ConsensusCover: Adaptive Input Selection for Neural Operators under Distribution Shift

Training neural operators requires deciding which candidate partial differential equation (PDE) inputs merit expensive solution labels. Under distribution shift, the best selection geometry depends on both the equation and the operator architecture: target-distribution coverage is stronger in five of six DeepONet/FNO settings, whereas field information is stronger for Darcy/FNO. We introduce ConsensusCover, an adaptive acquisition rule that resolves this conflict without target solutions or equation-specific switching. At each round, ConsensusCover ranks candidate inputs using two independent views: Fourier field modes and the current operator's readout Jacobian. Rank-consistent overlap beyond chance enters the next batch, while the remaining selections preserve target-distribution coverage through local probability transfer. We evaluate Darcy flow, Allen–Cahn and viscous Burgers with DeepONet and FNO. In a separate confirmation with 20 paired trials per setting and 32–256 labelled simulations, ConsensusCover cuts the worst-case gap in out-of-distribution learning-curve error from 55.44% for target coverage and 30.92% for field information to 6.62%. The paired-bootstrap 95% upper bound for this gap is 8.55%, and the geometric-mean gap across all six settings is 2.07%. The agreement signal activates most strongly on Darcy/FNO, showing how information consensus adapts PDE input selection across neural operators.