🤖 AI Summary
This work addresses the challenge of parameter estimation in physical system simulation models arising from complex residual distributions. The authors propose an end-to-end estimation framework that employs embedded normalizing flows to map intricate residuals onto a simple base distribution. Indirect constraints are imposed on this base distribution through empirical likelihood under moment conditions. Model and flow parameters are jointly optimized via implicit differentiation combined with first-order gradient methods. By innovatively integrating normalizing flows with constrained empirical likelihood, the approach establishes an information-theoretically interpretable and computationally tractable framework. The resulting inverse transformation serves as an invertible surrogate model, enhancing both the accuracy and efficiency of parameter estimation while enabling quantification of model bias and sensitivity analysis.
📝 Abstract
In this work, we present a simulation-based parameter estimation framework for a model defined by a computational simulation of a physical system. We specifically outline an estimation framework consisting of two closely-integrated steps that facilitate an overall end-to-end parameter estimation scheme. The first step involves utilizing an embedded normalizing flow which is used to transform the unknown complex distribution of the residual information into a simple base distribution corresponding to the transformed residual information. In the second step, an empirical-likelihood estimator, under moment restrictions, is utilized for imposing an indirect constrain on the base distribution, where such an instantiated task reasonably allows us to treat the transformed residual information as random variables arising from discretely distribution population with each transformed data point as a single-cell from a set of finite-cell contingencies. Moreover, we use first-order gradient methods for updating the estimated parameter values of the model defined by the computational simulation and the corresponding parametrized embedded normalizing flow, that call for all gradient-related information by leveraging implicitly differentiations of the empirical-likelihood function, which is constructed from the implied empirical probabilities under moment restrictions. Here, it is worth mentioning that the problem formulation presented in this work, which highlights an information-theoretic interpretation, allows to present a computational framework for algorithmic implementations. Finally, as a-by-product, the inverse of the parametrized embedded normalizing flow, w.r.t. the estimated parameter values, serves as a surrogate model for the computational simulation model, which provides useful information for quantifying model discrepancies and sensitivity analysis.