Efficient Prior Calibration From Indirect Data
O. Deniz Akyildiz, Mark Girolami, Andrew M. Stuart, Arnaud Vadeboncoeur
TL;DR
This work develops a data-informed prior learning framework for Bayesian inverse problems by expressing the prior as a pushforward of a Gaussian, $(T^\alpha)_\# \mu_0$, and matching induced data distributions to observations via a sliced-Wasserstein divergence. It introduces a residual-based neural operator to approximate the forward map $F^\dagger$, forming a bilevel optimization scheme that jointly learns the prior and the operator when $F^\dagger$ is expensive or unknown. Theoretical results connect the method to Bayes when $N=1$ and empiricalizations, and the approach is demonstrated through Darcy flow inverse problems with level-set and lognormal priors, showing accurate parameter recovery and favorable computational properties. The framework supports scalable prior calibration from indirect data and simultaneous operator learning, enabling efficient, data-driven Bayesian inversion for complex PDE systems.
Abstract
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.