A Label-Free Physics-to-Data Acceleration Framework for Parametric Time-Dependent PDEs with Latent-Space Differential-Operator Learning

Efficient solution of time-dependent parametric partial differential equations (PDEs) is central to computational science and engineering. Existing deep-learning-based accelerators span a physics-to-data spectrum, from physics-informed solvers with strong physical consistency but high computational cost to data-driven surrogates with efficient inference but strong dependence on high-fidelity datasets. These methods have largely evolved in isolation and are often connected only through explicit solution-field labels. We propose PHD-SF, a compact physics-to-data spectrum framework for accelerating time-dependent parametric PDEs. By combining an SD-TPD separation strategy with differential-operator learning, PHD-SF allows reusable model information, including spatial features, latent dynamical features, and spatial differential operators, to be generated, inherited, and enriched across three operating modes, PIDON, HIDON, and DIDON, within a unified DON-based architecture. This design enables label-free spectrum modeling without precomputed full-field solution labels or explicit label transfer among modes. PHD-SF further establishes a separated-solving, hyper-reduction-like enrichment, and direct-inference acceleration path, reducing the dependence on large-scale high-fidelity data generation and its offline cost. Results on four benchmark time-dependent PDEs show accurate cross-parameter solution and direct inference using only one or two representative parameter cases in the initial physics-informed stage. The total end-to-end cost is lower than that required to train a conventional PINN for a single parameter case, while supporting reusable cross-parameter inference. PHD-SF therefore provides an efficient solution path for many-query parametric PDEs.