标题： 神经期望算子 显示英文标题

标题： Neural Expectation Operators

摘要： 本文介绍了\textbf{度学习}，一种通过非线性期望建模模糊性的范式。 我们将神经期望算子定义为后向随机微分方程（BSDEs）的解，其驱动项由神经网络参数化。 主要数学贡献是针对满足状态变量$y$局部利普希茨条件和其鞅分量$z$二次增长的 BSDEs 的严格适定性定理。 该结果避免了经典的全局利普希茨假设，适用于常见的神经网络架构（例如，具有 ReLU 激活函数的架构），并且适用于指数可积的终端数据，这是此设置下的精确条件。 我们的主要创新在于在二次 BSDEs 的深度理论的抽象且通常限制性强的假设与机器学习世界之间建立一个建设性的桥梁，证明这些条件可以通过具体且可验证的神经网络设计来满足。 我们提供了通过架构设计强制执行关键公理性质（如凸性）的建设性方法。 该理论扩展到完全耦合的前向-后向 SDE 系统的分析以及大规模相互作用粒子系统的渐近分析，在此我们建立了大数定律（混沌传播）和中心极限定理。 这项工作为在模糊性下的数据驱动建模提供了基础数学框架。 显示英文摘要

摘要： This paper introduces \textbf{Measure Learning}, a paradigm for modeling ambiguity via non-linear expectations. We define Neural Expectation Operators as solutions to Backward Stochastic Differential Equations (BSDEs) whose drivers are parameterized by neural networks. The main mathematical contribution is a rigorous well-posedness theorem for BSDEs whose drivers satisfy a local Lipschitz condition in the state variable $y$ and quadratic growth in its martingale component $z$. This result circumvents the classical global Lipschitz assumption, is applicable to common neural network architectures (e.g., with ReLU activations), and holds for exponentially integrable terminal data, which is the sharp condition for this setting. Our primary innovation is to build a constructive bridge between the abstract, and often restrictive, assumptions of the deep theory of quadratic BSDEs and the world of machine learning, demonstrating that these conditions can be met by concrete, verifiable neural network designs. We provide constructive methods for enforcing key axiomatic properties, such as convexity, by architectural design. The theory is extended to the analysis of fully coupled Forward-Backward SDE systems and to the asymptotic analysis of large interacting particle systems, for which we establish both a Law of Large Numbers (propagation of chaos) and a Central Limit Theorem. This work provides the foundational mathematical framework for data-driven modeling under ambiguity.