System-Anchored Knee Estimation for Low-Cost Context Window Selection in PDE Forecasting

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.

We introduce a neural network-based framework for learning time series estimators through a process we term decision-theoretic pretraining. Analysts specify a generative world, a distribution over data-generating processes, and a target decision objective. A neural network trained on stratified simulations from this world approximates the corresponding optimal decision rule, yielding a neural estimator that provides forecasts, parameter estimates, predictive intervals, or model-selection for zero-shot inference on previously unseen time series. The joint specification of the generative world and objective enables the estimators to directly approximate process-level, finite-sample properties: near-optimal risk, bias control, minimax performance, and uniform calibration. Our experiments demonstrate that these neural estimators can outperform traditional baselines such as maximum likelihood estimation and model selection via AICc, for the same model structural model classes. Furthermore, even when trained purely on simulations of structural models, they achieve competitive or state-of-the-art forecasting accuracy on major real-world benchmarks, compared with statistical, neural or large pre-trained models. We illustrate the framework by addressing two longstanding challenges: finite-sample bias and miscalibration in AR(p) models, and the forecast combination puzzle. These applications highlight the approach's main advantage: its ability to approximate solutions to analytically intractable or computationally prohibitive time series problems, including complex structural equations or optimality criteria. Ultimately, by enabling explicit control over decision-theoretic trade-offs, the framework equips analysts with highly efficient estimation tools tailored to their specific analytical needs.

Motivated by the success of large language models (LLMs) with broad generalizability and robustness to noisy or unreliable pre-training data, we explore in-context prediction of PDE solution fields using approximate pre-training data. In addition, inspired by the posterior predictive mean-based inference mechanism of the in-context learning in prior-data fitted networks (PFNs), we propose PDE-PFN, a prior-data fitted neural solver that approximates the posterior predictive mean of PDE solutions via in-context learning. PDE-PFN builds on a PFN architecture with self- and cross-attention mechanisms of Transformer and is pre-trained on noisy approximate solutions generated by physics-informed neural networks, serving as diverse but not necessarily exact priors. Through experiments on a range of two-dimensional PDEs, we demonstrate that PDE-PFN achieves empirical generalization across heterogeneous equations, robustness under noisy priors, and zero-gradient-update in-context inference capability. PDE-PFN achieves competitive or improved predictive accuracy relative to the evaluated baselines while supporting flexible context and query representations.