This paper presents a framework for physics-informed learning in complex cyber-physical systems governed by differential equations with both unknown dynamics and algebraic invariants. First, we formalize the Hybrid Recurrent Physics-Informed Neural Network (HRPINN), a general-purpose architecture that embeds known physics as a hard structural constraint within a recurrent integrator to learn only residual dynamics. Second, we introduce the Projected HRPINN (PHRPINN), a novel extension that integrates a predict-project mechanism to strictly enforce algebraic invariants by design. The framework is supported by a theoretical analysis of its representational capacity. We validate HRPINN on a real-world battery prognostics DAE and evaluate PHRPINN on a suite of standard constrained benchmarks. The results demonstrate the framework's potential for achieving high accuracy and data efficiency, while also highlighting critical trade-offs between physical consistency, computational cost, and numerical stability, providing practical guidance for its deployment.

翻译：本文提出了一种用于复杂网络物理系统的物理信息学习框架，该系统由包含未知动力学和代数不变量的微分方程所控制。首先，我们形式化定义了混合循环物理信息神经网络（HRPINN），这是一种通用架构，它将已知物理知识作为硬结构约束嵌入循环积分器中，仅用于学习残差动力学。其次，我们引入了投影HRPINN（PHRPINN），这是一种新颖的扩展，通过集成预测-投影机制，在设计中严格强制实施代数不变量。该框架得到了其表示能力理论分析的支持。我们在一个真实世界的电池寿命预测微分代数方程上验证了HRPINN，并在一系列标准约束基准测试中评估了PHRPINN。结果表明，该框架在实现高精度和数据效率方面具有潜力，同时也突显了物理一致性、计算成本和数值稳定性之间的关键权衡，为其实际部署提供了实用指导。