A primary challenge of physics-informed machine learning (PIML) is its generalization beyond the training domain, especially when dealing with complex physical problems represented by partial differential equations (PDEs). This paper aims to enhance the generalization capabilities of PIML, facilitating practical, real-world applications where accurate predictions in unexplored regions are crucial. We leverage the inherent causality and temporal sequential characteristics of PDE solutions to fuse PIML models with recurrent neural architectures based on systems of ordinary differential equations, referred to as neural oscillators. Through effectively capturing long-time dependencies and mitigating the exploding and vanishing gradient problem, neural oscillators foster improved generalization in PIML tasks. Extensive experimentation involving time-dependent nonlinear PDEs and biharmonic beam equations demonstrates the efficacy of the proposed approach. Incorporating neural oscillators outperforms existing state-of-the-art methods on benchmark problems across various metrics. Consequently, the proposed method improves the generalization capabilities of PIML, providing accurate solutions for extrapolation and prediction beyond the training data.

翻译：物理信息机器学习（PIML）的主要挑战在于其超出训练域的泛化能力，尤其是在处理偏微分方程（PDEs）所代表的复杂物理问题时。本文旨在增强PIML的泛化能力，促进其在需要准确预测未探索区域的实用现实世界场景中的应用。我们利用PDE解的固有因果性和时间序列特征，将PIML模型与基于常微分方程系统的循环神经架构（称为神经振荡器）相融合。通过有效捕获长时间依赖并缓解梯度爆炸和梯度消失问题，神经振荡器促进了PIML任务中泛化能力的提升。涉及时间相关非线性PDEs和双调和梁方程的广泛实验证明了所提方法的有效性。在基准问题的各项指标上，融入神经振荡器的方法优于现有最先进方法。因此，所提方法提升了PIML的泛化能力，为超出训练数据的外推与预测提供了精确解。