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.