ENMA: Tokenwise Autoregression for Continuous Neural PDE Operators

Louis Serrano (Emmi AI) · Armand Kassaï Koupaï (Sorbonne Université - ISIR) · Lise Le Boudec (ISIR - Sorbonne University ) · Patrick Gallinari (Sorbonne Universite, Criteo AI Lab)

attention mechanismsauxiliary context trajectoriesenmaflow matching lossgenerative masked autoregressive transformergenerative models in-context learning latent space neural solversone-shot surrogate modelingparametric partial differential equationspde regimesspatio-temporal convolutional encoderspatio-temporal dynamicstokenwise generation

Solving time-dependent parametric partial differential equations (PDEs) remains a fundamental challenge for neural solvers, particularly when generalizing across a wide range of physical parameters and dynamics. When data is uncertain or incomplete—as is often the case—a natural approach is to turn to generative models. We introduce ENMA, a generative neural operator designed to model spatio-temporal dynamics arising from physical phenomena. ENMA predicts future dynamics in a compressed latent space using a generative masked autoregressive transformer trained with flow matching loss, enabling tokenwise generation. Irregularly sampled spatial observations are encoded into uniform latent representations via attention mechanisms and further compressed through a spatio-temporal convolutional encoder. This allows ENMA to perform in-context learning at inference time by conditioning on either past states of the target trajectory or auxiliary context trajectories with similar dynamics. The result is a robust and adaptable framework that generalizes to new PDE regimes and supports one-shot surrogate modeling of time-dependent parametric PDEs.