Generative model-based framework for parameter estimation and uncertainty quantification applied to a compartmental model in epidemiology

We propose a new method in which a generative network (GN) set within a reduced-order model (ROM) framework is used to solve inverse problems for partial differential equations (PDE). The aim is to match available measurements and estimate the corresponding uncertainties associated with the states and parameters of a numerical physical simulation. We train the GN using only unconditional simulations of the discretized PDE model. A GN is used here to exploit the ability of these methods to generate realistic outputs from complex probability distributions, such as the ones that represent the possible states and parameters of a physical problem. We compare the proposed method with the gold standard Markov chain Monte Carlo (MCMC). Additionally, we suggest the use of a novel type of time-stepping regularization to improve the representativeness of the physical solution, and we present a new way of evaluating the GN training, taking advantage of the real/generated sample structure. We apply the proposed approaches to a spatio-temporal compartmental model in epidemiology. The results show that the proposed GN-based ROM can efficiently quantify uncertainty and accurately match the measurements and the gold standard. This is achieved using only a limited number of unconditional simulations from the full-order numerical PDE model (40 simulations). The GN-based ROM operates 60 times faster than the gold standard (MCMC), while producing uncertainties that closely match those generated by the MCMC approach. The proposed method is a general framework for quantifying uncertainties in numerical physical simulations and is not restricted to the specific physics of this application.