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

arXiv: https://arxiv.org/abs/2605.03247 · HTML: https://mlsanity.com/paper/2605.03247/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 28 references that are in the map.
- First posted on arXiv 5 May 2026.

## Builds on (most cited first, 19 of 19)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [Outrageously Large Neural Networks: The Sparsely-Gated Mixture-of-Experts Layer](https://mlsanity.com/paper/1701.06538/) (2017)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Convolutional LSTM Network: A Machine Learning Approach for Precipitation Nowcasting](https://mlsanity.com/paper/1506.04214/) (2015)
- [The frontier of simulation-based inference](https://mlsanity.com/paper/1911.01429/) (2019)
- [PDEBENCH: An Extensive Benchmark for Scientific Machine Learning](https://mlsanity.com/paper/2210.07182/) (2022)
- [Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1801.06637/) (2018)
- [PDE-Refiner: Achieving Accurate Long Rollouts with Neural PDE Solvers](https://mlsanity.com/paper/2308.05732/) (2023)
- [BayesFlow: Learning complex stochastic models with invertible neural networks](https://mlsanity.com/paper/2003.06281/) (2020)
- [In-Context Operator Learning with Data Prompts for Differential Equation Problems](https://mlsanity.com/paper/2304.07993/) (2023)
- [Unisolver: PDE-Conditional Transformers Towards Universal Neural PDE Solvers](https://mlsanity.com/paper/2405.17527/) (2024)
- [On the Benefits of Memory for Modeling Time-Dependent PDEs](https://mlsanity.com/paper/2409.02313/) (2024)
- [Approximate Bayesian Neural Operators: Uncertainty Quantification for Parametric PDEs](https://mlsanity.com/paper/2208.01565/) (2022)
- [Semi-supervised Invertible Neural Operators for Bayesian Inverse Problems](https://mlsanity.com/paper/2209.02772/) (2022)
- [DISCO: learning to DISCover an evolution Operator for multi-physics-agnostic prediction](https://mlsanity.com/paper/2504.19496/) (2025)
- [Zebra: In-Context Generative Pretraining for Solving Parametric PDEs](https://mlsanity.com/paper/2410.03437/) (2024)

## Abstract (arXiv)

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.