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

• URL: http://arxiv.org/abs/2605.03247v1
• Date: Tue, 05 May 2026 00:37:01 GMT
• Title: Posterior-First Neural PDE Simulation: Inferring Hidden Problem State from a Single Field
• Abstract summary: A field-to-future predictor can collapse distinct latent problem states into the same deterministic interface.<n>We propose posterior-first neural PDE simulation: first infer a posterior over the minimal task-sufficient problem state, then condition prediction on that posterior.
• Score: 5.24655241578805
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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