Posterior-First Neural PDE Simulation: Inferring Hidden Problem State from a Single Field
For researchers in neural PDE simulation, this work provides a principled framework to handle ambiguity from single observations, improving rollout accuracy.
The paper addresses the problem of neural PDE simulators that receive only a single observed field at deployment, where deterministic predictors collapse distinct latent states. They propose posterior-first simulation, inferring a posterior over problem state before prediction, reducing rollout nRMSE from 0.175 to 0.132 on PDEBench tasks, closing 59.4% of the oracle gap.
Neural PDE simulators often receive only a single observed field at deployment. In this setting, a field-to-future predictor can collapse distinct latent problem states into the same deterministic interface, losing the ambiguity needed for reliable rollout and downstream decisions. We propose posterior-first neural PDE simulation: first infer a posterior over the minimal task-sufficient problem state, then condition prediction on that posterior. The resulting theory connects the object, the learning target, and the failure mode: Bayes downstream values factor through this posterior, refinement labels make it learnable by proper scoring rules, and deterministic collapse incurs an ambiguity barrier whenever the true posterior is non-Dirac. Synthetic exact-ambiguity experiments show that point-versus-posterior gaps track the predicted barrier. On metadata-hidden PDEBench tasks, posterior recovery reduces pooled rollout nRMSE from 0.175 to 0.132, closing 59.4% of the direct-to-oracle gap. These results suggest that single-observation neural PDE simulation should be posterior-first rather than monolithic field-to-future prediction.