Learning on dynamic statistical manifolds.

Area of Science:

• Computational fluid dynamics
• Applied mathematics
• Scientific computing

Background:

• Hyperbolic balance laws with random parameters are common in science and engineering.
• Quantifying and reducing predictive uncertainty is challenging due to nonlinearities and discontinuous solutions.

Purpose of the Study:

• To develop a computationally efficient method for uncertainty quantification and data assimilation.
• To address challenges posed by nonlinearities and non-Gaussian, discontinuous solutions in hyperbolic balance laws.

Main Methods:

• Utilizing the method of distributions for forward uncertainty propagation via a deterministic equation for the cumulative distribution function (CDF).
• Recasting the loss function in distributional terms, leveraging the equivalence between square error and Kullback-Leibler divergence for uncertainty reduction.
• Employing a Lagrangian constraint to enforce the CDF equation and sequential minimization for progressive parameter updates during data assimilation.

Main Results:

• Demonstrated a computationally efficient approach for uncertainty propagation in complex systems.
• Successfully reformulated data assimilation for uncertainty reduction using distributional loss functions.
• Showcased sequential minimization for adaptive parameter updates as new data becomes available.

Conclusions:

• The distributional method offers an effective solution for uncertainty quantification and data assimilation in hyperbolic balance laws.
• This approach ameliorates computational challenges associated with nonlinear, non-Gaussian, and discontinuous solutions.
• The method enables efficient and progressive reduction of predictive uncertainty through sequential data assimilation.