Cheap physics stops picking samples: PI-Init pretrains neural operators label-free, cutting Burgers error 53.3%

Cheap physical information need not choose which inputs get labeled; it can instead set the parameters from which a neural operator begins supervised training. The natural prior approach used physics residuals, uncertainty or feature scores to pick inputs for labeling, but acquisition experiments on Burgers showed these strategies held no consistent advantage over uniform random sampling at the same label budget. Random sampling reached the lowest mean test error under both fixed and variable viscosity on Burgers; across 960 pooled score-error pairs, normalized residual scores correlated with the probe model's current prediction error, showing that prediction difficulty is not labeling utility.

The method trains in two stages: pretrain on unlabeled PDE inputs with governing-equation residuals and initial or boundary constraints, then retain those parameters and reset the optimizer for supervised fine-tuning. The physics loss appears only in the label-free stage and is separated in time from the supervised loss, so the two objectives need no balancing and the operator architecture is unchanged. Across six PDEBench tasks, 2D Navier-Stokes and AirfRANS, with FNO as the primary architecture and WNO and LNO also tested, FNO error fell 53.3% on Burgers, 37.3% on 1D CFD, 62.7% on 1D Advection and 10.2% on 2D Navier-Stokes.

The gain concentrates in parameter directions that labeled data constrain weakly, so relative benefit grows as labels shrink. A linearized analysis splits prediction risk into an initialization-bias term and a label-variance term, and supervised optimization contracts initialization error only along directions with positive label curvature. On Burgers, PI-Init with 200 labeled trajectories had lower error than supervised-only training with 2,000, a 90% label reduction; label savings of about 80%, 70%, 25% and 33% were observed on 1D CFD, 2D Darcy Flow, AirfRANS and 2D Navier-Stokes respectively.

The demonstrated scope is eight PDE benchmarks, including geometry-dependent fluid dynamics, with the FNO, WNO and LNO architectures; pretraining uses exact or approximate governing-equation residuals, and supervised data come from uniformly sampled inputs with computed solutions. The method keeps the original operator architecture and inference operations, so it adds no inference complexity.

The analysis assumes a linearized operator model and a shared eigenbasis for physics and label curvature, and exact compatibility of finite-sample matrices is not generally guaranteed. When the physics objective is approximate, the benefit also depends on the mismatch between the physics minimizer and the supervised target, so stronger physics pretraining need not yield better downstream performance. Pretraining cost grows roughly linearly with cheap inputs and epochs, accounting for about 1.4%-8.1% of 500-epoch supervised training time and rising to 6.6%-39.8% with 50,000 inputs.