Posterior-First Neural PDE Simulation: Inferring Hidden Problem State from a Single Field

2026-05-04

AI Summary

Neural PDE simulators operating from a single observation field suffer from degraded predictions and unreliable rollouts due to their neglect of latent state uncertainty. This work proposes a posterior-first neural PDE simulation framework that first infers a task-informed posterior distribution over latent states from a single observation and then performs conditional prediction based on this posterior. By introducing the Bayesian posterior as the core intermediate representation in neural PDE simulation, the method reveals the ambiguity barrier inherent in deterministic prediction when the true posterior is non-Dirac and establishes a theoretical connection between learnability and proper scoring rules. Evaluated on the hidden metadata task in PDEBench, the proposed approach reduces rollout nRMSE from 0.175 to 0.132, closing 59.4% of the performance gap to an oracle model.

Abstract

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.

Problem

• neural PDE simulation
• hidden problem state
• single observation
• ambiguity
• posterior inference

Innovation

• posterior-first
• neural PDE simulation
• ambiguity barrier
• task-sufficient state
• proper scoring rules