用合成数据+强化学习，为粗粒度偏微分方程设计高效闭包模型。
Reinforcement Learning Closures for Underresolved Partial Differential Equations using Synthetic Data
- 通过构造合成数据，结合强化学习训练闭包模型。
- 在一维/二维伯格斯方程和二维平流方程上验证有效。
- 可在非齐次与齐次方程间泛化，适合数据稀缺场景。
偏微分方程（PDE）描述了从湍流、流行病到量子力学和金融市场等广泛现象。尽管计算科学不断进步，但真实应用中求解这些PDE仍因需解析多尺度时空结构而成本高昂。因此，实践者常采用粗粒度近似，以降低计算开销，但会损失细节。为此，闭包模型被用于表征未解析的时空相互作用。本文提出一种基于制造解法生成合成数据，并结合强化学习开发PDE闭包模型的框架。我们在一维和二维伯格斯方程以及二维平流方程上验证该方法的有效性。此外，我们证明了针对非齐次PDE训练的闭包模型可有效推广至齐次情形。结果表明，该方法有望为数据稀缺系统构建高精度且计算高效的闭包模型。
原文摘要 · Abstract (English)
Partial Differential Equations (PDEs) describe phenomena ranging from turbulence and epidemics to quantum mechanics and financial markets. Despite recent advances in computational science, solving such PDEs for real-world applications remains prohibitively expensive because of the necessity of resolving a broad range of spatiotemporal scales. In turn, practitioners often rely on coarse-grained approximations of the original PDEs, trading off accuracy for reduced computational resources. To mitigate the loss of detail inherent in such approximations, closure models are employed to represent unresolved spatiotemporal interactions. We present a framework for developing closure models for PDEs using synthetic data acquired through the method of manufactured solutions. These data are used in conjunction with reinforcement learning to provide closures for coarse-grained PDEs. We illustrate the efficacy of our method using the one-dimensional and two-dimensional Burgers' equations and the two-dimensional advection equation. Moreover, we demonstrate that closure models trained for inhomogeneous PDEs can be effectively generalized to homogeneous PDEs. The results demonstrate the potential for developing accurate and computationally efficient closure models for systems with scarce data.
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