Advances in Neural Controlled Differential Equations
2026-07-06 • Machine Learning
Machine Learning
AI summaryⓘ
The authors study how to improve models called Neural Controlled Differential Equations (NCDEs), which are good at understanding continuous-time data but are usually slow and hard to train. They propose three approaches: one that speeds up training by approximating solutions more efficiently, another that simplifies the math by using linear functions to allow faster calculations, and a third that structures these linear functions for even better efficiency. Together, these methods make NCDEs much faster to train while keeping high accuracy on various time series tasks. Their work helps make continuous-time models more practical for real-world use.
Neural Controlled Differential EquationsContinuous-time modelingTime seriesLog-ODE methodLinear vector fieldNeural rough differential equationsDynamical systemsParallel computationTraining efficiencyExpressive power
Authors
Benjamin Walker
Abstract
Many real-world systems evolve continuously, yet most machine learning models interpret time series as discrete sequences. Continuous-time approaches instead treat time series as samples from an underlying input path, a formulation that naturally accommodates irregularly sampled or oversampled data. Among these, Neural Controlled Differential Equations (NCDEs) are a maximally expressive class of models that parametrise a vector field using a neural network and evolve their hidden state by solving a dynamical system driven by the input path. NCDEs typically use a non-linear vector field, so their expressive power and continuous-time flexibility come at the cost of a forward pass that is both computationally expensive and inherently sequential, limiting their scalability and practical applicability. This thesis advances the training and scalability of NCDEs through three complementary contributions. First, building on neural rough differential equations, Log-NCDEs apply the Log-ODE method to efficiently approximate an NCDE's solution during training, improving both computational speed and empirical performance. Second, Linear NCDEs replace the non-linear vector field with a linear one, enabling closed-form solutions and parallel-in-time computation without sacrificing theoretical expressivity. Third, Structured Linear NCDEs use structured linear vector fields to further enhance efficiency while maintaining theoretical expressiveness and empirical performance. Collectively, these methods reduce the time per training step for an NCDE by up to three orders of magnitude while achieving state-of-the-art performance across diverse time series benchmarks.