用参数条件化训练深度学习模型，高效模拟混沌与随机动力系统。
Conditioning on PDE Parameters to Generalise Deep Learning Emulation of Stochastic and Chaotic Dynamics
- 模型基于PDE参数条件化，先在小范围预训练再微调以泛化到多种参数。
- 在柯莫戈洛夫-希瓦辛斯基方程和β平面湍流上实现快速准确模拟。
- 支持不同尺度和分辨率，适合需要参数扫描与不确定性分析的研究。
我们提出一种用于随机与混沌时空系统的深度学习代理模型，显式地根据底层偏微分方程（PDE）的参数进行条件化。该方法先在单一参数域上预训练模型，再通过一个小而多样化的数据集进行微调，从而实现对广泛参数值的良好泛化。通过引入局部注意力机制，网络能够处理不同尺寸与分辨率的域。这使得在较小域上进行计算高效的预训练成为可能，仅需少量额外数据即可学习扩展至更大域的能力。我们在混沌的柯莫戈洛夫-希瓦辛斯基方程和随机驱动的β平面湍流上展示了模型性能，其能在插值参数值下捕捉复杂现象。该代理模型相比传统数值积分大幅加速，便于参数空间探索；其概率变体还提供不确定性量化，支持稀有事件的统计研究。
原文摘要 · Abstract (English)
We present a deep learning emulator for stochastic and chaotic spatio-temporal systems, explicitly conditioned on the parameter values of the underlying partial differential equations (PDEs). Our approach involves pre-training the model on a single parameter domain, followed by fine-tuning on a smaller, yet diverse dataset, enabling generalisation across a broad range of parameter values. By incorporating local attention mechanisms, the network is capable of handling varying domain sizes and resolutions. This enables computationally efficient pre-training on smaller domains while requiring only a small additional dataset to learn how to generalise to larger domain sizes. We demonstrate the model's capabilities on the chaotic Kuramoto-Sivashinsky equation and stochastically-forced beta-plane turbulence, showcasing its ability to capture phenomena at interpolated parameter values. The emulator provides significant computational speed-ups over conventional numerical integration, facilitating efficient exploration of parameter space, while a probabilistic variant of the emulator provides uncertainty quantification, allowing for the statistical study of rare events.
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