Efficient Deconvolution in Populational Inverse Problems

TL;DR

This paper introduces a novel method for population-based inverse problems that jointly deconvolves noise and infers parameter distributions using large datasets, a specialized gradient descent, and active learning, demonstrated on physical models.

Contribution

It proposes a new approach combining a coupled deconvolution and parameter inference framework with an active learning scheme for efficient population inverse problem solving.

Findings

Successfully applied to porous medium flow, elastodynamics, and atmospheric models.

Achieved simultaneous noise deconvolution and parameter distribution inference.

Enhanced computational efficiency with a surrogate model trained via active learning.

Abstract

This work is focussed on the inversion task of inferring the distribution over parameters of interest leading to multiple sets of observations. The potential to solve such distributional inversion problems is driven by increasing availability of data, but a major roadblock is blind deconvolution, arising when the observational noise distribution is unknown. However, when data originates from collections of physical systems, a population, it is possible to leverage this information to perform deconvolution. To this end, we propose a methodology leveraging large data sets of observations, collected from different instantiations of the same physical processes, to simultaneously deconvolve the data corrupting noise distribution, and to identify the distribution over model parameters defining the physical processes. A parameter-dependent mathematical model of the physical process is employed. A loss function characterizing the match between the observed data and the output of the mathematical model is defined; it is minimized as a function of the both the parameter inputs to the model of the physics and the parameterized observational noise. This coupled problem is addressed with a modified gradient descent algorithm that leverages specific structure in the noise model. Furthermore, a new active learning scheme is proposed, based on adaptive empirical measures, to train a surrogate model to be accurate in parameter regions of interest; this approach accelerates computation and enables automatic differentiation of black-box, potentially nondifferentiable, code computing parameter-to-solution maps. The proposed methodology is demonstrated on porous medium flow, damped elastodynamics, and simplified models of atmospheric dynamics.