Efficient Prior Calibration From Indirect Data
Bayesian inversion is central to the quantification of uncertainty within problems arising from numerous applications in science and engineering. To formulate the approach, four ingredients are required: a forward model mapping the unknown parameter to an element of a solution space, often the solut...
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Zusammenfassung: | Bayesian inversion is central to the quantification of uncertainty within
problems arising from numerous applications in science and engineering. To
formulate the approach, four ingredients are required: a forward model mapping
the unknown parameter to an element of a solution space, often the solution
space for a differential equation; an observation operator mapping an element
of the solution space to the data space; a noise model describing how noise
pollutes the observations; and a prior model describing knowledge about the
unknown parameter before the data is acquired. This paper is concerned with
learning the prior model from data; in particular, learning the prior from
multiple realizations of indirect data obtained through the noisy observation
process. The prior is represented, using a generative model, as the pushforward
of a Gaussian in a latent space; the pushforward map is learned by minimizing
an appropriate loss function. A metric that is well-defined under empirical
approximation is used to define the loss function for the pushforward map to
make an implementable methodology. Furthermore, an efficient residual-based
neural operator approximation of the forward model is proposed and it is shown
that this may be learned concurrently with the pushforward map, using a bilevel
optimization formulation of the problem; this use of neural operator
approximation has the potential to make prior learning from indirect data more
computationally efficient, especially when the observation process is
expensive, non-smooth or not known. The ideas are illustrated with the Darcy
flow inverse problem of finding permeability from piezometric head
measurements. |
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DOI: | 10.48550/arxiv.2405.17955 |