On Neural Differential Equations (2202.02435v1)

Abstract: The conjoining of dynamical systems and deep learning has become a topic of great interest. In particular, neural differential equations (NDEs) demonstrate that neural networks and differential equation are two sides of the same coin. Traditional parameterised differential equations are a special case. Many popular neural network architectures, such as residual networks and recurrent networks, are discretisations. NDEs are suitable for tackling generative problems, dynamical systems, and time series (particularly in physics, finance, ...) and are thus of interest to both modern machine learning and traditional mathematical modelling. NDEs offer high-capacity function approximation, strong priors on model space, the ability to handle irregular data, memory efficiency, and a wealth of available theory on both sides. This doctoral thesis provides an in-depth survey of the field. Topics include: neural ordinary differential equations (e.g. for hybrid neural/mechanistic modelling of physical systems); neural controlled differential equations (e.g. for learning functions of irregular time series); and neural stochastic differential equations (e.g. to produce generative models capable of representing complex stochastic dynamics, or sampling from complex high-dimensional distributions). Further topics include: numerical methods for NDEs (e.g. reversible differential equations solvers, backpropagation through differential equations, Brownian reconstruction); symbolic regression for dynamical systems (e.g. via regularised evolution); and deep implicit models (e.g. deep equilibrium models, differentiable optimisation). We anticipate this thesis will be of interest to anyone interested in the marriage of deep learning with dynamical systems, and hope it will provide a useful reference for the current state of the art.

• The paper establishes neural differential equations as a unified framework that merges dynamical systems with deep learning for modeling complex, irregular data.
• The paper categorizes NDEs into neural ODEs, CDEs, and SDEs, each tailored to address specific challenges in time series, generative modeling, and stochastic processes.
• The paper demonstrates practical applications in fields like physics, finance, and biology while highlighting efficient numerical methods and open-source tools for scalable training.

Overview of "On Neural Differential Equations"

The thesis "On Neural Differential Equations" by Patrick Kidger presents a comprehensive exploration of the integration of differential equations and neural networks—a field experiencing heightened academic and practical interest. This work seeks to frame neural differential equations (NDEs) as an overview of dynamical systems and deep learning, offering avenues for applications in generative modeling, time series, dynamical systems, and more. As deep learning continues to evolve, the intersection with traditional mathematical modeling forms the crux of this thesis, promising both theoretical enrichment and practical innovations.

Key Elements

1. Neural Differential Equations (NDEs): Kidger's work discusses the foundational premise that neural networks and differential equations share a common ground. Many neural network architectures, like residual networks, can be viewed as discretizations of differential equations. NDEs help bridge the theoretical constructs of dynamical models with the data-driven efficiency of deep learning, enabling the modeling of complex, irregular data while leveraging advanced function approximation capabilities.
2. Variants of NDEs: The thesis categorizes NDEs into neural ordinary differential equations (neural ODEs), neural controlled differential equations (neural CDEs), and neural stochastic differential equations (neural SDEs). Each type addresses unique challenges:
   • Neural ODEs are explored for their applicability in tasks like hybrid modeling and conditional generation, linking them to residual networks.
   • Neural CDEs focus on continuous updates, offering significant advantages for handling irregular time series and evolving hidden states in alignment with new data.
   • Neural SDEs compensate for stochasticity, providing a framework for generative modeling with a probabilistic underpinning, mimicking how randomness is integrated into classical dynamical systems models.
3. Applications and Implications: NDEs are applied across various domains, from physics-informed models that enhance traditional analytical approaches with data-driven insights to finance and biological systems that benefit from faster, adaptive simulations. These models offer the dual benefit of flexibility through neural networks and structure through differential equations, promoting advancements in engineering, scientific computing, and even artistic domains with generative capacities.
4. Theoretical and Numerical Aspects: The discourse extends into the numerical solutions and computational techniques essential for utilizing NDEs effectively. This includes discussions on the efficiency of adjoint methods for backpropagation through differential equations, the importance of solver choice, and the use of reversible solvers and checkpointing for memory-efficient training.
5. Code and Software Development: The thesis emphasizes the pivotal role of open-source software development in democratizing access to NDEs. Tools like Diffrax and torchdiffeq are highlighted as vital components supporting the community in extending work on differential equations and modeling in machine learning frameworks.

Future Directions and Challenges

• Hybrid Models: Further development in NDEs lies in enhancing hybrid models that seamlessly blend domain-specific knowledge with machine learning to capture intricate behaviors absent from pure data-driven or theoretical models.
• Universal Differential Equations: Expanding upon neural augmentations in traditional equations could drive a new approach to modeling, where universal differential equations cater to complex systems, learning parts of the dynamics directly from data.
• Symbolic Regression: The potential to render NDE models interpretable through symbolic regression remains a promising yet challenging facet, with implications for expanding human understanding of learned representations.
• Neural PDEs: A future arena with boundless promise is neural PDEs, which extend these concepts to partial differential equations presenting opportunities in spatial-temporal modeling.

Conclusion

Patrick Kidger’s thesis is a substantial contribution to the field of neural differential equations, bridging gaps between deep learning and classical mathematical modeling. The document serves not only as a pivotal survey of an emergent field but also as a guide for future research paths, highlighting the prospect of NDEs to innovate within both scientific and applied machine learning communities.