发表机构
Politecnico di Milano（米兰理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究提出潜动力学网络扩展，通过自动解码和元学习推断初始潜状态，实现可变初始条件下PDE解算子的高效、分辨率无关学习，并在多种物理现象中验证了准确性。
AI 中文摘要
在多查询场景中，数据驱动的代理模型为模拟由偏微分方程（PDEs）控制的物理系统提供了一种高效替代高保真求解器的方法。在此背景下，潜动力学网络（LDNet）最近在预测时空系统响应方面表现出卓越性能，它将神经常微分方程与非线性降维相结合。然而，原始公式假设初始条件固定，这限制了其在许多系统从不同起始状态演化的实际应用中的适用性。在本工作中，我们克服了这一限制，同时保留了原始LDNet的端到端训练过程及其无编码器特性，这保持了其对空间分辨率和网格拓扑的固有独立性。我们直接从一小部分早期时间观测中推断初始潜状态，将潜状态初始化视为一个适应问题，并研究了两种策略：一种自动解码公式和一种元学习方法，其中初始潜状态作为任务特定的上下文变量。我们展示了所提出方法在多种物理现象中的准确性，涵盖对流扩散、流体动力学和固体力学。元学习显著加速了潜状态推断，并诱导出更平滑、条件更好的优化景观，且自发地将潜空间组织成反映底层动力学物理意义特征的结构化表示。基于坐标的解码器使得能够从空间子采样数据中训练，同时在推断时恢复高分辨率解场。所得到的方法为具有可变初始条件的时间相关PDE的多查询模拟提供了一个高效且分辨率无关的代理建模框架。
英文摘要
In many-query scenarios, data-driven surrogate models provide an efficient alternative to high-fidelity solvers for simulating physical systems governed by Partial Differential Equations (PDEs). In this context, the Latent Dynamics Network (LDNet) has recently demonstrated remarkable performance in predicting the response of spatio-temporal systems, combining Neural Ordinary Differential Equations with nonlinear dimensionality reduction. However, the original formulation assumes a fixed initial condition, limiting its applicability to many real-world applications where a system evolves from varying starting states. In this work, we overcome this limitation while keeping the end-to-end training procedure of the original LDNet and its encoder-free nature, which preserves its intrinsic independence from spatial resolution and grid topology. We infer the initial latent state directly from a small set of early-time observations, treating latent-state initialization as an adaptation problem, and investigate two strategies: an auto-decoding formulation and a meta-learning approach in which the initial latent state acts as a task-specific context variable. We demonstrate the accuracy of the proposed methods across diverse physical phenomena, spanning advection-diffusion, fluid dynamics, and solid mechanics. Meta-learning markedly accelerates latent-state inference and induces smoother, better-conditioned optimization landscapes, and spontaneously organizes the latent space into a structured representation that reflects physically meaningful features of the underlying dynamics. The coordinate-based decoder enables training from spatially subsampled data while recovering high-resolution solution fields at inference. The resulting approach provides an efficient and resolution-independent surrogate modeling framework for many-query simulations of time-dependent PDEs with varying initial conditions.
