PDE-PFN: Prior-Data Fitted Neural PDE Solver

Abstract: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.

Machine Learning (cs.LG)

arXiv:2610.13334 [cs.LG]

(or arXiv:2610.13334v1 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2610.13334 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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