Universal Differential Equations for Stable Multi-Step Volatility Time Series Forecasting

Abstract

Neural differential equations such as Neural ODEs, Neural CDEs, and Universal Differential Equations (UDEs) model temporal evolution as a continuous-time flow rather than a fixed-step recurrence. Even for regularly sampled data, this formulation differs fundamentally from discrete-time architectures: it learns smooth vector fields governing instantaneous rates of change, reducing discretization bias and improving long-horizon stability. We present a systematic study of Universal Differential Equations for financial volatility forecasting, a domain characterized by regime shifts, heavy tails, and jump discontinuities. UDEs extend Neural ODEs by embedding mechanistic structure within learned dynamics, using neural networks to parameterize coefficients in partially known differential equations instead of learning the system purely from data. Our UDE variants incorporate volatility’s empirical regularities while retaining neural flexibility for regime adaptation. Across market regimes, they outperform both continuous-time baselines and discrete-time models, achieving higher accuracy and greater long-horizon stability while remaining interpretable. These results suggest that UDEs grounded in mechanistic structure and neural flexibility offer a principled route to stable, interpretable multi-step forecasting in nonstationary domains.