Learning Multiplicative-Time Operators for Long-Horizon PDE Prediction

Long-horizon prediction of dissipative PDEs challenges learned fixed-step solvers: solutions change scale while rollout length grows with physical time. We formulate prediction as learning a multiplicative-time evolution operator. For equations with known scaling covariance, normalising space and amplitude makes a fixed ratio of physical times correspond to an autonomous transition of the full state. Our multiplicative-time renormalisation-group evolution method (MTRG) learns this transition from early-time trajectories and composes it with conserved-mass correction to reach distant horizons at logarithmic depth. The basic method fixes the ratio; a ratio-conditioned extension represents a range of intervals. We establish the exact composition law and derive a conditional error recurrence separating approximation, discretisation and propagation. Tests on diffusion and Burgers equations evaluate accuracy across distant horizons and the contributions of conservation and composition consistency. Comparisons with analytic profiles identify regimes where evolving the full state remains useful beyond the supervision window, before the limiting shape becomes the better predictor. MTRG turns known scaling structure into a reusable learned transition for this finite-time evolution.