The stochastic inverse problem (StocIP), which aims to align push-forward and observed output distributions by estimating probability distributions of unknown system inputs, often faces optimization challenges and the curse of dimensionality. A novel deep network called StocIPNet which comprises an affine-embedded reparameterization subnetwork (ReparNet) and a complex system metamodeling subnetwork (MetaNet) is proposed to alleviate these issues. The ReparNet subnetwork embeds the affine transformation to convert the statistical parameters of the physical random vector into learnable weights and biases, effectively implementing the reparameterization trick by separating random and deterministic elements in the stochastic sampling operation to preserve differentiability. In parallel, the MetaNet subnetwork offers a computationally efficient alternative to time-consuming forward solvers, facilitating the generation of push-forward distributions. The entire StocIPNet utilizes the kernel maximum mean discrepancy (MMD) as a distribution-free loss function, quantifying the discrepancy between push-forward and observed output distributions. By leveraging the ReparNet’s advantage of reformulating the sampling process as a differentiable transformation and combining two subnetworks seamlessly, the StocIP is reconfigured into a pure network training paradigm preserving differentiability perfectly, which allows for direct modeling and efficient inference of uncertainty within the network using automatic differentiation, backpropagation and gradient-based optimization methods, enabling ease of scaling to high-dimensional problems. The proposed framework has been theoretically demonstrated to be equivalent to the maximum likelihood method, ensuring its solid probabilistic interpretable foundation. The proposed framework is applied to perform stochastic model updating on a numerical and an experimental structure, which effectively demonstrates the framework’s remarkable effectiveness and high efficiency in treating high-dimensional StocIPs.

中文翻译：

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StocIPNet：一种新颖的概率可解释网络，具有仿射嵌入重参数化层，适用于高维随机逆问题


随机反问题（StocIP）旨在通过估计未知系统输入的概率分布来对齐前推和观察到的输出分布，通常面临优化挑战和维数灾难。为了缓解这些问题，提出了一种名为 StocIPNet 的新型深度网络，该网络由仿射嵌入重参数化子网络（ReparNet）和复杂系统元建模子网络（MetaNet）组成。 ReparNet 子网络嵌入仿射变换，将物理随机向量的统计参数转换为可学习的权重和偏差，通过在随机采样操作中分离随机元素和确定性元素来保持可微分性，从而有效地实现重新参数化技巧。同时，MetaNet 子网络为耗时的前向求解器提供了一种计算高效的替代方案，有助于生成前推分布。整个StocIPNet利用内核最大平均差异（MMD）作为无分布损失函数，量化前推和观察到的输出分布之间的差异。通过利用 ReparNet 将采样过程重新表述为可微分变换并无缝组合两个子网络的优势，StocIP 被重新配置为完美保留可微分性的纯网络训练范式，从而允许使用自动微分直接建模并有效推断网络内的不确定性、反向传播和基于梯度的优化方法，可以轻松缩放高维问题。 所提出的框架已在理论上证明与最大似然法等效，确保了其坚实的概率可解释基础。所提出的框架应用于对数值和实验结构进行随机模型更新，有效地证明了该框架在处理高维StocIPs方面的显着有效性和高效率。