Derived Fields Increase the Recoverable Information in Budgeted Neural PDE Surrogate

For neural surrogates of time-dependent partial differential equations (PDEs), storage and memory budgets require training data to be obtained by coarsening and quantizing very-high-resolution numerical simulations. Fine-scale information can thus be lost before it reaches the model. We analyze this effect in periodic incompressible Navier-Stokes: quantization spreads error evenly over the retained modes, so the finest retained band, which holds the least energy, is the most corrupted, and velocity and vorticity are distorted differently under the same operator. Building on this analysis, we propose Derived-Field Optimization (DerivOpt), which chooses which primitive and derived fields to carry from a supplied library, and how many bits each receives, by exactly minimizing a closed-form posterior risk under a calibrated channel model. This optimization problem always has an optimum and requires no knowledge of the governing equation. On PDEBench, DerivOpt not only lowers the pooled rollout nRMSE but also substantially reduces the fine-band error. These gains are already present in the predictor-free input state and add to those of a multiscale architecture. Which fields are carried, and at what precision, is thus an optimizable design variable upstream of the predictor that determines how much information a fixed budget leaves recoverable.