PDE-Refiner：利用神经偏微分方程求解器实现准确的长期推演
PDE-Refiner: Achieving Accurate Long Rollouts with Neural PDE Solvers
August 10, 2023
作者: Phillip Lippe, Bastiaan S. Veeling, Paris Perdikaris, Richard E. Turner, Johannes Brandstetter
cs.AI
摘要
时间依赖的偏微分方程（PDEs）在科学和工程中随处可见。最近，主要是由于传统解决方案的高计算成本，基于深度神经网络的替代方法引起了越来越多的关注。这种神经网络PDE求解器的实际效用取决于它们能够在长时间范围内提供准确、稳定的预测，这是一个众所周知的难题。在这项工作中，我们对常见的时间展开策略进行了大规模分析，确定了忽略非主导空间频率信息的问题，这些信息通常与PDE解中的高频相关，是限制稳定、准确展开性能的主要缺陷。基于这些见解，我们从最近扩散模型的进展中汲取灵感，引入了PDE-Refiner；这是一种新颖的模型类别，通过多步细化过程实现对所有频率分量更准确的建模。我们在复杂流体动力学的具有挑战性的基准测试中验证了PDE-Refiner，展示了稳定和准确的展开，始终优于最先进的模型，包括神经网络、数值和混合神经-数值结构。我们进一步证明，PDE-Refiner极大地提高了数据效率，因为去噪目标隐含地引入了一种新形式的频谱数据增强。最后，PDE-Refiner与扩散模型的联系使我们能够准确高效地评估模型的预测不确定性，从而让我们能够估计替代模型何时变得不准确。
English
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.