从分布到随机过程：测度值映射的神经逼近
From Distributions to Stochastic Processes: Neural Approximation of Measure-Valued Maps
另 4 家 · 查看机构详情
- Shanghai Center for Mathematical Sciences（上海数学中心）
- Department of Mathematics, Fudan University（复旦大学数学系）
- Center for Applied Mathematics, Fudan University（复旦大学应用数学中心）
- Alibaba Group（阿里巴巴集团）
- Shanghai, China（（此处为城市信息，非机构，按要求不提取））
- Hangzhou, China（（此处为城市信息，非机构，按要求不提取））
- Fudan University（复旦大学）
机构由 AI 辅助整理，请以论文原文为准。
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中文总结 AI 辅助
本文提出分布到分布及随机过程映射的神经逼近理论框架，并在奥恩斯坦-乌伦贝克首达时间与杜芬振子响应路径预测中验证其有效性，优于基线模型。
中文摘要 AI 辅助
当输入和输出由样本群体而非单个观测表示时，学习概率分布之间的映射自然出现。我们为分布到分布的学习发展了一个逼近论框架，并将其扩展到随机过程之间的映射。对于有限维概率律的$W_2$-紧族上的连续算子，我们利用有限律统计量、一个单纯形值神经映射和一个共享原子输出支撑集，在2-Wasserstein度量下建立了均匀神经逼近，这保证了有效的概率测度。我们进一步通过有限秩正交投影将该原理扩展到可分希尔伯特空间上的概率律。这些结果确立了学习概率律之间变换（而非确定性向量或函数）的表征可行性。为展示实际相关性，我们研究了两个自然定义在分布层面的问题：奥恩斯坦-乌伦贝克过程的首达时间分布预测和杜芬振子的非线性响应路径律。由于该理论与模型无关且比任何单一实际架构更广泛，实验使用任务自适应的神经模型而非精确复现理论构造。在两个问题中，所提出的模型均优于固定特征MLP基线和分布空间核回归。这些实验通过展示随机系统中分布到分布变换的实际可学习性，补充了理论。
英文摘要
Learning mappings between probability distributions arises naturally when inputs and outputs are represented by populations of samples rather than individual observations. We develop an approximation-theoretic framework for distribution-to-distribution learning and extend it to mappings between stochastic processes. For continuous operators on $W_2$-compact families of finite-dimensional probability laws, we establish uniform neural approximation in the 2-Wasserstein metric using finite law statistics, a simplex-valued neural map, and a shared atomic output support that guarantees valid probability measures. We further extend this principle to probability laws on separable Hilbert spaces through finite-rank orthogonal projections. These results establish the representational feasibility of learning transformations between probability laws rather than deterministic vectors or functions. To demonstrate practical relevance, we study two problems naturally defined at the distribution level: prediction of first-passage-time distributions for an Ornstein--Uhlenbeck process and nonlinear response-path laws of a Duffing oscillator. Because the theory is model-agnostic and broader than any single practical architecture, the experiments use task-adapted neural models rather than reproducing the theoretical construction exactly. In both problems, the proposed models outperform a fixed-feature MLP baseline and distribution-space kernel regression. These experiments complement the theory by demonstrating the practical learnability of distribution-to-distribution transformations in random systems.
