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Physics-Informed Learning: Distributed Parameter Systems, Hidden Markov Models, and the Viterbi Algorithm

We present an approach for a physics-informed Hidden Markov Model with a special emphasis on including prior physical knowledge in the Viterbi algorithm. To this end, we develop a covariate-dependent (forward) hidden Markov model (HMM) with two emission or observed variables that represent two observable time series y and u. The goal is to predict the time series u based on the observations y. The hidden Markov chain serves as a forward model with the two observed quantities, assumed to be described by a bivariate normal distribution which is a function of the state of the hidden Markov chain. In addition, the bivariate normal and hidden Markov chain parameters are allowed to depend on various covariates via built-in regression models. The model can then be trained using the Baum-Welch/Expectation Maximization (EM) algorithm and a training set of the time series y and u and covariate measurements. The resulting fit model, together with observations of time series y only and the Viterbi algorithm, are then used to find the most likely path through the hidden Markov chain and the corresponding emissions with the intent of solving the inverse problem of obtaining an estimate of u from measurements of y alone. At this point we propose to augment the Viterbi algorithm to also include prior physical knowledge in the form of a forward model y=ℒu, creating a physics-informed Viterbi algorithm. We illustrate the proposed method in the context of a problem involving a transdermal alcohol biosensor and provide numerical examples.