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）的主要挑战在于其在训练域之外的泛化能力，尤其是在处理由偏微分方程（PDEs）表示的复杂物理问题时尤为突出。本文旨在增强PIML的泛化能力，促进其在实际应用场景中的部署——在这些场景中，对未探索区域进行准确预测至关重要。我们利用PDE解固有的因果性和时间序列特性，将PIML模型与基于常微分方程系统的循环神经网络架构（称为神经振荡器）相融合。通过有效捕捉长时间依赖关系并缓解梯度爆炸和梯度消失问题，神经振荡器促进了PIML任务中泛化性能的提升。涉及含时非线性PDEs及双调和梁方程的广泛实验验证了所提方法的有效性。在基准测试问题中，整合神经振荡器的方法在各评估指标上均优于现有最先进方法。因此，所提方法改善了PIML的泛化能力，为训练数据之外的区域提供了准确的外推与预测解。