Controlled Differential Equations

Controlled Differential Equations (CDEs) are a mathematical framework where the dynamics of a system are driven by an external control signal — $dz_t = f(z_t) , dX_t$ where $X_t$ is the control path, enabling neural network models that naturally handle irregular, streaming time series data.

How CDEs Work

• Control Path: The input time series $X$ is treated as a continuous path that "drives" the system.
• Dynamics: The hidden state $z_t$ evolves according to the response function $f$ applied to increments of $X$.
• Rough Path Theory: CDEs are grounded in rough path theory, providing rigorous mathematical foundations.
• Solution Map: The CDE solution is a continuous function of the input path — providing well-defined gradients.

Why It Matters

• Irregular Sampling: CDEs naturally handle irregularly sampled time series without interpolation or imputation.
• Streaming Data: State updates are driven by new data arrivals — natural for online/streaming applications.
• Mathematical Foundation: CDEs provide the theoretical underpinning for Neural CDEs and related architectures.

CDEs are dynamical systems driven by data streams — a mathematical framework where the input signal continuously drives the system evolution.