Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

ICODE: Modeling Dynamical Systems with Extrinsic Input Information

Zhaoyi Li, Wenjie Mei, Ke Yu, Yang Bai, Shihua Li

TL;DR

The paper introduces Input Concomitant Neural ODEs (ICODEs), a contraction-guaranteed neural ODE framework that directly incorporates real-time external inputs via an affine-in-input dynamics $\dot{x} = f_0(x) + \sum_j g_j(x) u_j$. It provides sufficient contraction conditions using a uniform metric $M(x) = L(x)^T L(x)$ and the matrix $R = L J(x,u) L^{-1} + \dot{L} L^{-1}$ to ensure exponential convergence of trajectories, regardless of initialization. Through extensive experiments on mechanical, electrical, chemical, and PDE systems, including 1D/2D heat conduction and R-F dynamics, ICODEs consistently outperform NODE-based baselines (NODE, CDE, ANODE), especially under atypical or rapidly changing inputs, with Transformer-enhanced variants further boosting performance. The work demonstrates a practical and robust approach for learning physically informed dynamical systems with external inputs, offering stability guarantees and scalability for applications in robotics, physics, and control engineering.

Abstract

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 \emph{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.

ICODE: Modeling Dynamical Systems with Extrinsic Input Information

TL;DR

The paper introduces Input Concomitant Neural ODEs (ICODEs), a contraction-guaranteed neural ODE framework that directly incorporates real-time external inputs via an affine-in-input dynamics . It provides sufficient contraction conditions using a uniform metric and the matrix to ensure exponential convergence of trajectories, regardless of initialization. Through extensive experiments on mechanical, electrical, chemical, and PDE systems, including 1D/2D heat conduction and R-F dynamics, ICODEs consistently outperform NODE-based baselines (NODE, CDE, ANODE), especially under atypical or rapidly changing inputs, with Transformer-enhanced variants further boosting performance. The work demonstrates a practical and robust approach for learning physically informed dynamical systems with external inputs, offering stability guarantees and scalability for applications in robotics, physics, and control engineering.

Abstract

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 \emph{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.

Paper Structure

This paper contains 18 sections, 2 theorems, 27 equations, 19 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Neural ODEs with External Input
  3. Experimental Results on Learning and Prediction of Physical Systems
  4. Single-link Robot
  5. DC-to-DC Converter
  6. Motion of Rigid Body
  7. Rabinovich-Fabrikant Equation
  8. Glycolytic-glycogenolytic Pathway Model
  9. Swing Equation
  10. Heat Conduction Equation
  11. One-Dimensional Case
  12. Two-Dimensional Case
  13. Further Statistical Studies
  14. Experiments on Input Role
  15. Input Span
  16. ...and 3 more sections

Key Result

Theorem 1

The NODEs eq:used_NDOE are contracting if there exists a uniformly positive definite metric $M(x) = L(x)^\top L(x)$, where $L(x) \in \mathbb{R}^{n \times n}$ is a smooth and invertible transformation, such that for the transformed system of $\delta y = L(x) \delta x$, we have for some constants $c>0$, where $I_n$ denotes the $n \times n$ identity matrix, and the vector $\delta x$ denotes the sta

Figures (19)

  • Figure 1: The schematic of ICODEs. Using the ODE integration methods, it illustrates how the state $x$ evolves through the external input $u$ and the neural networks (NNs) $k_1, \dots, k_m$, and the NN consisting of the subnetworks $f_1(\cdot)$, ..., $f_M(\cdot)$.
  • Figure 2: Comparison of training and prediction errors at each time step on the test set for different models on the task of the single-link robot. (a) Training loss in MSE under $u_3$. (b) Prediction loss in MSE under $u_3$. (c) Inputs $u$.
  • Figure 3: System trajectories of the single-link robot on phase plane (horizontal coordinate: $q$, and vertical coordinate: $\dot{q}$).
  • Figure 4: Comparison of training loss and prediction error across time steps for various models applied to the DC-to-DC converter task under two input conditions (top and bottom).
  • Figure 5: Comparison of prediction errors at each time step on the test set for different models on the task of a rigid body.
  • ...and 14 more figures

Theorems & Definitions (4)

  • Theorem 1
  • Proof 1: Proof of Theorem 1
  • Corollary 1
  • Proof 2: Proof of Corollary 1