解决机器学习与偏微分方程耦合系统的数值不稳定性问题
Stabilizing PDE--ML coupled systems
- 通过分析黏性Burgers'方程的机器学习耦合系统，定位不稳定性根源
- 提出稳定策略，使耦合系统在数值求解中保持收敛
- 结合莫里-茨万格理论提升稳定系统的精度，适用于复杂系统研究
大型偏微分方程（PDE）系统中使用机器学习代理模型时，数值求解常出现不稳定性，现有方法多聚焦于提升代理模型精度或引入结构，但效果有限。本文以黏性Burgers'-ML系统为原型，识别出不稳定性来源，并提出相应稳定化策略。为进一步提升稳定系统精度，探索基于莫里-茨万格（Mori--Zwanzig）形式化的改进方法，为更复杂系统提供可推广的解决方案。
原文摘要 · Abstract (English)
A long-standing obstacle in the use of machine-learnt surrogates with larger PDE systems is the onset of instabilities when solved numerically. Efforts towards ameliorating these have mostly concentrated on improving the accuracy of the surrogates or imbuing them with additional structure, and have garnered limited success. In this article, we study a prototype problem and draw insights that can help with more complex systems. In particular, we focus on a viscous Burgers'-ML system and, after identifying the cause of the instabilities, prescribe strategies to stabilize the coupled system. To improve the accuracy of the stabilized system, we next explore methods based on the Mori--Zwanzig formalism.
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