On Robustness of Neural Ordinary Differential Equations (1910.05513v4)

Abstract: Neural ordinary differential equations (ODEs) have been attracting increasing attention in various research domains recently. There have been some works studying optimization issues and approximation capabilities of neural ODEs, but their robustness is still yet unclear. In this work, we fill this important gap by exploring robustness properties of neural ODEs both empirically and theoretically. We first present an empirical study on the robustness of the neural ODE-based networks (ODENets) by exposing them to inputs with various types of perturbations and subsequently investigating the changes of the corresponding outputs. In contrast to conventional convolutional neural networks (CNNs), we find that the ODENets are more robust against both random Gaussian perturbations and adversarial attack examples. We then provide an insightful understanding of this phenomenon by exploiting a certain desirable property of the flow of a continuous-time ODE, namely that integral curves are non-intersecting. Our work suggests that, due to their intrinsic robustness, it is promising to use neural ODEs as a basic block for building robust deep network models. To further enhance the robustness of vanilla neural ODEs, we propose the time-invariant steady neural ODE (TisODE), which regularizes the flow on perturbed data via the time-invariant property and the imposition of a steady-state constraint. We show that the TisODE method outperforms vanilla neural ODEs and also can work in conjunction with other state-of-the-art architectural methods to build more robust deep networks.

Robustness of Neural Ordinary Differential Equations

The paper "On Robustness of Neural Ordinary Differential Equations" addresses the gap in understanding the robustness properties of Neural Ordinary Differential Equations (ODEs) in comparison to conventional Convolutional Neural Networks (CNNs). Neural ODEs have gained traction in various domains due to their continuous-time dynamical representation capability, which offers advantages like invertibility and parameter efficiency. However, their robustness against input perturbations remained unexplored until this paper.

Empirical Evaluation of Neural ODE Robustness

The authors conducted empirical evaluations focusing on the robustness of Neural ODE-based networks (ODENets) against various types of perturbations, including random Gaussian noise and adversarial attacks like FGSM and PGD. These experiments were conducted across multiple datasets including MNIST, SVHN, and a subset of ImageNet called ImgNet10.

Key Observations:

• Robustness Comparison: ODENets consistently outperformed comparable CNN models in robustness tests across all datasets and types of perturbations.
• Intrinsic Properties: The enhanced robustness of Neural ODEs emanates from the non-intersecting property of integral curves in continuous-time systems, which provides intrinsic regularization absent in CNNs.

Enhancing Robustness: Time-Invariant Steady Neural ODE (TisODE)

To further improve the robustness of Neural ODEs, the authors propose the Time-Invariant Steady Neural ODE (TisODE). This approach introduces a steady-state constraint into the neural ODE model, which further regularizes the flow and improves robustness against perturbations.

Methodology:

• Removal of Time Dependence: The dynamics function of the ODE is modified to be time-invariant, ensuring that the integral curve is horizontally translatable based on initial conditions.
• Steady-State Constraint: A regularization term is introduced into the loss function to stabilize the output, effectively bounding the deviation from neighborhood integral curves.

This modification demonstrated superior robustness in classification tasks compared to vanilla neural ODEs, particularly against stronger adversarial attacks.

Complementary Architectural Techniques

Additionally, the paper illustrates the versatility of TisODE by integrating it with existing architectural robustness techniques such as feature denoising and input randomization. In conjunction with these methods, TisODE further boosted model robustness, showcasing its effectiveness as a drop-in module for enhancing deep model reliability.

Implications and Future Directions

The findings have significant practical implications in deploying deep learning systems in applications demanding high reliability, such as healthcare and autonomous systems. The ability of Neural ODEs to inherently resist perturbations, complemented by the TisODE method, can inspire broader adoption of differential equation-based models in AI research.

Looking ahead, future research can explore optimizing neural ODEs for scalability in large datasets, exploring various differential equation solvers for improved performance, and integrating stochastic dynamics to expand their applicability in domains where noise and randomness are prevalent.

In summary, the paper expands the knowledge on the robustness of Neural ODEs and proposes an effective enhancement technique, positioning these models as strong contenders in building robust machine learning architectures.