Feedback Favors the Generalization of Neural ODEs

TL;DR

The paper tackles the generalization limitations of neural ODEs in continuous-time prediction by introducing feedback neural networks that real-time-correct latent dynamics. It first proves convergence for a linear feedback form, then learns a nonlinear neural feedback via domain randomization, yielding a two-DOF network that preserves nominal accuracy while enhancing robustness to unseen dynamics. The approach is validated through trajectory prediction of irregular objects and model predictive control of a quadrotor, where the feedback-augmented models outperform state-of-the-art baselines in both accuracy and adaptability under uncertainty. The work demonstrates tangible improvements in real-time adaptive control and continuous-time prediction, offering a practical pathway to deploy neural ODEs in dynamic, uncertain environments. Future directions include automated gain optimization and broader application to complex, real-world tasks.

Abstract

The well-known generalization problem hinders the application of artificial neural networks in continuous-time prediction tasks with varying latent dynamics. In sharp contrast, biological systems can neatly adapt to evolving environments benefiting from real-time feedback mechanisms. Inspired by the feedback philosophy, we present feedback neural networks, showing that a feedback loop can flexibly correct the learned latent dynamics of neural ordinary differential equations (neural ODEs), leading to a prominent generalization improvement. The feedback neural network is a novel two-DOF neural network, which possesses robust performance in unseen scenarios with no loss of accuracy performance on previous tasks.} A linear feedback form is presented to correct the learned latent dynamics firstly, with a convergence guarantee. Then, domain randomization is utilized to learn a nonlinear neural feedback form. Finally, extensive tests including trajectory prediction of a real irregular object and model predictive control of a quadrotor with various uncertainties, are implemented, indicating significant improvements over state-of-the-art model-based and learning-based methods.

Figures (23)

• Figure 1: Neural network architectures. Left: Neural ODE developed in chen2018neural. Right: Proposed feedback neural network.
• Figure 2: The learned latent dynamics are modified through accumulative evaluation errors to approach the truth one.
• Figure 3: The multi-step prediction.
• Figure 4: Prediction errors of the spiral curve with different levels of feedback gains and uncertainties to show practical implications of Theorem \ref{['thm1']} udner Assumption \ref{['error_dyna']}. The right image is a partial enlargement of the left one. The blue star denotes the case without uncertainty, and the uncertainty increases along both the left and right directions. When the gain is set as $0$, the feedback neural network will equal the neural ODE. The related simulation setup is detailed in Appendix \ref{['appen_gain']}.
• Figure 5: A toy example is presented to intuitively illustrate the developed linear feedback. The mission is to predict the future trajectory of a spiral curve with a given initial state $\{\bm{x}(t), \bm{y}(t)\}$. The neural ODE is trained on a given training set (a), yielding an approving learning result (c). Note that the pentagrams denote start points. The trained network is then transferred to a test set (b), which model is significantly different from the training one. With the linear feedback mechanism, the feedback neural network can achieve a better approximated accuracy of the change rate (e), in comparison with the neural ODE (d). As a result, a smaller multi-step prediction error (f) can be attained by benefiting from the feedback neural network. (g) shows that the noise amplification issue in multi-step prediction can be alleviated by the gain-decay strategy. (h) further presents the prediction results with different prediction steps $N$. $N$ in (f)-(g) is set as $50$.

Theorems & Definitions (4)

• Theorem 1
• proof
• Theorem 2
• proof