Zebra: In-Context and Generative Pretraining for Solving Parametric PDEs
Authors: name: Louis Serrano; name: Armand Kassaï Koupaï; name: Thomas X Wang; name: Pierre Erbacher; name: Patrick Gallinari
Published: 2024-10-04 13:52:02

Abstract: Solving time-dependent parametric partial differential equations (PDEs) is
challenging, as models must adapt to variations in parameters such as
coefficients, forcing terms, and boundary conditions. Data-driven neural
solvers either train on data sampled from the PDE parameters distribution in
the hope that the model generalizes to new instances or rely on gradient-based
adaptation and meta-learning to implicitly encode the dynamics from
observations. This often comes with increased inference complexity. Inspired by
the in-context learning capabilities of large language models (LLMs), we
introduce Zebra, a novel generative auto-regressive transformer designed to
solve parametric PDEs without requiring gradient adaptation at inference. By
leveraging in-context information during both pre-training and inference, Zebra
dynamically adapts to new tasks by conditioning on input sequences that
incorporate context trajectories or preceding states. This approach enables
Zebra to flexibly handle arbitrarily sized context inputs and supports
uncertainty quantification through the sampling of multiple solution
trajectories. We evaluate Zebra across a variety of challenging PDE scenarios,
demonstrating its adaptability, robustness, and superior performance compared
to existing approaches.