We develop an algebraic framework for sequential data assimilation of partially observed dynamical systems. In this framework, Bayesian data assimilation is embedded in a nonabelian operator algebra, which provides a representation of observables by multiplication operators and probability densities by density operators (quantum states). In the algebraic approach, the forecast step of data assimilation is represented by a quantum operation induced by the Koopman operator of the dynamical system. Moreover, the analysis step is described by a quantum effect, which generalizes the Bayesian observational update rule. Projecting this formulation to finite-dimensional matrix algebras leads to computational schemes that are i) automatically positivity-preserving and ii) amenable to consistent data-driven approximation using kernel methods for machine learning. Moreover, these methods are natural candidates for implementation on quantum computers. Applications to the Lorenz 96 multiscale system and the El Niño Southern Oscillation in a climate model show promising results in terms of forecast skill and uncertainty quantification.

Cultures around the world show varying levels of conservatism. While maintaining traditional ideas prevents wrong ones from being embraced, it also slows or prevents adaptation to new times. Without exploration there can be no improvement, but often this effort is wasted as it fails to produce better results, making it better to exploit the best known option. This tension is known as the exploration/exploitation issue, and it occurs at the individual and group levels, whenever decisions are made. As such, it is has been investigated across many disciplines. We extend previous work by approximating a continuum of traits under local exploration, employing the method of adaptive dynamics, and studying multiple fitness functions. In this work, we ask how nature would solve the exploration/exploitation issue, by allowing natural selection to operate on an exploration parameter in a variety of contexts, thinking of exploration as mutation in a trait space with a varying fitness function. Specifically, we study how exploration rates evolve by applying adaptive dynamics to the replicator-mutator equation, under two types of fitness functions. For the first, payoffs are accrued from playing a two-player, two-action symmetric game, we consider representatives of all games in this class, including the Prisoner’s Dilemma, Hawk-Dove, and Stag Hunt games, finding exploration rates often evolve downwards, but can also undergo neutral selection as well depending on the games parameters or initial conditions. Second, we study time dependent fitness with a function having a single oscillating peak. By increasing the period, we see a jump in the optimal exploration rate, which then decreases towards zero as the frequency of environmental change increases. These results establish several possible evolutionary scenarios for exploration rates, providing insight into many applications, including why we can see such diversity in rates of cultural change.

Under constant selection, each trait has a fixed fitness, and small mutation rates allow populations to efficiently exploit the optimal trait. Therefore, it is reasonable to expect that mutation rates will evolve downwards. However, we find that this need not be the case, examining several models of mutation. While upwards evolution of the mutation rate has been found with frequency- or time-dependent fitness, we demonstrate its possibility in a much simpler context. This work uses adaptive dynamics to study the evolution of the mutation rate, and the replicator–mutator equation to model trait evolution. Our approach differs from previous studies by considering a wide variety of methods to represent mutation. We use a finite string approach inspired by genetics as well as a model of local mutation on a discretization of the unit intervals, handling mutation beyond the endpoints in three ways. The main contribution of this work is a demonstration that the evolution of the mutation rate can be significantly more complicated than what is usually expected in relatively simple models.