Neural networks are a commonly used approach to replace physical models with computationally cheap surrogates. Parametric uncertainty quantification can be included in training, assuming that an accurate prior distribution of the model parameters is available. Here we study the common opposite situation, where direct screening or random sampling of model parameters leads to exhaustive training times and evaluations at unphysical parameter values. Our solution is to decouple uncertainty quantification from network architecture. Instead of sampling network weights, we introduce the model-parameter distribution as an input to network training via Markov chain Monte Carlo (MCMC). In this way, the surrogate achieves the same uncertainty quantification as the underlying physical model, but with substantially reduced computation time. The approach is fully agnostic with respect to the neural network choice. In our examples, we present a quantile emulator for prediction and a novel autoencoder-based ODE network emulator that can flexibly estimate different trajectory paths corresponding to different ODE model parameters. Moreover, we present a mathematical analysis that provides a transparent way to relate potential performance loss to measurable distribution mismatch.

翻译：神经网络是替代物理模型、构建计算廉价代理的常用方法。若假设模型参数的先验分布准确，参数不确定性量化可被纳入训练过程。本文研究常见的相反情形：直接筛选或随机采样模型参数会导致训练时间过长，且在非物理参数值处进行评估。我们的解决方案是将不确定性量化与网络架构解耦。通过马尔可夫链蒙特卡洛（MCMC）将模型参数分布作为网络训练的输入，而非采样网络权重。由此，代理模型在实现与底层物理模型相同不确定性量化的同时，大幅减少了计算时间。该方法对神经网络的选择完全不可知。在示例中，我们提出了用于预测的分位数模拟器，以及一种新型基于自编码器的常微分方程网络模拟器，该模拟器能灵活估计对应不同常微分方程模型参数的不同轨迹路径。此外，我们提出的数学分析为将潜在性能损失与可测量的分布失配建立关联提供了透明途径。