The 'signature method' refers to a collection of feature extraction techniques for multimodal sequential data, derived from the theory of controlled differential equations. Variations exist as many authors have proposed modifications to the method, so as to improve some aspect of it. Here, we introduce a generalised signature method that contains these variations as special cases, and groups them conceptually into augmentations, windows, transforms, and rescalings. Within this framework we are then able to propose novel variations, and demonstrate how previously distinct options may be combined. We go on to perform an extensive empirical study on 26 datasets as to which aspects of this framework typically produce the best results. Combining the top choices produces a canonical pipeline for the generalised signature method, which demonstrates state-of-the-art accuracy on benchmark problems in multivariate time series classification. * Equal contribution Preprint. Under review.

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.

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JAX and PyTorch are two popular Python autodifferentiation frameworks. JAX is based around pure functions and functional programming. PyTorch has popularised the use of an object-oriented (OO) class-based syntax for defining parameterised functions, such as neural networks. That this seems like a fundamental difference means current libraries for building parameterised functions in JAX have either rejected the OO approach entirely (Stax) or have introduced OO-tofunctional transformations, multiple new abstractions, and been limited in the extent to which they integrate with JAX (Flax, Haiku, Objax). Either way this OO/functional difference has been a source of tension. Here, we introduce 'Equinox', a small neural network library showing how a PyTorch-like class-based approach may be admitted without sacrificing JAX-like functional programming. We provide two main ideas. One: parameterised functions are themselves represented as 'PyTrees', which means that the parameterisation of a function is transparent to the JAX framework. Two: we filter a PyTree to isolate just those components that should be treated when transforming ('jit', 'grad' or 'vmap'-ing) a higher-order function of a parameterised function -such as a loss function applied to a model. Overall Equinox resolves the above tension without introducing any new programmatic abstractions: only PyTrees and transformations, just as with regular JAX. Equinox is available at https://github.com/patrick-kidger/equinox.