We present Prior-Fitted Scientific Foundation Models that target two complementary goals: (i) fast, accurate PDE solution operators and (ii) weather nowcasting from heterogeneous observations. For PDE solving, we tackle key limitations of many SciML methods—reliance on explicit equations, rigid input structures, and weak cross-PDE generalization—by developing a foundation-model-style neural solver. Inspired by large language models and prior-data fitted networks (PFNs), we propose PDE-PFN, a prior-fitted solver that approximates the posterior predictive distribution of PDE solutions via in-context Bayesian inference. Built on a PFN/Transformer backbone, PDE-PFN is pre-trained on low-cost approximate priors from physics-informed neural networks, enabling robust generalization across diverse 2D PDE families and zero-shot inference under noisy priors. For nowcasting, we target operational, real-world spatial resolution rather than downsampled benchmarks. We fuse radar, satellite imagery, and AWS/in-situ observations into compact tokens, and perform prompt-based forecasting: a shared spatiotemporal backbone conditions on prompts (e.g., lead time, region, variables, and available sensors) to predict future fields, remaining robust to missing channels and irregular sampling.