Uncertainty quantification (UQ) in scientific machine learning (SciML) combines the powerful predictive power of SciML with methods for quantifying the reliability of the learned models. However, two major challenges remain: limited interpretability and expensive training procedures. We provide a new interpretation for UQ problems by establishing a new theoretical connection between some Bayesian inference problems arising in SciML and viscous Hamilton-Jacobi partial differential equations (HJ PDEs). Namely, we show that the posterior mean and covariance can be recovered from the spatial gradient and Hessian of the solution to a viscous HJ PDE. As a first exploration of this connection, we specialize to Bayesian inference problems with linear models, Gaussian likelihoods, and Gaussian priors. In this case, the associated viscous HJ PDEs can be solved using Riccati ODEs, and we develop a new Riccati-based methodology that provides computational advantages when continuously updating the model predictions. Specifically, our Riccati-based approach can efficiently add or remove data points to the training set invariant to the order of the data and continuously tune hyperparameters. Moreover, neither update requires retraining on or access to previously incorporated data. We provide several examples from SciML involving noisy data and \textit{epistemic uncertainty} to illustrate the potential advantages of our approach. In particular, this approach's amenability to data streaming applications demonstrates its potential for real-time inferences, which, in turn, allows for applications in which the predicted uncertainty is used to dynamically alter the learning process.

翻译：不确定性量化（UQ）在科学机器学习（SciML）中结合了SciML强大的预测能力与量化学习模型可靠性的方法。然而，仍存在两大挑战：可解释性有限以及训练过程成本高昂。我们通过建立SciML中某些贝叶斯推理问题与粘性Hamilton-Jacobi偏微分方程（HJ PDEs）之间新的理论联系，为UQ问题提供了新解释。具体而言，我们证明后验均值与协方差可从粘性HJ PDE解的空间梯度与Hessian矩阵恢复。作为对该联系的首次数值探索，我们专门针对具有线性模型、高斯似然和高斯先验的贝叶斯推理问题。在此情形下，关联的粘性HJ PDE可通过Riccati常微分方程求解，并由此发展出一种新型基于Riccati的方法，该方法在连续更新模型预测时具有计算优势。具体来说，我们的Riccati方法可以高效地向训练集添加或移除数据点，且对数据顺序保持不变性，同时能连续调整超参数。此外，两种更新均无需重新训练或访问先前纳入的数据。我们提供了多个涉及噪声数据与\textit{认知不确定性}的SciML实例，以展示该方法潜在优势。特别地，该方法对数据流应用的适应性证明了其在实时推理中的潜力，进而可应用于利用预测不确定性动态改变学习过程的场景。