这篇论文用RTS平滑器引导神经网络学缺失的微分方程，部分观测下就能恢复动力学，还能保持模型解释性。
本文提出混合神经-物理框架，将已知ODE组分保留、未知组分用神经网络表示。方法分两阶段：先用RTS平滑器从部分观测推断潜在状态，再用平滑轨迹通过反向传播学习神经网络参数。在包含线性、非线性和刚性动力学的基准系统上评估，该方法能从不完整测量中学习缺失ODE组件，同时保持可解释性，并改善潜在状态重建和长时预测。
RTS Smoother-Guided Learning of Physics-Based Neural Differential Models
Ordinary differential equations (ODEs) are widely used to model dynamical systems in physics, biology, neuroscience, and physiology, but in many applications some equations of the dynamics are unknown and only a subset of the state variables are measured. We propose a hybrid neural--physics framework in which the known components of the ODE are kept explicit and the missing components are represented by a neural network. The proposed method consists of two stages where we alternate between state and parameter estimation and iterate until a predetermined criterion is met. Specifically, in the first step, we treat the model parameters as being known and we infer the latent states from the available measurements using a Rauch--Tung--Striebel (RTS) smoother. In the second stage, we treat the smoothed trajectories as being known and use them to estimate the neural networks' parameters through backpropagation. We evaluate the method on benchmark systems spanning linear, nonlinear, and stiff dynamics under partial state observation. Across these settings, the proposed method learns missing ODE components from incomplete measurements while exploiting and retaining interpretable mechanistic structure and improving latent-state reconstruction and long-horizon prediction.