PhyGRU: a Physics-Biased Variant for Gated Recurrent Unit

The modeling of dynamical systems often takes place in regimes where data are limited and physical knowledge is only partially available. In such conditions, simplified physical models are often insufficient, while fully data-driven recurrent networks struggle to generalize. Moreover, many existing hybrid solutions require substantial architectural modifications or intervention, making them difficult to integrate into existing models or pipelines without major restructuring. To address this, PhyGRU is introduced, a physics-biased variant of the Gated Recurrent Unit in which the learned candidate state is replaced by an explicit time integration of a parametric physical model, optionally augmented by low-dimensional latent dynamics. The standard GRU gating mechanism is preserved, enabling interpolation between the previous state and a physics-informed candidate. PhyGRU targets scenarios with limited data and partial physical knowledge, where neither simplified physical models nor fully data-driven recurrent networks are sufficient. The approach requires only minimal architectural modifications relative to a standard GRU, at the cost of increased per-step computational time due to explicit state integration. Experiments, on three controlled dynamical systems of increasing complexity, under time-invariant and time-varying parameter regimes, indicate that PhyGRU can yield more physically coherent predicted trajectories when the assumed physical prior is reasonably aligned with the true dynamics, while performance degrade as the mismatch between the prior and the true system increases.