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Neural networks (NNs) are currently changing the computational paradigm onhow to combine data with mathematical laws in physics and engineering in aprofound way, tackling challenging inverse and ill-posed problems not solvablewith traditional methods. However, quantifying errors and uncertainties inNN-based inference is more complicated than in traditional methods. This isbecause in addition to aleatoric uncertainty associated with noisy data, thereis also uncertainty due to limited data, but also due to NN hyperparameters,overparametrization, optimization and sampling errors as well as modelmisspecification. Although there are some recent works on uncertaintyquantification (UQ) in NNs, there is no systematic investigation of suitablemethods towards quantifying the total uncertainty effectively and efficientlyeven for function approximation, and there is even less work on solving partialdifferential equations and learning operator mappings betweeninfinite-dimensional function spaces using NNs. In this work, we present acomprehensive framework that includes uncertainty modeling, new and existingsolution methods, as well as evaluation metrics and post-hoc improvementapproaches. To demonstrate the applicability and reliability of our framework,we present an extensive comparative study in which various methods are testedon prototype problems, including problems with mixed input-output data, andstochastic problems in high dimensions. In the Appendix, we include acomprehensive description of all the UQ methods employed, which we will makeavailable as open-source library of all codes included in this framework.

Uncertainty Quantification in Scientific Machine Learning: Methods, Metrics, and Comparisons