这篇论文提出OrthoReg，解决了混合建模中神经网络容易学走符号部分的问题，实验效果明显，写代码的朋友可以看看。
OrthoReg提出了一种正交正则化方法，直接惩罚符号组件与神经组件的重叠，防止符号结构被神经残差吸收。相比标准L2正则化，该方法在符号组件通过稀疏发现学习时仍能保持互补分解。在部分库不匹配的基准动力系统上，OrthoReg改善了符号恢复准确性和分布外泛化性能。
OrthoReg: Orthogonal Regularization for Hybrid Symbolic-Neural Dynamical Systems
Dynamical systems are fundamental to modeling the natural world, yet modeling them involves a persistent trade-off: manually prescribed mechanistic models are interpretable by design but often overly simplistic and misspecified; in contrast, flexible data-driven neural methods lack physical insight. Hybrid modeling aims for the best of both worlds by combining a prescribed or symbolic, physics-based component with a flexible neural network. A critical challenge, however, is that the neural component may relearn mechanistic parts, yielding redundant and uninterpretable models, especially when the symbolic structure itself is discovered from data. Existing methods based on standard $L^2$ regularization rely on a projection argument that breaks when the symbolic component is learned through sparse discovery, allowing the neural augmentation to overlap with symbolic structure. We introduce \textbf{OrthoReg} (Orthogonal Regularization), which directly penalizes overlap between the symbolic and neural components, preventing symbolic structure from being absorbed by the neural residual. This yields a complementary decomposition: the symbolic part captures what the library can express, and the neural part captures what remains. On benchmark dynamical systems with partial library mismatch, OrthoReg improves symbolic recovery and out-of-distribution behavior.