PDE-Refiner：利用神經偏微分方程求解器實現準確的長期模擬
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
時變偏微分方程（PDE）在科學和工程領域中無處不在。最近，主要是由於傳統解決方法的高計算成本，基於深度神經網絡的替代方案引起了日益增加的關注。這些神經網絡PDE求解器的實際應用取決於它們能夠在長時間範圍內提供準確、穩定的預測，這是一個眾所周知的難題。在這項工作中，我們對常見的時間展開策略進行了大規模分析，識別了忽略非主導空間頻率信息的問題，這種信息通常與PDE解的高頻率相關，是限制穩定、準確展開性能的主要問題。基於這些見解，我們從最近擴散模型的進展中汲取靈感，引入了PDE-Refiner；這是一種新型模型類別，通過多步細化過程實現對所有頻率成分更準確的建模。我們在複雜流體動力學的具有挑戰性的基準測試中驗證了PDE-Refiner，展示了穩定且準確的展開，始終優於最先進的模型，包括神經網絡、數值和混合神經-數值結構。我們進一步證明，PDE-Refiner極大地提高了數據效率，因為去噪目標隱含地引入了一種新形式的頻譜數據增強。最後，PDE-Refiner與擴散模型的聯繫使我們能夠準確高效地評估模型的預測不確定性，從而使我們能夠估計替代方案何時變得不準確。
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.