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

TL;DR

This paper introduces a posterior-first approach for neural PDE simulation, inferring a distribution over hidden states from a single observed field to improve prediction accuracy and reduce ambiguity.

Contribution

It proposes a novel posterior inference framework that enhances neural PDE simulators by capturing uncertainty over hidden states, addressing deterministic collapse issues.

Findings

Posterior recovery reduces pooled rollout nRMSE from 0.175 to 0.132.

Synthetic experiments confirm the predicted ambiguity barrier.

Posterior-first approach closes 59.4% of the direct-to-oracle gap on PDEBench.

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.