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The Koopman operator provides a linear perspective on non-linear dynamics by focusing on the evolution of observables in an invariant subspace. Observables of interest are typically linearly reconstructed from the Koopman eigenfunctions. Despite the broad use of Koopman operators over the past few years, there exist some misconceptions about the applicability of Koopman operators to dynamical systems with more than one fixed point. In this work, an explanation is provided for the mechanism of lifting for the Koopman operator of nonlinear systems with multiple attractors. Considering the example of the Duffing oscillator, we show that by exploiting the inherent symmetry between the basins of attraction, a linear reconstruction with three degrees of freedom in the Koopman observable space is sufficient to globally linearize the system.

Many data science students and practitioners don't see the value in making time to learn and adopt good coding practices as long as the code "works". However, code standards are an important part of modern data science practice, and they play an essential role in the development of data acumen. Good coding practices lead to more reliable code and save more time than they cost, making them important even for beginners. We believe that principled coding is vital for quality data science practice. To effectively instill these practices within academic programs, instructors and programs need to begin establishing these practices early, to reinforce them often, and to hold themselves to a higher standard while guiding students. We describe key aspects of good coding practices for data science, illustrating with examples in R and in Python, though similar standards are applicable to other software environments. Practical coding guidelines are organized into a top ten list.

Machine learning techniques, in particular the so-called normalizing flows, are becoming increasingly popular in the context of Monte Carlo simulations as they can effectively approximate target probability distributions. In the case of lattice field theories (LFT) the target distribution is given by the exponential of the action. The common loss function's gradient estimator based on the "reparametrization trick" requires the calculation of the derivative of the action with respect to the fields. This can present a significant computational cost for complicated, non-local actions like e.g. fermionic action in QCD. In this contribution, we propose an estimator for normalizing flows based on the REINFORCE algorithm that avoids this issue. We apply it to two dimensional Schwinger model with Wilson fermions at criticality and show that it is up to ten times faster in terms of the wall-clock time as well as requiring up to $30\%$ less memory than the reparameterization trick estimator. It is also more numerically stable allowing for single precision calculations and the use of half-float tensor cores. We present an in-depth analysis of the origins of those improvements. We believe that these benefits will appear also outside the realm of the LFT, in each case where the target probability distribution is computationally intensive.

This paper proposes a hierarchy of numerical fluxes for the compressible flow equations which are kinetic-energy and pressure equilibrium preserving and asymptotically entropy conservative, i.e., they are able to arbitrarily reduce the numerical error on entropy production due to the spatial discretization. The fluxes are based on the use of the harmonic mean for internal energy and only use algebraic operations, making them less computationally expensive than the entropy-conserving fluxes based on the logarithmic mean. The use of the geometric mean is also explored and identified to be well-suited to reduce errors on entropy evolution. Results of numerical tests confirmed the theoretical predictions and the entropy-conserving capabilities of a selection of schemes have been compared.

We review common situations in Bayesian latent variable models where the prior distribution that a researcher specifies differs from the prior distribution used during estimation. These situations can arise from the positive definite requirement on correlation matrices, from sign indeterminacy of factor loadings, and from order constraints on threshold parameters. The issue is especially problematic for reproducibility and for model checks that involve prior distributions, including prior predictive assessment and Bayes factors. In these cases, one might be assessing the wrong model, casting doubt on the relevance of the results. The most straightforward solution to the issue sometimes involves use of informative prior distributions. We explore other solutions and make recommendations for practice.

Early sensory systems in the brain rapidly adapt to fluctuating input statistics, which requires recurrent communication between neurons. Mechanistically, such recurrent communication is often indirect and mediated by local interneurons. In this work, we explore the computational benefits of mediating recurrent communication via interneurons compared with direct recurrent connections. To this end, we consider two mathematically tractable recurrent linear neural networks that statistically whiten their inputs -- one with direct recurrent connections and the other with interneurons that mediate recurrent communication. By analyzing the corresponding continuous synaptic dynamics and numerically simulating the networks, we show that the network with interneurons is more robust to initialization than the network with direct recurrent connections in the sense that the convergence time for the synaptic dynamics in the network with interneurons (resp. direct recurrent connections) scales logarithmically (resp. linearly) with the spectrum of their initialization. Our results suggest that interneurons are computationally useful for rapid adaptation to changing input statistics. Interestingly, the network with interneurons is an overparameterized solution of the whitening objective for the network with direct recurrent connections, so our results can be viewed as a recurrent linear neural network analogue of the implicit acceleration phenomenon observed in overparameterized feedforward linear neural networks.

Neural dynamical systems with stable attractor structures, such as point attractors and continuous attractors, are hypothesized to underlie meaningful temporal behavior that requires working memory. However, working memory may not support useful learning signals necessary to adapt to changes in the temporal structure of the environment. We show that in addition to the continuous attractors that are widely implicated, periodic and quasi-periodic attractors can also support learning arbitrarily long temporal relationships. Unlike the continuous attractors that suffer from the fine-tuning problem, the less explored quasi-periodic attractors are uniquely qualified for learning to produce temporally structured behavior. Our theory has broad implications for the design of artificial learning systems and makes predictions about observable signatures of biological neural dynamics that can support temporal dependence learning and working memory. Based on our theory, we developed a new initialization scheme for artificial recurrent neural networks that outperforms standard methods for tasks that require learning temporal dynamics. Moreover, we propose a robust recurrent memory mechanism for integrating and maintaining head direction without a ring attractor.

Joint models (JM) for longitudinal and survival data have gained increasing interest and found applications in a wide range of clinical and biomedical settings. These models facilitate the understanding of the relationship between outcomes and enable individualized predictions. In many applications, more complex event processes arise, necessitating joint longitudinal and multistate models. However, their practical application can be hindered by computational challenges due to increased model complexity and large sample sizes. Motivated by a longitudinal multimorbidity analysis of large UK health records, we have developed a scalable Bayesian methodology for such joint multistate models that is capable of handling complex event processes and large datasets, with straightforward implementation. We propose two blockwise inference approaches for different inferential purposes based on different levels of decomposition of the multistate processes. These approaches leverage parallel computing, ease the specification of different models for different transitions, and model/variable selection can be performed within a Bayesian framework using Bayesian leave-one-out cross-validation. Using a simulation study, we show that the proposed approaches achieve satisfactory performance regarding posterior point and interval estimation, with notable gains in sampling efficiency compared to the standard estimation strategy. We illustrate our approaches using a large UK electronic health record dataset where we analysed the coevolution of routinely measured systolic blood pressure (SBP) and the progression of multimorbidity, defined as the combinations of three chronic conditions. Our analysis identified distinct association structures between SBP and different disease transitions.

We consider a linear implicit-explicit (IMEX) time discretization of the Cahn-Hilliard equation with a source term, endowed with Dirichlet boundary conditions. For every time step small enough, we build an exponential attractor of the discrete-in-time dynamical system associated to the discretization. We prove that, as the time step tends to 0, this attractor converges for the symmmetric Hausdorff distance to an exponential attractor of the continuous-in-time dynamical system associated with the PDE. We also prove that the fractal dimension of the exponential attractor (and consequently, of the global attractor) is bounded by a constant independent of the time step. The results also apply to the classical Cahn-Hilliard equation with Neumann boundary conditions.

Differential geometric approaches are ubiquitous in several fields of mathematics, physics and engineering, and their discretizations enable the development of network-based mathematical and computational frameworks, which are essential for large-scale data science. The Forman-Ricci curvature (FRC) - a statistical measure based on Riemannian geometry and designed for networks - is known for its high capacity for extracting geometric information from complex networks. However, extracting information from dense networks is still challenging due to the combinatorial explosion of high-order network structures. Motivated by this challenge we sought a set-theoretic representation theory for high-order network cells and FRC, as well as their associated concepts and properties, which together provide an alternative and efficient formulation for computing high-order FRC in complex networks. We provide a pseudo-code, a software implementation coined FastForman, as well as a benchmark comparison with alternative implementations. Crucially, our representation theory reveals previous computational bottlenecks and also accelerates the computation of FRC. As a consequence, our findings open new research possibilities in complex systems where higher-order geometric computations are required.