[Submitted on 5 May 2026 (v1), last revised 27 Oct 2026 (this version, v2)]
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, a task-to-target-to-interface paradigm: the declared downstream tasks determine the minimal task-sufficient problem state to infer, and the posterior over that state conditions the rollout and downstream queries. This posterior can be supplied by amortized posterior estimators, including simulation-based inference with learned summaries; our contribution is the target, the interface, and their theory. We show that Bayes downstream values factor through this posterior, proper scoring rules on refinement labels identify it as their population-risk minimizer, 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. Matched controls attribute the gains to task-aligned targeting and posterior deployment, and the gains persist with a latent-diffusion backbone. These results suggest that single-observation neural PDE simulation should be posterior-first rather than monolithic field-to-future prediction.
Submission history
From: Wenshuo Wang [view email]
[v1] Tue, 5 May 2026 00:37:01 UTC (199 KB)
[v2] Tue, 27 Oct 2026 12:03:14 UTC (195 KB)
