Are Partial Differential Equation Foundation Models Really Foundational? Controversies in Generalization, Refinement, and Scientific Trustworthiness

Foundation models are increasingly proposed as surrogates and discovery tools for PDE-governed systems, but it remains unclear when such ‘PDE foundation models’ are genuinely foundational rather than merely large, task-specific surrogates. This paper proposes a PDE-specific notion of foundation models as learned solution operators trained via predominantly physics-based self-supervision on broad PDE families and adapted to downstream tasks such as forward simulation, inverse problems, control, and uncertainty quantification. Within this framework, we formulate key controversies-generalization under distribution shift, the absence of systematic refinement mechanisms, physics and structure preservation, computational cost, uncertainty quantification, and reproducibility-as operator-theoretic and statistical questions about approximation, convergence, and error decomposition. These formulations delineate regimes where PDE foundation models are most promising (e.g., many-query, inverse, and multiscale scenarios) and where classical numerical methods remain preferable (few-query, safety-critical, or poorly sampled regimes), and they motivate community standards for baselines, benchmark PDE suites, and reproducible datasets and code to make claims about PDE foundation models independently auditable.