AI 中文总结
提出PHD-SF框架，结合SD-TPD分离策略与微分算子学习，实现无标签的时变参数化PDE加速，仅用1-2个参数案例即可完成准确跨参数解，成本低于传统PINN。
AI 中文摘要
时变参数化偏微分方程（PDE）的高效求解是计算科学与工程的核心。现有深度学习加速器横跨物理到数据的谱系：从物理一致性强但计算成本高的物理信息求解器，到推理高效但高度依赖高保真数据集的数据驱动代理。这些方法大多独立发展，通常仅通过显式解场标签关联。我们提出PHD-SF，一种用于加速时变参数化PDE的紧凑物理到数据谱系框架。通过将SD-TPD分离策略与微分算子学习相结合，PHD-SF可在统一的基于DON的架构内，于PIDON、HIDON和DIDON三种运行模式间生成、继承并丰富可复用的模型信息，包括空间特征、隐式动力学特征及空间微分算子。该设计实现了无标签谱系建模，无需预计算全场解标签或模式间的显式标签传递。PHD-SF还建立了分离求解、类超约简的增强及直接推理加速路径，降低了对大规模高保真数据生成及其离线成本的依赖。在四个基准时变PDE上的结果表明，仅利用初始物理信息阶段的1或2个代表性参数案例，即可实现准确的跨参数解和直接推理。总端到端成本低于训练传统PINN单个参数案例所需的成本，同时支持可复用的跨参数推理。因此，PHD-SF为多查询参数化PDE提供了一条高效的求解路径。
英文摘要
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.
Comments56 pages, 11 figures, 4 tables
