从观测数据中学习随机偏微分方程的随机动力学，无需噪声或初值信息。
Latent-Variable Learning of SPDEs via Wiener Chaos
- 用截断的威纳混沌展开结合谱伽辽金投影，将无限维方程简化为有限维隐变量系统。
- 在有界和无界一维空间上，合成数据实验表现优于现有方法。
- 适合研究含随机扰动的物理过程建模，如气候、流体模拟等场景。
我们研究从时空观测数据中学习线性随机偏微分方程（SPDEs）的解分布问题，其驱动为加性高斯白噪声。现有深度学习方法通常需假设已知驱动噪声或初始条件，或依赖无法捕捉内在随机性的确定性代理模型。本文提出一种结构化隐变量框架，仅需解的实现实例观测，即可学习潜在的随机驱动动力学。方法结合谱伽辽金投影与截断威纳混沌展开，实现确定性演化与随机扰动的可解释分离，将无限维SPDE约化为一组参数化的常微分方程，描述隐变量时序动态。通过变分学习联合推断隐动态与随机扰动，可在训练时不显式观测或模拟噪声的前提下恢复其统计结构。在合成数据上的实验表明，在相同建模假设下，该方法在有界与无界一维空间域均达到当前最优性能。
原文摘要 · Abstract (English)
We study the problem of learning the law of linear stochastic partial differential equations (SPDEs) with additive Gaussian forcing from spatiotemporal observations. Most existing deep learning approaches either assume access to the driving noise or initial condition, or rely on deterministic surrogate models that fail to capture intrinsic stochasticity. We propose a structured latent-variable formulation that requires only observations of solution realizations and learns the underlying randomly forced dynamics. Our approach combines a spectral Galerkin projection with a truncated Wiener chaos expansion, yielding a principled separation between deterministic evolution and stochastic forcing. This reduces the infinite-dimensional SPDE to a finite system of parametrized ordinary differential equations governing latent temporal dynamics. The latent dynamics and stochastic forcing are jointly inferred through variational learning, allowing recovery of stochastic structure without explicit observation or simulation of noise during training. Empirical evaluation on synthetic data demonstrates state-of-the-art performance under comparable modeling assumptions across bounded and unbounded one-dimensional spatial domains.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。