ISTerre seminar

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Differentiable spectral methods for geophysical fluid dynamics

Friday 10 October 2025 - 09h30

Keaton Burns - MIT Math
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Global spectral methods are a classical technique for solving partial differential equations (PDEs), including Fourier series in periodic domains and orthogonal polynomials in bounded domains. While traditional collocation-based approaches for polynomials are slow at scale, modern spectral methods reformulate these systems in sparse operator form, enabling fast and accurate solvers with near FFT-like performance in geometries such as channels, disks, and spheres. These sparse methods are also amenable to automatic differentiation (autodiff), enabling efficient PDE-based optimization as well as forward modeling.

Here we will present Dedalus, an open-source Python framework for solving general PDEs using modern sparse spectral methods. Unlike traditional domain-specific solvers, Dedalus does not hard-code particular equation sets, but instead provides a symbolic equation specification system that allows users to define their own PDEs. The framework then automatically constructs an optimally sparse, MPI-parallelized solver tailored to the chosen equations and geometry. This flexibility has enabled Dedalus to support a wide range of interdisciplinary applications, including many for which specialized solvers do not yet exist.

We will also introduce a new autodiff system that can compute the discrete adjoint of any Dedalus solver. Adjoint methods are central to optimization in CFD, including parameter estimation and nonlinear stability analyses, but are traditionally cumbersome to derive and implement. Reverse-mode autodiff, by contrast, automates these computations at the compiler level and has become a mainstay of differential programming in machine learning. Our approach efficiently implements reverse-mode differentiation by leveraging Dedalus's symbolic equation representation, requiring no differentiable compiler and preserving full MPI parallelism. We will demonstrate the capabilities of this system with applications including optimal perturbations in shear flows and kinematic dynamos.

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Organizing team : Géodynamo

Amphithéâtre Killian, Maison des Géosciences, 38400 Saint Martin d'Hères

Informations de visio :

https://univ-grenoble-alpes-fr.zoom.us/j/91690538748?pwd=buNDS0rAZw9rG4vS1rWxX53zHYdQaw.1