Vectorized Conditional Neural Fields: A Framework for Solving Time-dependent Parametric Partial Differential Equations

Abstract

Transformer models are increasingly used for solving Partial Differential Equations (PDEs). Several adaptations have been proposed all of which, however, suffer from the typical problem of Transformers such as quadratic memory and time complexity. Furthermore, all prevalent architectures for PDE solving lack at least one of several desirable properties of an ideal surrogate model such as (i) generalization to PDE parameters not seen during training, (ii) spatial and temporal zero-shot super-resolution, (iii) continuous temporal extrapolation, (iv) applicability to PDEs of different dimensionalities, and (v) efficient inference for longer temporal rollouts. To address these limitations, we propose Vectorized Conditional Neural Fields (VCNeFs) which represent the solution of time-dependent PDEs as neural fields. Contrary to prior methods, however, VCNeFs compute, for a set of multiple spatio-temporal query points, their solutions in parallel and model their dependencies through attention mechanisms. Moreover, VCNeF can condition the neural field on both the initial conditions and the parameters of the PDEs. An extensive set of experiments demonstrate that VCNeFs are competitive with and often outperform existing ML-based surrogate models.

Jan Hagnberger

Student in Data Science & AI

My research interests include machine learning for science, deep learning, transformers and partial differential equations.