Training physics-based machine-learning parameterizations with gradient-free ensemble Kalman methods

Lopez‐Gomez, Ignacio; Christopoulos, Costa; Ervik, Haakon Ludvig Langeland; Dunbar, Oliver R. A.; Cohen, Yair; Schneider, Tapio

doi:10.1002/essoar.10510937.1

Cited by 1 publication

(1 citation statement)

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“…Moreover, the forward solvers are often given as a black box (e.g., off-the-shelf solvers [13] or multiphysics systems requiring coupling of different solvers [14,15]), and may not be differentiable due to the numerical methods used (e.g., embedded boundary method [16,17] and adaptive mesh refinement [18,19]) or because of the inherently discontinuous physics (e.g. in fracture [20] or cloud modeling [21,22]).…”

Section: Orientationmentioning

confidence: 99%

Efficient Derivative-free Bayesian Inference for Large-Scale Inverse Problems

Huang¹,

Huang²,

Reich³

et al. 2022

Preprint

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We consider Bayesian inference for large scale inverse problems, where computational challenges arise from the need for repeated evaluations of an expensive forward model. This renders most Markov chain Monte Carlo approaches infeasible, since they typically require O(10 4 ) model runs, or more. Moreover, the forward model is often given as a black box or is impractical to differentiate. Therefore derivative-free algorithms are highly desirable. We propose a framework, which is built on Kalman methodology, to efficiently perform Bayesian inference in such inverse problems. The basic method is based on an approximation of the filtering distribution of a novel mean-field dynamical system into which the inverse problem is embedded as an observation operator. Theoretical properties of the mean-field model are established for linear inverse problems, demonstrating that the desired Bayesian posterior is given by the steady state of the law of the filtering distribution of the mean-field dynamical system, and proving exponential convergence to it. This suggests that, for nonlinear problems which are close to Gaussian, sequentially computing this law provides the basis for efficient iterative methods to approximate the Bayesian posterior. Ensemble methods are applied to obtain interacting particle system approximations of the filtering distribution of the mean-field model; and practical strategies to further reduce the computational and memory cost of the methodology are presented, including low-rank approximation and a bi-fidelity approach. The effectiveness of the framework is demonstrated in several numerical experiments, including proofof-concept linear/nonlinear examples and two large-scale applications: learning of

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