NeuralPDE: Learning Latent PDE Dynamics for Time-Series Forecasting

Time-series analysis extracts information from historical observations and predicts their future behavior. Many existing models make predictions only by learning historical data patterns, and the underlying evolution laws, which are often described by partial differential equations (PDEs) or other dynamical rules in real world physical systems, may be overlooked. We therefore propose NeuralPDE, which encodes historical observations to latent functions of an auxiliary variable and learns their evolution through an explicit advection–diffusion–reaction PDE on an auxiliary domain, with a learned readout then decodes the evolving functions to future predictions. NeuralPDE learns PDE-based evolution laws of complex systems in latent space to improve the stability and accuracy of multi-step forecasting. The roles of the transport and diffusion terms in NeuralPDE are theoretically verified through establishing energy dissipation for the linear component. We apply NeuralPDE on real-world time series and NeuralPDE demonstrates competitiveness in time-series multi-step forecasting tasks in comparison with existing models.