让神经网络严格满足物理约束，提升工程模型可靠性。
Physics-Informed Neural Networks with Hard Nonlinear Equality and Inequality Constraints
- 通过KKT条件的可微投影，强制实现等式与不等式约束
- 在非凸池化与化工过程模拟中实现机器精度级约束满足
- 无需权衡数据与物理残差，适合需要高可靠性的建模场景
传统物理信息神经网络（PINNs）无法保证约束的严格满足，这在工程系统中可能导致模型预测不可靠。本文提出KKT-Hardnet，一种能将线性与非线性等式及不等式约束精确到机器精度的神经网络架构。其核心是通过求解距离最小化问题的Karush-Kuhn-Tucker（KKT）条件，实现对可行域的可微投影。进一步地，采用对数-指数变换重构非线性KKT条件，形成仅含线性和指数项的稀疏系统。实验表明，相较于多层感知机和PINNs，KKT-Hardnet在非凸池化问题和真实化工过程仿真中均实现严格约束满足，且无需平衡数据与物理残差，从而支持将领域知识融入机器学习，构建复杂系统的可靠混合模型。
原文摘要 · Abstract (English)
Traditional physics-informed neural networks (PINNs) do not guarantee strict constraint satisfaction. This is problematic in engineering systems where minor violations of governing laws can degrade the reliability and consistency of model predictions. In this work, we introduce KKT-Hardnet, a neural network architecture that enforces linear and nonlinear equality and inequality constraints up to machine precision. It leverages a differentiable projection onto the feasible region by solving Karush-Kuhn-Tucker (KKT) conditions of a distance minimization problem. Furthermore, we reformulate the nonlinear KKT conditions via a log-exponential transformation to construct a sparse system with linear and exponential terms. We apply KKT-Hardnet to nonconvex pooling problem and a real-world chemical process simulation. Compared to multilayer perceptrons and PINNs, KKT-Hardnet achieves strict constraint satisfaction. It also circumvents the need to balance data and physics residuals in PINN training. This enables the integration of domain knowledge into machine learning towards reliable hybrid modeling of complex systems.
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