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

NeurIPS 2026 · Probabilistic and scientific modeling › Dynamical systems · approach: Latent-space control and planning · 0 citations (OpenAlex)

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Proposes posterior-first neural PDE simulation, inferring a posterior over minimal task-sufficient problem states from one observed field before conditioning rollout, with theory on Bayes factorization, proper scoring, and a point-latent ambiguity barrier, validated on a synthetic exact-ambiguity benchmark and metadata-hidden PDEBench.

Problem

Neural PDE simulators receiving only a single observed field at deployment collapse distinct hidden problem states into one deterministic prediction, losing ambiguity needed for reliable rollout and decisions.

Approach

Insert an explicit posterior over a task-sufficient latent problem state between the observed field and the rollout predictor, trained with proper scoring rules, and show deterministic point latents incur an irreducible ambiguity barrier when the true posterior is non-Dirac.

Key results

• On metadata-hidden PDEBench, pooled nRMSE drops from 0.1747 (Direct-Point) to 0.1319 for Posterior-z, closing 59.4% of the direct-to-oracle gap (oracle 0.1027).
• On the synthetic benchmark, Point-z vs Posterior-z gap tracks the ambiguity barrier (ratio 0.89-0.90 across ambiguous regimes); future NLL falls from 1.18 to 1.02.
• Mean hidden-metadata probe score rises from 0.628 (direct) to 0.800 (posterior-z), approaching oracle 0.901.

Surprising findings

• Replacing the summary-conditioning interface by an explicit posterior mixture changes results only marginally (synthetic G_rem 0.381 to 0.365) at about 5.6x cost.

Limitations

• Experiments use a countable, supervised semantic refinement; continuous hidden states and weakly supervised posteriors are untested.
• Rollout consumes a posterior summary embedding rather than an exact mixture over conditional simulators.
• Posterior-to-utility bound depends on task-family and stability assumptions.

Experimental setup

Datasets: PDEBench Gray-Scott-style reaction-diffusion synthetic generator

Benchmarks: PDEBench (DR, DS, SW, INS) metadata-hidden protocol synthetic exact-ambiguity benchmark

Metrics: nRMSE GapClose G_rem future NLL hidden-metadata probe accuracy/R^2 ambiguity barrier

Baselines: Direct-Point Direct-Dist Direct-MTL Direct-Large Point-z RichContext-Posterior-z ABNO DeepEns-Direct GateMoE-Direct

Builds on: FNO PDE-Refiner PDEBench Approximate Bayesian Neural Operators

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