Transformer models are increasingly used for solving Partial Differential Equations (PDEs). Several adaptations have been proposed, all of which suffer from the typical problems 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) support for 1D, 2D, and 3D PDEs, 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 demonstrates that VCNeFs are competitive with and often outperform existing ML-based surrogate models.

翻译：Transformer模型越来越多地用于求解偏微分方程（PDEs）。目前已提出若干改进方案，但均存在Transformer的典型问题，如二次内存与时间复杂度。此外，所有主流PDE求解架构至少缺失理想代理模型的若干理想特性之一，例如：（i）泛化至训练期间未见的PDE参数；（ii）空间与时间零样本超分辨率；（iii）连续时间外推；（iv）支持一维、二维与三维PDEs；（v）长时程推演的高效推理。为突破这些局限，我们提出向量化条件神经场（VCNeFs），将时间依赖PDE的解表示为神经场。然而与现有方法不同，VCNeFs可并行计算多个时空查询点集的解，并通过注意力机制建模其依赖关系。此外，VCNeF能将神经场同时条件化于初始条件与PDE参数。大量实验表明，VCNeFs与现有基于机器学习的代理模型相比具有竞争力，且往往表现更优。