BROC: Block-Recursive Output Correction for Frozen Neural PDE Surrogates

Simulation-trained neural PDE surrogates often suffer systematic performance degradation when deployed on measured physical streams or subjected to shifted target dynamics. We introduce Block-Recursive Output Correction (BROC), a residual observer that calibrates a trained and frozen surrogate using matured sparse measurements. Unlike conventional test-time adaptation (TTA) methods that rely on iterative gradient steps, BROC updates a spatially shared temporal component error map via a recursive block, requiring neither backpropagation nor neural parameter updates. Theoretically, we prove the exact optimality of the recursive update and strict temporal non-anticipation. We also establish a finite-sample certificate for full-field improvement when the shared residual signal dominates delay drift, probe error, and measurement noise. Across 20 system–surrogate–checkpoint cells in RealPDEBench, BROC achieves the best accuracy in every cell, reducing MSE by 32.56% relative to the Frozen surrogate and outperforming recent adaptation baselines by 23.84%–29.84%. Under controlled shifts of physical parameters across 12 PDEBench cells, it removes 38.31%–80.47% of Frozen error and outperforms recent baselines by 25.05%–65.87%. Moreover, its analytic calibration core executes in 5.317 ms on RealPDEBench and 4.348 ms on PDEBench, yielding respective 5.02×–11.97× and 7.74×–17.08× speedups over iterative output adapters. Compared with the baseline requiring surrogate re-execution, BROC achieves speedups of 74.90× and 26.75×. Comprehensive stress tests show that BROC remains effective under sparse, dynamic, delayed, and noisy sensing, maintains substantial gains in autoregressive rollouts, supports protective gating, and improves physical fidelity.