# RTS Smoother-Guided Learning of Physics-Based Neural Differential Models

arXiv: https://arxiv.org/abs/2607.15180 · HTML: https://mlsanity.com/paper/2607.15180/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 27 references that are in the map.
- First posted on arXiv 16 Jul 2026.

## Builds on (most cited first, 19 of 19)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Universal Differential Equations for Scientific Machine Learning](https://mlsanity.com/paper/2001.04385/) (2020)
- [GRU-ODE-Bayes: Continuous modeling of sporadically-observed time series](https://mlsanity.com/paper/1905.12374/) (2019)
- [Latent ODEs for Irregularly-Sampled Time Series](https://mlsanity.com/paper/1907.03907/) (2019)
- [KalmanNet: Neural Network Aided Kalman Filtering for Partially Known Dynamics](https://mlsanity.com/paper/2107.10043/) (2021)
- [Lipschitz Continuity in Model-based Reinforcement Learning](https://mlsanity.com/paper/1804.07193/) (2018)
- [On Numerical Integration in Neural Ordinary Differential Equations](https://mlsanity.com/paper/2206.07335/) (2022)
- [Unsupervised Learned Kalman Filtering](https://mlsanity.com/paper/2110.09005/) (2021)
- [RTSNet: Learning to Smooth in Partially Known State-Space Models (Preprint)](https://mlsanity.com/paper/2110.04717/) (2021)
- [Learning the Hodgkin-Huxley Model with Operator Learning Techniques](https://mlsanity.com/paper/2406.02173/) (2024)
- [Learning Physics Informed Neural ODEs With Partial Measurements](https://mlsanity.com/paper/2412.08681/) (2024)
- [Recognition Models to Learn Dynamics from Partial Observations with Neural ODEs](https://mlsanity.com/paper/2205.12550/) (2022)
- [Neural Extended Kalman Filters for Learning and Predicting Dynamics of Structural Systems](https://mlsanity.com/paper/2210.04165/) (2022)
- [Learning Governing Equations of Unobserved States in Dynamical Systems](https://mlsanity.com/paper/2404.18572/) (2024)
- [Hybrid Neural Network Augmented Physics-based Models for Nonlinear Filtering](https://mlsanity.com/paper/2204.06471/) (2022)
- [Learning with Imperfect Models: When Multi-step Prediction Mitigates Compounding Error](https://mlsanity.com/paper/2504.01766/) (2025)
- [Cubature Kalman Filter Based Training of Hybrid Differential Equation Recurrent Neural Network Physiological Dynamic Models](https://mlsanity.com/paper/2110.06089/) (2021)
- [Identifying multi-compartment Hodgkin-Huxley models with high-density extracellular voltage recordings](https://mlsanity.com/paper/2506.20233/) (2025)
- [Interpretable Augmented Physics-Based Model for Estimation and Tracking](https://mlsanity.com/paper/2505.19953/) (2025)

## Abstract (arXiv)

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.