用静态数据训练物理约束神经网络，预测系统动态。
Physics Informed Constrained Learning of Dynamics from Static Data
- 基于非时序或部分观测数据，通过物理约束学习系统演化规律。
- 在代谢通量分析任务中，优于现有所有数据驱动方法。
- 适合缺乏完整时间序列数据的物理建模场景，如生物代谢研究。
物理信息神经网络（PINN）通过将物理定律融入神经网络架构，以约束方式建模系统动力学，克服数据稀缺与高维挑战。现有框架依赖完整的时序数据，而此类数据获取对许多系统而言成本高昂。本文提出一种新范式——约束学习（Constrained Learning），可利用非时序或部分观测数据近似一阶导数或运动行为。我们建立了该方法的计算原理与通用数学框架，并引入基于消息传递优化的约束学习方法（MPOCtrL），旨在平衡物理模型拟合与观测数据匹配。代码已开源。在合成与真实数据上的实验表明，MPOCtrL能有效揭示观测数据与系统内在物理属性间的非线性关系；尤其在代谢通量分析任务中，性能超越所有现有数据驱动估计器。
原文摘要 · Abstract (English)
A physics-informed neural network (PINN) models the dynamics of a system by integrating the governing physical laws into the architecture of a neural network. By enforcing physical laws as constraints, PINN overcomes challenges with data scarsity and potentially high dimensionality. Existing PINN frameworks rely on fully observed time-course data, the acquisition of which could be prohibitive for many systems. In this study, we developed a new PINN learning paradigm, namely Constrained Learning, that enables the approximation of first-order derivatives or motions using non-time course or partially observed data. Computational principles and a general mathematical formulation of Constrained Learning were developed. We further introduced MPOCtrL (Message Passing Optimization-based Constrained Learning) an optimization approach tailored for the Constrained Learning framework that strives to balance the fitting of physical models and observed data. Its code is available at github link: https://github.com/ptdang1001/MPOCtrL Experiments on synthetic and real-world data demonstrated that MPOCtrL can effectively detect the nonlinear dependency between observed data and the underlying physical properties of the system. In particular, on the task of metabolic flux analysis, MPOCtrL outperforms all existing data-driven flux estimators.
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