Uncertainty Quantification in Scientific Machine Learning: Methods, Metrics, and Comparisons (2201.07766v1)

Abstract: Neural networks (NNs) are currently changing the computational paradigm on how to combine data with mathematical laws in physics and engineering in a profound way, tackling challenging inverse and ill-posed problems not solvable with traditional methods. However, quantifying errors and uncertainties in NN-based inference is more complicated than in traditional methods. This is because in addition to aleatoric uncertainty associated with noisy data, there is also uncertainty due to limited data, but also due to NN hyperparameters, overparametrization, optimization and sampling errors as well as model misspecification. Although there are some recent works on uncertainty quantification (UQ) in NNs, there is no systematic investigation of suitable methods towards quantifying the total uncertainty effectively and efficiently even for function approximation, and there is even less work on solving partial differential equations and learning operator mappings between infinite-dimensional function spaces using NNs. In this work, we present a comprehensive framework that includes uncertainty modeling, new and existing solution methods, as well as evaluation metrics and post-hoc improvement approaches. To demonstrate the applicability and reliability of our framework, we present an extensive comparative study in which various methods are tested on prototype problems, including problems with mixed input-output data, and stochastic problems in high dimensions. In the Appendix, we include a comprehensive description of all the UQ methods employed, which we will make available as open-source library of all codes included in this framework.

• The paper introduces a systematic framework that quantifies total uncertainty in SciML by integrating Bayesian, ensemble, and physics-informed methods to address diverse error sources.
• It presents comprehensive methodological comparisons and evaluates performance using metrics for function approximation, PDE solutions, and stochastic models.
• The study demonstrates enhanced predictive accuracy through robust numerical experiments, highlighting transformative applications in aerospace, biomedical, and climate modeling.

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

Scientific machine learning (SciML) is innovatively transforming computational paradigms by integrating neural networks (NNs) with mathematical frameworks in physics and engineering. This paper delineates a comprehensive framework for understanding uncertainty quantification (UQ) within this domain, emphasizing its importance in handling inverse and ill-posed problems traditionally deemed unsolvable. While acknowledging the existing strides made, it spotlights the scarcity of systematic methodologies for effectively and efficiently quantifying total uncertainties — a critical gap this research aims to address.

Overview

The paper stresses that the challenge with NN-based inference lies in the multifaceted error and uncertainty components introduced not only from noisy and limited data (aleatoric uncertainty) but from various model-specific characteristics. These include overparameterization, optimization, and sampling errors, as well as model misspecification — collectively encompassing epistemic uncertainties. Here, a robust and systematic framework for total uncertainty quantification is proposed.

The main contributions are segmented into several key methodological areas:

• Uncertainty Modeling: Addressing error sources by proposing a combined modeling framework that leverages data, physical laws, and learned priors.
• Solution Methods: Introducing new and existing uncertainty quantification methods, evaluating them through comparative studies in SciML applications.
• Evaluation Metrics: Proposing metrics for evaluating effectiveness in UQ, focusing on their applicability in function approximation, PDE solutions, and stochastic modeling.
• Practical Applications: Demonstration of methodology through various scientific prototypes, indicating broad potential for these approaches in real-world issues.

Strong Numerical Results and Key Insights

The paper reports strong numerical results demonstrating that informed combinations of Bayesian methods, ensembles, and functional priors significantly enhance the understanding and representation of uncertainties. For instance, empirical evaluations with heteroscedastic and stochastic noise shed light on superiority in accuracy when using functional priors over standard Bayesian Neural Networks (BNNs). Importantly, the presented Physics-Informed Generative Adversarial Networks (PI-GAN) and Bayesian physics-informed approaches exhibit promising results in predictive accuracy by incorporating knowledge from historical data into the UQ framework.

Implications and Future Directions

The implications of this research are manifold:

• Practical Implementation: SciML methodologies enhanced with robust UQ frameworks will significantly impact applications in areas where model certainty is paramount, such as in aerospace, biomedical engineering, and climate modeling.
• Theoretical Advancement: The framework paves the way for more refined interpretations of NN predictions by explicitly modeling diverse uncertainties. This advancement also entails improving the theoretical grounding of UQ within SciML, enhancing its reliability.
• Future Developments in AI: Prospective progressions involve exploring scalability of UQ methods to handle larger datasets and complex systems, integrating deeper GP architectures, and developing multi-fidelity models that handle varying uncertainty levels across different scales.

In conclusion, this paper marks an essential step in systematically embedding uncertainty quantification into scientific machine learning. By doing so, it contributes significantly to both theoretical advancements and practical applications of NNs across disciplines reliant on computational modeling of complex systems. The presented methods set a promising groundwork for future enhancements, particularly in balancing computational efficiency with the nuanced representation of uncertainties inherent to scientific problems.