arXiv：cs.LG· Stefano Maria Pizzamiglio, Stefano Pagani, Francesco Regazzoni·· 4 小时前AI 评分32
用潜在动力学网络学习可变初始条件的 PDE 解算子
Learning PDE solution operators with variable initial conditions via Latent Dynamics Networks
AI 导读
研究者扩展了 Latent Dynamics Network（LDNet），使其支持可变初始条件，同时保留原有端到端训练和无编码器特性。方法从少量早期观测推断初始潜在状态，比较了自解码与元学习两种策略；元学习显著加速潜在状态推断并带来更平滑的优化景观，在平流扩散、流体动力学和固体力学等物理现象上验证了准确性。基于坐标的解码器支持从空间下采样数据训练，并在推理时恢复高分辨率解场。
正文
Abstract: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.
| Subjects: | Machine Learning (cs.LG); Numerical Analysis (math.NA) |
| Cite as: | arXiv:2610.08475 [cs.LG] |
| (or arXiv:2610.08475v1 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2610.08475 arXiv-issued DOI via DataCite (pending registration) |
Submission history
From: Stefano Maria Pizzamiglio [view email]
[v1]
Tue, 6 Oct 2026 14:55:39 UTC (7,916 KB)
来源：arXiv：cs.LG · arxiv.org