Neural Conditional Simulation for PDE Refinement

Refining a cheap coarse PDE solution to high fidelity is a conditional simulation problem, not a regression one. Under squared-error loss the optimal predictor is the conditional mean, which by the law of total variance discards all conditional variability and so cannot serve as a realization of the fine field. For Gaussian fields the answer is classical. Kriging supplies the conditional mean, and Matheron's construction adds an innovation drawn from the Schur-complement covariance to produce an exact sample in one pass. The fields that matter in practice, such as turbulence, stochastically forced dynamics, and precipitation, break these assumptions with innovations that are input-dependent, heteroscedastic, and heavy-tailed. We introduce Neural Conditional Simulation (NCS), which keeps the classical recipe but learns the innovation. NCS is initialized at the exact Gaussian simulator, trained end to end with strictly proper scoring rules, and samples through a coarse-to-fine cascade at one network pass per level. Because the Gaussian simulator is embedded as a zero-correction fallback and reported as a baseline, every result isolates the nonlinear conditional-simulation gain, which highlights the improvement that learning adds beyond exact Gaussian conditioning. On the near-Gaussian Allen-Cahn and Navier-Stokes benchmarks, this gain is small, as expected, and NCS matches the calibration of the best classical baseline to within a few percent. On the GARD-LENS precipitation ensemble, where the Gaussian assumptions break, NCS is the only method that is globally variance-calibrated at competitive CRPS and spectral error (Fortin-corrected $\sigma /s=0.96$): across $132$ noise-schedule settings of the diffusion baselines, every setting with $\sigma /s\ge 0.9$ has higher CRPS and more than twice the spectral error. The heavy precipitation tail, however, is under-reproduced by NCS trained with the energy score ($\kappa /{\kappa }_{\mathrm{data}}=0.17$); tuned consistency models reach it only while under-dispersed ($\sigma /s\le 0.57$). Our supply-versus-reach diagnostic attributes this deficit to the objective rather than the source. A scale-free flatness term closes $93\%$ of an analogous deficit on a controlled system. On GARD-LENS, where the deficit depends on amplitude, an amplitude-conditional objective recovers most of the tail while keeping calibration and spectrum within their guards. With the same backbone and no more parameters than CorrDiff, NCS matches multi-step diffusion baselines such as EDM and CorrDiff in accuracy and proper score while using two network passes instead of $35$ which reduces per-member wall-clock time by $17$-$36\times$.