Time-dependent partial differential equations (PDEs) are ubiquitous in science and engineering. Recently, mostly due to the high computational cost of traditional solution techniques, deep neural network based surrogates have gained increased interest. The practical utility of such neural PDE solvers relies on their ability to provide accurate, stable predictions over long time horizons, which is a notoriously hard problem. In this work, we present a large-scale analysis of common temporal rollout strategies, identifying the neglect of non-dominant spatial frequency information, often associated with high frequencies in PDE solutions, as the primary pitfall limiting stable, accurate rollout performance. Based on these insights, we draw inspiration from recent advances in diffusion models to introduce PDE-Refiner; a novel model class that enables more accurate modeling of all frequency components via a multistep refinement process. We validate PDE-Refiner on challenging benchmarks of complex fluid dynamics, demonstrating stable and accurate rollouts that consistently outperform state-of-the-art models, including neural, numerical, and hybrid neural-numerical architectures. We further demonstrate that PDE-Refiner greatly enhances data efficiency, since the denoising objective implicitly induces a novel form of spectral data augmentation. Finally, PDE-Refiner's connection to diffusion models enables an accurate and efficient assessment of the model's predictive uncertainty, allowing us to estimate when the surrogate becomes inaccurate.

翻译：时间依赖偏微分方程在科学和工程领域中无处不在。近年来，主要由于传统求解方法高昂的计算成本，基于深度神经网络的替代模型引起了越来越多的关注。这类神经PDE求解器的实际效用取决于其在长时间范围内提供准确稳定预测的能力，而这正是公认的难题。在本工作中，我们对常见的时域推演策略进行了大规模分析，发现忽视非主导空间频率信息（通常与PDE解中的高频分量相关）是限制稳定、准确推演性能的主要障碍。基于这些发现，我们从扩散模型的最新进展中汲取灵感，提出了PDE-Refiner——一种新型模型类别，通过多步精化过程实现对所有频率分量的更精确建模。我们在复杂流体动力学的挑战性基准测试上验证了PDE-Refiner，展示了其稳定且准确的推演能力，并始终优于包括神经、数值及混合神经-数值架构在内的最先进模型。我们进一步证明，PDE-Refiner能大幅提升数据效率，因为去噪目标隐式引入了一种新颖的谱数据增强形式。最后，PDE-Refiner与扩散模型的关联使得对模型预测不确定性的精确高效评估成为可能，从而使我们能够估计替代模型何时变得不准确。