorporating both the terminal thermal state and surface velocity evolution, overcomes limitations observed with other techniques, particularly in the presence of observational noise. The accuracy of these neural operators extends to reconstructing mantle convection states approximately 150 to 200 million years into the past, sufficient for resolving long-term tectonic processes.

The authors acknowledge that the current work requires considerable computational resources for training, but note ongoing advancements in neural network architectures and GPU hardware are addressing this. Future research may focus on developing three-dimensional neural operators for global convection modelling, building on recent progress in learning complex physics at high resolution.