Augmented Neural ODEs (1904.01681v3)

Abstract: We show that Neural Ordinary Differential Equations (ODEs) learn representations that preserve the topology of the input space and prove that this implies the existence of functions Neural ODEs cannot represent. To address these limitations, we introduce Augmented Neural ODEs which, in addition to being more expressive models, are empirically more stable, generalize better and have a lower computational cost than Neural ODEs.

• The paper demonstrates that standard NODEs are limited by topology preservation, preventing them from representing complex function mappings.
• The paper introduces Augmented Neural ODEs, which augment the state space to enable richer function representation without sacrificing model invertibility.
• Empirical results show that ANODEs require fewer function evaluations and offer improved training stability and generalization across tasks.

Augmented Neural ODEs: Enhancing the Expressive Power of Continuous Models

The concept of Neural Ordinary Differential Equations (NODEs) marks an intriguing synthesis between neural networks and differential equations, specifically exploring the continuous limit of discrete deep learning models such as Residual Networks (ResNets). Since their introduction, NODEs have been positioned as promising frameworks for a variety of applications, including continuous-time data modeling and efficient normalizing flows. However, a pivotal limitation arises from their inherent characteristic that limits their ability to express certain classes of functions. This paper posits that such limitations stem from NODEs' preservation of the input space topology, which fundamentally restricts them from representing functions necessary for some practical tasks.

To address these significant limitations, the researchers propose Augmented Neural ODEs (ANODEs). This new model presents an extension to NODEs by augmenting the dimensional space in which the ordinary differential equation (ODE) operates, thus enabling the representation of more complex functions. Such an extension not only allows ANODEs to overcome expressiveness constraints observed in standard NODEs but also facilitates enhanced model stability, improved generalization performance, and reduced computational requirements in practice.

Key Contributions and Findings

1. Expressiveness Limitation of NODEs: The paper offers a detailed analysis of the limitations inherent to NODEs resulting from their property of preserving input space topology. The authors prove that NODEs cannot represent arbitrary functions due to their inability to learn mappings requiring changes in topological features, such as discontinuities or singularity-inducing transformations.
2. Introduction of Augmented Neural ODEs: By increasing the dimensionality of the learning space, ANODEs provide a significant extension of expressive power. These augmented models leverage additional dimensions to maneuver around the topological constraints faced by NODEs, thereby increasing the complexity of functions they can effectively represent.
3. Computational Efficiency: With the more expressive flows ANODEs can compute, the number of function evaluations (NFEs) required decreases drastically compared to standard NODEs. This reduction addresses one of the classical criticisms associated with NODEs around training efficiency and provides a pathway to adopt these models more broadly in computationally demanding tasks.
4. Better Generalization and Stability: Empirical results indicate that due to their ability to learn simpler, more natural flows, ANODEs exhibit robust training stability and generalized performance across various datasets, including common image datasets like MNIST and CIFAR-10. The introduction of augmentation serves not only as a method to increase expressiveness but also facilitates easier convergence and minimization of loss in complex data scenarios.
5. Comparison with ResNets: While ResNets intrinsically bypass some NODE limitations through discretization that allows trajectories to intersect, ANODEs achieve these intersection properties through augmentation. Hence, this method maintains the beneficial attributes of NODEs, such as invertibility and minimal parameterization, while allowing function representation beyond NODEs' capabilities.

Practical and Theoretical Implications

The integration of augmented dimensions into NODEs suggests a potential pathway for designing neural architectures that capture the advantages of continuous problem representations without succumbing to the broader misunderstandings or critiques related to topology preservation. This innovation proves beneficial for applications in normalizing flows and beyond, suggesting a possibility to revisit the implementation of NODEs in contexts demanding highly expressive yet efficient function representation.

Further research investigating the impact of augmentation on other NODE applications (e.g., continuous normalizing flows) or exploring automated augmentation learning methods could yield even broader advancements. An understanding of how ANODEs perform relative to more conventional architectures could profoundly influence neural network-based modeling in domains requiring fidelity to continuous mathematical representations, such as systems biology or control systems.

Overall, this paper presents a substantial advancement in overcoming expressiveness bottlenecks associated with continuous neural network models, by proposing a methodology that is theoretically sound, practically effective, and capable of being adopted in various domains of artificial intelligence.