KKT-HardNet 教程：把物理约束嵌入神经网络做混合建模与控制
A Tutorial on Physics Constrained Machine Learning for Hybrid Modeling, Optimization, and Control
Texas A&M 把物理约束硬塞进神经网络的开源框架，推理快还能严格满足约束，搞工程建模和优化控制的可以看看。
Texas A&M 团队发布 KKT-HardNet 的分步教程，这是一个物理约束机器学习（PCML）框架，通过微分投影让神经网络在推理时严格满足代数约束。框架支持线性和非线性的等式与不等式约束，并允许自定义网络骨干和投影机制。为提升端到端训练效率，它利用隐函数定理对投影层做反向传播，避免高开销的自动微分，还提供精确的降阶 Fischer-Burmeister 重构和原始-对偶内点法作为备选投影求解策略。在约束满足度和推理时间两项指标上，KKT-HardNet 比基线方法好多个数量级，代码和基准数据集已在 GitHub 开源。
A Tutorial on Physics Constrained Machine Learning for Hybrid Modeling, Optimization, and Control
Physics-constrained machine learning (PCML) integrates domain knowledge, first-principles, and physical constraints into AI/ML models. In this work, we present a step-by-step tutorial of a modular PCML framework, KKT-HardNet, and illustrate how it could be used for hybrid modeling, process simulation, optimization and model predictive control in a single platform. KKT-HardNet combines fast inference capabilities of neural networks with guaranteed satisfaction of algebraic constraints through differential projection. It can handle heterogeneous data and enforce both linear and nonlinear equality and inequality constraints. The modular design supports custom neural-network backbones and user-defined projection mechanisms for PCML. To enable efficient end-to-end training, KKT-HardNet exploits implicit function theorem to backpropagate through the projection layer. This avoids computationally expensive automatic differentiation. This work also introduces an exact reduced-order Fischer-Burmeister reformulation and a primal-dual interior-point method as an alternative projection solve strategy. We also provide a diverse set of benchmarking problems for PCML in engineering applications. KKT-HardNet achieves orders of magnitude better performance in terms of both constraint satisfaction and inference time. The KKT-HardNet code and the benchmark datasets are available at https://github.com/souls-tamu/kkt-hardnet .