Fourier Neural Operators for Geodynamic Modeling: A Hybrid Surrogate–Solver Framework

Viven Sharma

Keywords: Geodynamic Modeling, Rayleigh-Taylor instability, Fourier Neural Operator, Scientific Machine Learning

TL;DR: We present a Fourier neural operator-based hybrid surrogate solver framework for accelerating geodynamic numerical modeling.

Abstract:

Geodynamic research, particularly the study of buoyancy-driven instabilities, relies
heavily on large-scale numerical simulations. However, the substantial wall-time
and memory requirements of traditional methods like the finite difference (FD)
model hinder comprehensive ensemble studies, inverse problems, and rapid hy-
pothesis testing. This paper introduces a hybrid computational framework that
leverages the capabilities of Fourier Neural Operators (FNOs), a class of deep
learning models that learn resolution-invariant mappings between function spaces.
An FNO is trained on a dataset of density-field evolutions for the Rayleigh–Taylor
instability (RTI), generated by a high-fidelity FD solver. The trained FNO is then
deployed in two distinct modes: as a standalone, ultra-fast surrogate model for
rapid inference and as a non-intrusive, physics-aware initializer within a tightly
coupled FD loop to accelerate the convergence of nonlinear solves. We present the
end-to-end data-generation and training pipeline, detail the FNO architecture, and
quantify the framework’s performance against a U-Net and DeepONet baselines.
Results demonstrate that the FNO accurately reproduces the complex morphology
of RT instabilities while achieving inference speeds several orders of magnitude
faster than the FD solver. Furthermore, we outline a clear pathway for extending
this approach to the irregular and spherical domains characteristic of realistic geo-
dynamic problems by using geometry-aware FNO variants. This hybrid strategy
retains the physical accuracy of FD methods where needed, while unlocking the
superior performance gains of operator learning for parameter-space exploration
and numerical-model simulation.

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Journal Notes:

In the present work I implemented a vanilla Fourier Neural Operator into a hybrid solver framework. The implementation is demonstrated to work on fixed timestep, iso-viscous problems that require significant (~20k samples) amount of training data. I have in-parallel implemented an advanced neural operator framework similar to Markov Neural Operator, that trains on ~1000 samples and also works for variable timestep problems.
If given the opportunity, I'd submit that work for this special issue.

Submission Number: 80