We introduce a training-efficient framework for time-series learning that combines random features with controlled differential equations (CDEs). In this approach, large randomly parameterized CDEs act as continuous-time reservoirs, mapping input paths to rich representations. Only a linear readout layer is trained, resulting in fast, scalable models with strong inductive bias. Building on this foundation, we propose two variants: (i) Random Fourier CDEs (RF-CDEs): these lift the input signal using random Fourier features prior to the dynamics, providing a kernel-free approximation of RBF-enhanced sequence models; (ii) Random Rough DEs (R-RDEs): these operate directly on rough-path inputs via a log-ODE discretization, using log-signatures to capture higher-order temporal interactions while remaining stable and efficient. We prove that in the infinite-width limit, these model induces the RBF-lifted signature kernel and the rough signature kernel, respectively, offering a unified perspective on random-feature reservoirs, continuous-time deep architectures, and path-signature theory. We evaluate both models across a range of time-series benchmarks, demonstrating competitive or state-of-the-art performance. These methods provide a practical alternative to explicit signature computations, retaining their inductive bias while benefiting from the efficiency of random features.

Backpropagation through (neural) SDE solvers is traditionally approached in two ways: discretise-then-optimise, which offers accurate gradients but incurs prohibitive memory costs due to storing the full computational graph (even when mitigated by checkpointing); and optimise-then-discretise, which achieves constant memory cost by solving an auxiliary backward SDE, but suffers from slower evaluation and gradient approximation errors. Algebraically reversible solvers promise both memory efficiency and gradient accuracy, yet existing methods such as the Reversible Heun scheme are often unstable under complex models and large step sizes. We address these limitations by introducing a novel class of stable, near-reversible Runge--Kutta schemes for neural SDEs. These Explicit and Effectively Symmetric (EES) schemes retain the benefits of reversible solvers while overcoming their instability, enabling memory-efficient training without severe restrictions on step size or model complexity. Through numerical experiments, we demonstrate the superior stability and reliability of our schemes, establishing them as a practical foundation for scalable and accurate training of neural SDEs.

One of the most notable examples of such systems is that of stochastic spiking neural networks (SSNNs).

Stochastic optimal control problems typically involve a dynamical system described by a stochastic differential equation (SDE) dx t = b (t, x t, u t)dt + σ (t, x t) dB t, t [0, T], (1.1) in Stratonovich or Itˆ o form, where x t is the state of the system at time t, u t is the control input, b is the drift, σ is the diffusion, B is a Brownian motion, T is the final time, and consist of optimizing an objective E[null T 0 f ( t, x t, u t)dt + g (x T)] over a set of control input trajectories subject to state and control constraints. By now, a rich literature on stochastic optimal control is available, with optimality conditions characterized by the dynamic programming principle as Hamilton-Jacobi-Bellman (HJB) partial differential equations (PDEs) [6-8], and by the Pontryagin Maximum Principle (PMP) as forward-backward stochastic differential equations (FBSDEs) [8-11]. For problems with linear dynamics and linear-quadratic costs, both approaches lead to tractable solutions characterized by stochastic Riccati equations [7,12,13]. However, for general nonlinear problems, solving HJB-PDEs or FBSDEs remains computationally challenging for high-dimensional state spaces, despite recent progress [14-17]. In practice, an effective approach consists of optimizing over a class of solutions u θ t parameterized by finitely-many parameters θ R k [18,19] (see [20,21] for machine learning applications). However, restricting solutions to a finite-dimensional space may obscure the structure of solutions and lead to suboptimality.

Building on the functional-analytic framework of operator-valued kernels and un-truncated signature kernels, we propose a scalable, provably convergent signature-based algorithm for a broad class of high-dimensional, path-dependent hedging problems. We make minimal assumptions about market dynamics by modelling them as general geometric rough paths, yielding a fully model-free approach. Furthermore, through a representer theorem, we provide theoretical guarantees on the existence and uniqueness of a global minimum for the resulting optimization problem and derive an analytic solution under highly general loss functions. Similar to the popular deep hedging approach, but in a more rigorous fashion, our method can also incorporate additional features via the underlying operator-valued kernel, such as trading signals, news analytics, and past hedging decisions, closely aligning with true machine-learning practice.

These notes expound the recent use of the signature transform and rough path theory in data science and machine learning. We develop the core theory of the signature from first principles and then survey some recent popular applications of this approach, including signature-based kernel methods and neural rough differential equations. The notes are based on a course given by the two authors at Imperial College London.

Signature kernels are at the core of several machine learning algorithms for analysing multivariate time series. The kernel of two bounded variation paths (such as piecewise linear interpolations of time series data) is typically computed by solving a Goursat problem for a hyperbolic partial differential equation (PDE) in two independent time variables. However, this approach becomes considerably less practical for highly oscillatory input paths, as they have to be resolved at a fine enough scale to accurately recover their signature kernel, resulting in significant time and memory complexities. To mitigate this issue, we first show that the signature kernel of a broader class of paths, known as \emph{smooth rough paths}, also satisfies a PDE, albeit in the form of a system of coupled equations. We then use this result to introduce new algorithms for the numerical approximation of signature kernels. As bounded variation paths (and more generally geometric $p$-rough paths) can be approximated by piecewise smooth rough paths, one can replace the PDE with rapidly varying coefficients in the original Goursat problem by an explicit system of coupled equations with piecewise constant coefficients derived from the first few iterated integrals of the original input paths. While this approach requires solving more equations, they do not require looking back at the complex and fine structure of the initial paths, which significantly reduces the computational complexity associated with the analysis of highly oscillatory time series.