一种用于非稳态非线性偏微分方程的基于区域分解的自回归深度学习模型
A domain decomposition-based autoregressive deep learning model for unsteady and nonlinear partial differential equations
AI总结：
提出 transient-CoMLSim 区域分解深度学习框架，结合 CNN 自编码器、潜空间自回归与课程学习，提升非稳态非线性 PDE 预测的精度、外推性和稳定性。
摘要：
本文提出一种基于区域分解的深度学习（DL）框架，名为 transient-CoMLSim，用于精确建模非稳态非线性偏微分方程（PDEs）。该框架包含两个关键组成部分：（a）基于卷积神经网络（CNN）的自编码器架构，以及（b）由全连接层组成的自回归模型。与现有在整个计算域上运行的最先进方法不同，我们基于 CNN 的自编码器为在子域上表示的解场和条件场计算低维基。时间推进完全在潜空间中执行，根据解变量和条件变量嵌入的时间历史生成解变量的嵌入。该方法不仅降低了计算复杂度，还增强了可扩展性，使其非常适合大规模模拟。此外，为提高滚动预测的稳定性，我们在自回归模型训练期间采用课程学习（CL）方法。区域分解策略使其能够扩展到分布外的域尺寸，同时保持预测精度——这一特性不易集成到流行的基于 DL 的物理模拟方法中。我们将模型与两种广泛使用的 DL 架构 Fourier Neural Operator（FNO）和 U-Net 进行基准比较，结果表明，在广泛用例中，我们的框架在精度、对未见时间步的外推以及稳定性方面均优于它们。
英文摘要：
In this paper, we propose a domain-decomposition-based deep learning (DL) framework, named transient-CoMLSim, for accurately modeling unsteady and nonlinear partial differential equations (PDEs). The framework consists of two key components: (a) a convolutional neural network (CNN)-based autoencoder architecture and (b) an autoregressive model composed of fully connected layers. Unlike existing state-of-the-art methods that operate on the entire computational domain, our CNN-based autoencoder computes a lower-dimensional basis for solution and condition fields represented on subdomains. Timestepping is performed entirely in the latent space, generating embeddings of the solution variables from the time history of embeddings of solution and condition variables. This approach not only reduces computational complexity but also enhances scalability, making it well-suited for large-scale simulations. Furthermore, to improve the stability of our rollouts, we employ a curriculum learning (CL) approach during the training of the autoregressive model. The domain-decomposition strategy enables scaling to out-of-distribution domain sizes while maintaining the accuracy of predictions -- a feature not easily integrated into popular DL-based approaches for physics simulations. We benchmark our model against two widely-used DL architectures, Fourier Neural Operator (FNO) and U-Net, and demonstrate that our framework outperforms them in terms of accuracy, extrapolation to unseen timesteps, and stability for a wide range of use cases.
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