Random Controlled Differential Equations

Abstract

We introduce a training-efficient framework for time-series learning in which large randomly parameterized controlled and rough differential equations act as continuous-time reservoirs. These random dynamical systems map input paths to rich path-dependent representations, while only a linear readout layer is trained, yielding 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 discretisation, using log-signatures to capture higher-order temporal interactions while remaining stable and efficient. We prove that in the infinite-width limit, these models induce 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 superior performance. These methods provide a practical alternative to explicit signature computations, retaining their inductive bias while benefiting from the efficiency of random features. Code is publicly available at: https://github.com/FrancescoPiatti/RandomSigJax.