ICODE: Modeling Dynamical Systems with Extrinsic Input Information

2024-11-21

arXiv.org

AI Summary

Modeling dynamical systems driven by nonsmooth or piecewise-continuous external inputs remains challenging for conventional neural ODEs, which typically treat inputs as static parameters or discrete-time features, failing to capture rigorous continuous-time input-state coupling. Method: We propose Input Concomitant Neural ODEs (ICODEs), explicitly embedding time-varying input signals into the right-hand side of the neural ODE differential equation to enforce strict continuous-time co-evolution of states and inputs. We establish the first sufficient contraction condition for input-driven neural ODEs, guaranteeing global exponential convergence of all trajectories to a unique equilibrium under arbitrary bounded inputs. Our framework integrates neural differential equations, Lyapunov-based stability analysis, and differentiable ODE solvers. Results: Evaluated on six real-world systems—including a single-link robot, DC–DC converter, rigid-body dynamics, Rabinovich–Fabrikant system, glycolysis pathway, and heat equation—ICODEs achieve significantly improved prediction accuracy and cross-input generalization, while demonstrating exceptional robustness to anomalous inputs.

Technology Category

• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Intelligent Robots: State Estimation
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Responsible Web: Machine-in-the-loop, human agency and autonomy

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.

Problem

• Modeling dynamical systems with external inputs
• Incorporating real-time input information into learning
• Ensuring model convergence to a fixed point

Innovation

• Incorporates real-time input information into models
• Ensures model trajectories converge to fixed points
• Validated on diverse real-world dynamic systems