一种用于无限维主方程的测度一致算子学习方法
A Measure-Consistent Operator Learning Method for Infinite-Dimensional Master Equations
AI总结：
针对平均场博弈中无限维主方程，提出测度一致算子学习方法（MCOL），利用经验测度与对称池化结构近似值函数及其测度导数，避免额外积分网格，数值实验验证了准确性。
摘要：
平均场博弈论中的主方程描述了依赖于时间、状态（空间）和群体分布的反馈值函数。其数值近似具有挑战性，因为未知函数定义在概率测度空间上，且方程涉及内在测度导数和非局部群体项。本文提出了一种用于无限维主方程的测度一致算子学习方法（MCOL）。群体分布由经验测度表示，并通过对称池化结构编码，使得网络输入直接由表示测度的粒子构建。相同的粒子用于非局部残差项的经验求积，避免了额外的求积网格或辅助积分点。一个关键特征是残差中出现的内在导数由定义值函数近似的相同测度依赖表示所诱导。因此，值函数、其测度导数和经验残差都绑定到一个共同的测度表示上，导致结构耦合的值-导数近似。我们还引入了一个误差分解，将神经近似误差与经验离散误差分开。对几个主方程的数值实验表明，MCOL能够准确近似值函数、内在测度导数和反馈量，并且在输入测度变化时保持鲁棒性。
英文摘要：
Master equations in mean field game theory characterize feedback value functions that depend on time, state (space), and the population distribution. Their numerical approximation is challenging because the unknown is defined on a space of probability measures and the equation involves intrinsic measure derivatives and nonlocal population terms. This paper proposes a measure-consistent operator learning method (MCOL) for infinite-dimensional master equations. The population distribution is represented by an empirical measure and encoded through a symmetric pooling structure, so that the network input is built directly from the particles representing the measure. The same particles are used in the empirical quadrature of the nonlocal residual terms, avoiding additional quadrature grids or auxiliary integration points. A key feature is that the intrinsic derivative appearing in the residual is induced by the same measure-dependent representation that defines the approximation of the value function. Consequently, the value function, its measure derivative, and the empirical residual are tied to a common measure representation, leading to a structurally coupled value-derivative approximation. We also introduce an error decomposition separating neural approximation error from empirical discretization error. Numerical experiments on several master equations show that MCOL accurately approximates the value function, intrinsic measure derivatives, and feedback quantities, and remains robust under changes in the input measures.
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