Uncertainty quantification

UQ distinguishes between aleatoric uncertainty, which arises from inherent randomness and cannot be reduced, and epistemic uncertainty, which stems from lack of knowledge and can be decreased with more data or better models.¹ The forward problem propagates input uncertainties through a model to quantify output variability, while the inverse problem uses observations to infer unknown parameters and reduce epistemic uncertainty.² Common techniques include Monte Carlo sampling, polynomial chaos expansions, and Gaussian process emulators, each offering different trade-offs between accuracy and computational cost.³ Sensitivity analysis, such as variance-based Sobol indices, identifies which inputs contribute most to output uncertainty, guiding model refinement and data collection.

UQ is integral to nuclear reactor safety analysis, where it quantifies the margin of safety under postulated accident scenarios, and to climate modeling, where it characterizes the spread of future temperature projections across different emission pathways.⁴ In computational fluid dynamics, UQ assesses the reliability of aerodynamic predictions for aircraft design, while in structural engineering it informs the probability of failure under uncertain loads and material properties. The oil and gas industry uses UQ to evaluate reservoir performance and optimize extraction strategies, and medical device manufacturers apply it to verify that implants meet regulatory safety standards under patient variability.

A central challenge is the curse of dimensionality: as the number of uncertain inputs grows, the computational cost of accurate UQ escalates rapidly. Surrogate models, such as sparse grids and neural network emulators, approximate the original simulator to make UQ tractable for high-fidelity simulations.³ Multilevel Monte Carlo methods reduce variance by combining simulations at different resolutions, achieving significant speedups for problems with smooth responses.⁵ Verification and validation are closely linked to UQ, ensuring that numerical codes correctly solve the governing equations and that models adequately represent reality within quantified uncertainty bounds.²

UQ has roots in the 1960s nuclear safety studies, where the concept of 'best estimate plus uncertainty' was developed to replace overly conservative safety margins.⁴ The field also intersects with philosophy of science, particularly in debates about the nature of probability and model confirmation. In recent years, UQ has been applied to machine learning, where it quantifies prediction confidence in deep neural networks, a practice now essential for autonomous vehicles and medical diagnostics.⁶ A niche application is in forensic science, where UQ helps assess the reliability of evidence interpretation, and in archaeology, where it quantifies uncertainty in radiocarbon dating calibrations.

Aleatoric uncertainty

Irreducible uncertainty due to inherent randomness in a system.

Epistemic uncertainty

Uncertainty due to lack of knowledge, reducible with more data or better models.

Polynomial chaos expansion

A spectral method that represents model output as a series of orthogonal polynomials in the uncertain inputs.

Sobol indices

Variance-based sensitivity measures that attribute output variance to individual inputs or interactions.