论文标题

部分可观测时空混沌系统的无模型预测

Meta Learning of Interface Conditions for Multi-Domain Physics-Informed Neural Networks

论文作者

Li, Shibo, Penwarden, Michael, Xu, Yiming, Tillinghast, Conor, Narayan, Akil, Kirby, Robert M., Zhe, Shandian

论文摘要

物理知识的神经网络(PINN)正在作为部分微分方程(PDE)的流行无网求解器。最近的扩展分解了域,应用不同的PINN来解决每个子域中的问题,并在界面处缝制子域。因此,它们可以进一步缓解问题的复杂性,降低计算成本并允许并行化。但是,多域Pins的性能对界面条件的选择敏感。尽管已经提出了很多条件,但没有建议如何根据特定问题选择条件。为了解决这一差距,我们建议对界面条件(Metalic)的元学习,这是一种简单,有效而强大的方法,用于动态确定解决参数PDE家族的适当界面条件。具体而言,我们开发了两个上下文多臂强盗(MAB)模型。第一个适用于整个培训课程,在线更新高斯流程(GP)奖励,鉴于PDE参数和接口条件可以预测性能。我们证明了UCB和Thompson抽样的次线性遗憾,从理论上讲,这可以保证我们的单位单位的有效性。第二个将训练分为两个阶段,一个是随机阶段,另一个是确定性阶段。我们更新每个阶段的GP奖励,以在两个阶段启用不同的条件选择,以进一步增强灵活性和性能。我们已经显示了金属在四个台式PDE家族上的优势。

Physics-informed neural networks (PINNs) are emerging as popular mesh-free solvers for partial differential equations (PDEs). Recent extensions decompose the domain, apply different PINNs to solve the problem in each subdomain, and stitch the subdomains at the interface. Thereby, they can further alleviate the problem complexity, reduce the computational cost, and allow parallelization. However, the performance of multi-domain PINNs is sensitive to the choice of the interface conditions. While quite a few conditions have been proposed, there is no suggestion about how to select the conditions according to specific problems. To address this gap, we propose META Learning of Interface Conditions (METALIC), a simple, efficient yet powerful approach to dynamically determine appropriate interface conditions for solving a family of parametric PDEs. Specifically, we develop two contextual multi-arm bandit (MAB) models. The first one applies to the entire training course, and online updates a Gaussian process (GP) reward that given the PDE parameters and interface conditions predicts the performance. We prove a sub-linear regret bound for both UCB and Thompson sampling, which in theory guarantees the effectiveness of our MAB. The second one partitions the training into two stages, one is the stochastic phase and the other deterministic phase; we update a GP reward for each phase to enable different condition selections at the two stages to further bolster the flexibility and performance. We have shown the advantage of METALIC on four bench-mark PDE families.

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