Learning Neural PDE Solvers from Physics Terms That Matter

Neural PDE solvers are commonly trained on high-fidelity simulations of the full governing equations, which is costly and limits generalization to unseen physical regimes. Cheap simulations of simplified PDEs can help, but prior methods often fix a single basic form and assume the same physical term remains dominant. This is unreliable: the relative importance of advection, diffusion, reaction, and other operators can change with system parameters and evolving states.We present a physics-guided framework that estimates each PDE operator's contribution and uses Adaptive Dominant-Operator Routing to select the basic form of the current regime, then generates inexpensive auxiliary simulations from it. When no operator is clearly dominant, Confidence-Aware Multi-Fidelity Training down-weights cheap data according to routing confidence. Experiments on 1D advection-diffusion, 2D reaction-diffusion, and 2D Navier-Stokes cover distinct dominance regimes. The method jointly decides which simplified physics to simulate and how much to trust it, targeting better data efficiency, long-term prediction, and out-of-distribution generalization.