# ICODE: Modeling Dynamical Systems with Extrinsic Input Information

arXiv: https://arxiv.org/abs/2411.13914 · HTML: https://mlsanity.com/paper/2411.13914/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 39 references that are in the map.
- First posted on arXiv 21 Nov 2024.

## Builds on (most cited first, 22 of 22)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Neural Controlled Differential Equations for Irregular Time Series](https://mlsanity.com/paper/2005.08926/) (2020)
- [Lagrangian Neural Networks](https://mlsanity.com/paper/2003.04630/) (2020)
- [Augmented Neural ODEs](https://mlsanity.com/paper/1904.01681/) (2019)
- [On Neural Differential Equations](https://mlsanity.com/paper/2202.02435/) (2022)
- [Stable Neural ODE with Lyapunov-Stable Equilibrium Points for Defending Against Adversarial Attacks](https://mlsanity.com/paper/2110.12976/) (2021)
- [LyaNet: A Lyapunov Framework for Training Neural ODEs](https://mlsanity.com/paper/2202.02526/) (2022)
- [Vid-ODE: Continuous-Time Video Generation with Neural Ordinary Differential Equation](https://mlsanity.com/paper/2010.08188/) (2020)
- [OVERT: An Algorithm for Safety Verification of Neural Network Control Policies for Nonlinear Systems](https://mlsanity.com/paper/2108.01220/) (2021)
- [Will Artificial Intelligence supersede Earth System and Climate Models?](https://mlsanity.com/paper/2101.09126/) (2021)
- [Physics-informed neural networks for the shallow-water equations on the sphere](https://mlsanity.com/paper/2104.00615/) (2021)
- [Stable Neural Flows](https://mlsanity.com/paper/2003.08063/) (2020)
- [Stochastic Physics-Informed Neural Ordinary Differential Equations](https://mlsanity.com/paper/2109.01621/) (2021)
- [Stabilized Neural Differential Equations for Learning Dynamics with Explicit Constraints](https://mlsanity.com/paper/2306.09739/) (2023)
- [Neural network stochastic differential equation models with applications to financial data forecasting](https://mlsanity.com/paper/2111.13164/) (2021)
- [A Neural ODE Interpretation of Transformer Layers](https://mlsanity.com/paper/2212.06011/) (2022)
- [Robust Classification using Contractive Hamiltonian Neural ODEs](https://mlsanity.com/paper/2203.11805/) (2022)
- [Data-driven Methods Applied to Soft Robot Modeling and Control: A Review](https://mlsanity.com/paper/2305.12137/) (2023)
- [Towards Complex Dynamic Physics System Simulation with Graph Neural ODEs](https://mlsanity.com/paper/2305.12334/) (2023)
- [Learning Stabilizable Dynamical Systems via Control Contraction Metrics](https://mlsanity.com/paper/1808.00113/) (2018)
- [ControlSynth Neural ODEs: Modeling Dynamical Systems with Guaranteed Convergence](https://mlsanity.com/paper/2411.02292/) (2024)

## Built on by (most cited first, 1 of 1)

- [Fine-Tuning Flow Matching via Maximum Likelihood Estimation of Reconstructions](https://mlsanity.com/paper/2510.02081/) (2025)

## Abstract (arXiv)

Learning models of dynamical systems with external inputs, which may be, for example, nonsmooth or piecewise, is crucial for studying complex phenomena and predicting future state evolution, which is essential for applications such as safety guarantees and decision-making. In this work, we introduce Input Concomitant Neural ODEs (ICODEs), which incorporate precise real-time input information into the learning process of the models, rather than treating the inputs as hidden parameters to be learned. The sufficient conditions to ensure the model's contraction property are provided to guarantee that system trajectories of the trained model converge to a fixed point, regardless of initial conditions across different training processes. We validate our method through experiments on several representative real dynamics: Single-link robot, DC-to-DC converter, motion dynamics of a rigid body, Rabinovich-Fabrikant equation, Glycolytic-glycogenolytic pathway model, and heat conduction equation. The experimental results demonstrate that our proposed ICODEs efficiently learn the ground truth systems, achieving superior prediction performance under both typical and atypical inputs. This work offers a valuable class of neural ODE models for understanding physical systems with explicit external input information, with potentially promising applications in fields such as physics and robotics. Our code is available online at https://github.com/EEE-ai59/ICODE.git.