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Abstract Mathematical models of cognition are often memoryless and ignore potential fluctuations of their parameters. However, human cognition is inherently dynamic. Thus, we propose to augment mechanistic cognitive models with a temporal dimension and estimate the resulting dynamics from a superstatistics perspective. Such a model entails a hierarchy between a low-level observation model and a high-level transition model. The observation model describes the local behavior of a system, and the transition model specifies how the parameters of the observation model evolve over time. To overcome the estimation challenges resulting from the complexity of superstatistical models, we develop and validate a simulation-based deep learning method for Bayesian inference, which can recover both time-varying and time-invariant parameters. We first benchmark our method against two existing frameworks capable of estimating time-varying parameters. We then apply our method to fit a dynamic version of the diffusion decision model to long time series of human response times data. Our results show that the deep learning approach is very efficient in capturing the temporal dynamics of the model. Furthermore, we show that the erroneous assumption of static or homogeneous parameters will hide important temporal information.

Abstract The inference of physical parameters from measured distributions constitutes a core task in physics data analyses. Among recent deep learning methods, so-called conditional invertible neural networks provide an elegant approach owing to their probability-preserving bijective mapping properties. They enable training the parameter-observation correspondence in one mapping direction and evaluating the parameter posterior distributions in the reverse direction. Here, we study the inference of cosmic-ray source properties from cosmic-ray observations on Earth using extensive astrophysical simulations. We compare the performance of conditional invertible neural networks (cINNs) with the frequently used Markov Chain Monte Carlo (MCMC) method. While cINNs are trained to directly predict the parameters’ posterior distributions, the MCMC method extracts the posterior distributions through a likelihood function that matches simulations with observations. Overall, we find good agreement between the physics parameters derived by the two different methods. As a result of its computational efficiency, the cINN method allows for a swift assessment of inference quality.

Simulating abundances of stable water isotopologues, that is, molecules differing in their isotopic composition, within climate models allows for comparisons with proxy data and, thus, for testing hypotheses about past climate and validating climate models under varying climatic conditions. However, many models are run without explicitly simulating water isotopologues. We investigate the possibility of replacing the explicit physics-based simulation of oxygen isotopic composition in precipitation using machine learning methods. These methods estimate isotopic composition at each time step for given fields of surface temperature and precipitation amount. We implement convolutional neural networks (CNNs) based on the successful UNet architecture and test whether a spherical network architecture outperforms the naive approach of treating Earth’s latitude-longitude grid as a flat image. Conducting a case study on a last millennium run with the iHadCM3 climate model, we find that roughly 40% of the temporal variance in the isotopic composition is explained by the emulations on interannual and monthly timescale, with spatially varying emulation quality. The tested CNNs outperform simple baseline models such as random forest and pixel-wise linear regression substantially. A modified version of the standard UNet architecture for flat images yields results that are as good as the predictions by the spherical CNN. Variations in the implementation of isotopes between climate models likely contribute to an observed deterioration of emulation results when testing on data obtained from different climate models than the one used for training. Future work toward stable water-isotope emulation might focus on achieving robust climate–oxygen isotope relationships or exploring the set of possible predictor variables.

Mathematical models in epidemiology are an indispensable tool to determine the dynamics and important characteristics of infectious diseases. Apart from their scientific merit, these models are often used to inform political decisions and interventional measures during an ongoing outbreak. However, reliably inferring the epidemical dynamics by connecting complex models to real data is still hard and requires either laborious manual parameter fitting or expensive optimization methods which have to be repeated from scratch for every application of a given model. In this work, we address this problem with a novel combination of epidemiological modeling with specialized neural networks. Our approach entails two computational phases: In an initial training phase, a mathematical model describing the epidemic is used as a coach for a neural network, which acquires global knowledge about the full range of possible disease dynamics. In the subsequent inference phase, the trained neural network processes the observed data of an actual outbreak and infers the parameters of the model in order to realistically reproduce the observed dynamics and reliably predict future progression. With its flexible framework, our simulation-based approach is applicable to a variety of epidemiological models. Moreover, since our method is fully Bayesian, it is designed to incorporate all available prior knowledge about plausible parameter values and returns complete joint posterior distributions over these parameters. Application of our method to the early Covid-19 outbreak phase in Germany demonstrates that we are able to obtain reliable probabilistic estimates for important disease characteristics, such as generation time, fraction of undetected infections, likelihood of transmission before symptom onset, and reporting delays using a very moderate amount of real-world observations. Author summary Emerging infections and epidemic outbreaks are associated with large uncertainties concerning data integrity that challenge the timely detection of disease characteristics and dynamics. Robust parameter inference for mathematical models aiming to describe these dynamics is essential to predict the progression of an epidemic and inform on appropriate public health interventions. In this study, we present a novel method based on invertible neural networks that allows inference of important epidemiological characteristics in case of limited data, thereby allowing for reliable uncertainty quantification. The method circumvents common challenges associated with sparse data by using simulation-based training of an expressive generative neural network. Applying our method to data of the early Covid-19 epidemic in Germany, we are able to obtain reliable estimates on important disease characteristics, such as the proportion of infected individuals remaining undetected, despite limited observations during early outbreak dynamics.