时间记忆多输出PINN动态不确定性量化验证
Time-Memory Multi-Output PINNs for Dynamic Uncertainty Quantification
摘要: 针对一维线性加性噪声随机热方程，构建时间统计记忆增强的多输出物理信息神经网络(TC-MO-PINN)框架。在无显式时间记忆基线模型的基础上，引入上一时间层均值记忆(B2)和均值–标准差联合记忆(B3)，结合教师继承、残差修正与时间连续性约束。以闭式解析均值、标准差和分位数为基准，从分布中心、尺度校准、区间覆盖、分位数结构和误差传播稳定性多维度评价各模型。结果表明，均值记忆有效降低分布中心误差，均值–标准差联合记忆进一步增强均值传播稳定性；去除解析统计矩锚定后，联合记忆模型仍优于基线，表明时间统计记忆对随机动力系统不确定性量化具有独立贡献。
Abstract: A time-statistical-memory enhanced multi-output physics-informed neural network (TC-MO-PINN) framework is proposed for a one-dimensional linear stochastic heat equation with additive noise. Starting from a memory-free baseline (B1), a previous-time mean memory model (B2) and a joint mean—standard-deviation memory model (B3) are introduced, combined with teacher model inheritance, residual correction, and temporal continuity constraints. Using closed-form analytical mean, standard deviation and quantiles as benchmarks, models are evaluated on distribution center accuracy, scale calibration, interval coverage, quantile agreement, and error propagation stability. Results show that mean memory reduces distribution center error, while standard-deviation joint memory further suppresses late-time mean drift. A control experiment without analytical moment anchoring confirms that the joint memory model is still superior to the baseline, and time-statistical memory provides an independent contribution to stable uncertainty quantification of stochastic dynamical systems.
文章引用：袁瑞阳, 吕艳. 时间记忆多输出PINN动态不确定性量化验证[J]. 统计学与应用, 2026, 15(7): 127-139. https://doi.org/10.12677/sa.2026.157155

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