The Bandt-Pompe permutation entropy framework, alongside the complexity-entropy causality plane, has become a standard tool for characterizing the dynamical properties of time series. However, observational noise distorts ordinal pattern probability distributions in ways that can systematically misplace time series within the causality plane, compromising dynamical classification. This effect is particularly relevant for geophysical signals, which are typically poorly and irregularly sampled, and have a low signal-to-noise level. In this work, we characterize the distortions on ordinal pattern statistics using the formalism of majorization. We provide theoretical results and propose corrective strategies that restore discriminability under realistic measurement conditions. To achieve this, we introduce methodology that allows the characterization of noisy dynamical series and further allows the quantification of observational noise without the need of a fitting procedure. Finally, we illustrate our methodology by analyzing paleomagnetic records to determine if the geological evolution of the Earth dipole is better described by a stochastic or chaotic system.

Single spacecraft missions do not measure the pristine solar wind continuously because of the spacecrafts' orbital trajectory. The infrequent spatiotemporal cadence of measurement fundamentally limits conclusions about solar wind-magnetosphere coupling throughout the solar system. At Mars, such single spacecraft missions result in limitations for assessing the solar wind's role in causing lower altitude observations such as auroral dynamics or atmospheric loss. In this work, we detail the development of a virtual solar wind monitor from the Mars Atmosphere and Volatile Evolution (MAVEN) mission; a single spacecraft. This virtual solar wind monitor provides a continuous estimate of the solar wind upstream from Mars with uncertainties. We specifically employ Gaussian process regression to estimate the upstream solar wind and uncertainty estimations that scale with the data sparsity of our real observations. This proxy enables continuous solar wind estimation at Mars with representative uncertainties for the majority of the time since since late 2014. We conclude by discussing suggested uses of this virtual solar wind monitor for statistical studies of the Mars space environment and heliosphere.

The differentiable programming paradigm is a cornerstone of modern scientific computing. It refers to numerical methods for computing the gradient of a numerical model's output. Many scientific models are based on differential equations, where differentiable programming plays a crucial role in calculating model sensitivities, inverting model parameters, and training hybrid models that combine differential equations with data-driven approaches. Furthermore, recognizing the strong synergies between inverse methods and machine learning offers the opportunity to establish a coherent framework applicable to both fields. Differentiating functions based on the numerical solution of differential equations is non-trivial. Numerous methods based on a wide variety of paradigms have been proposed in the literature, each with pros and cons specific to the type of problem investigated. Here, we provide a comprehensive review of existing techniques to compute derivatives of numerical solutions of differential equations. We first discuss the importance of gradients of solutions of differential equations in a variety of scientific domains. Second, we lay out the mathematical foundations of the various approaches and compare them with each other. Third, we cover the computational considerations and explore the solutions available in modern scientific software. Last but not least, we provide best-practices and recommendations for practitioners. We hope that this work accelerates the fusion of scientific models and data, and fosters a modern approach to scientific modelling.

The Fermat distance has been recently established as a useful tool for machine learning tasks when a natural distance is not directly available to the practitioner or to improve the results given by Euclidean distances by exploding the geometrical and statistical properties of the dataset. This distance depends on a parameter $\alpha$ that greatly impacts the performance of subsequent tasks. Ideally, the value of $\alpha$ should be large enough to navigate the geometric intricacies inherent to the problem. At the same, it should remain restrained enough to sidestep any deleterious ramifications stemming from noise during the process of distance estimation. We study both theoretically and through simulations how to select this parameter.

Sampling strategies used in paleomagnetic studies play a crucial role in dictating the accuracy of our estimates of properties of the ancient geomagnetic field. However, there has been little quantitative analysis of optimal paleomagnetic sampling strategies and the community has instead defaulted to traditional practices that vary between laboratories. In this paper, we quantitatively evaluate the accuracy of alternative paleomagnetic sampling strategies through numerical experiment and an associated analytical framework. Our findings demonstrate a strong correspondence between the accuracy of an estimated paleopole position and the number of sites or independent readings of the time-varying paleomagnetic field, whereas larger numbers of in-site samples have a dwindling effect. This remains true even when a large proportion of the sample directions are spurious. This approach can be readily achieved in sedimentary sequences by distributing samples stratigraphically, considering each sample as an individual reading. However, where the number of potential independent sites is inherently limited the collection of additional in-site samples can improve the accuracy of the paleopole estimate (although with diminishing returns with increasing samples per site). Where an estimate of the magnitude of paleosecular variation is sought, multiple in-site samples should be taken, but the optimal number is dependent on the expected fraction of outliers. We provide both analytical formulas and a series of interactive Jupyter notebooks allowing optimal sampling strategies to be derived from user-informed expectations.

Abstract Our understanding of Earth's paleogeography relies heavily on paleomagnetic apparent polar wander paths (APWPs), which represent the time‐dependent position of Earth's spin axis relative to a given block of lithosphere. However, conventional approaches to APWP construction have significant limitations. First, the paleomagnetic record contains substantial noise that is not integrated into APWPs. Second, parametric assumptions are adopted to represent spatial and temporal uncertainties even where the underlying data do not conform to the assumed distributions. The consequences of these limitations remain largely unknown. Here, we address these challenges with a bottom‐up Monte Carlo uncertainty propagation scheme that operates on site‐level paleomagnetic data. To demonstrate our methodology, we present an extensive compilation of site‐level Cenozoic paleomagnetic data from North America, which we use to generate a high‐resolution APWP. Our results demonstrate that even in the presence of substantial noise, polar wandering can be assessed with unprecedented temporal and spatial resolution.

Glaciers and ice sheets lose their mass by ablation (the output term of their surface mass balance) and discharging into a water body (dynamic loss). The latter is associated with multiple physical characteristics such as bed geometry, inland thinning, terminus stability, and basal conditions. Better assessing the dynamic loss, especially its spatiotemporal variability within a drainage basin, will help improve our understanding of the underlying processes and quantify the future contribution of sea level rise. We propose a new inverse model to decompose glacier elevation change and optimize the dynamic mass loss components for each pixel of the elevation data grid. The model unmixes the observed elevation change from remote sensing data using the modeled surface mass balance and the ice flux as constraints. We use two approaches to design the ice flux term; one is based on glacier surface velocity and the conservation of mass, and the other builds on the flow law and the Shallow Ice Approximation. We test the model for selected marine-terminating glacier outlets in the Greenland ice sheet. If the surface velocity can be decomposed into short-term (seasonal) and multi-year signals, our model may be able to further resolve the dynamic loss components of different physical processes.

The Mars Atmosphere and Volatile EvolutioN (MAVEN) mission has been orbiting Mars since 2014 and now has over 10,000 orbits which we use to characterize Mars' dynamic space environment. Through global field line tracing with MAVEN magnetic field data we find an altitude dependent draping morphology that differs from expectations of induced magnetospheres in the vertical (Z Mars Sun-state, MSO) direction. We quantify this difference from the classical picture of induced magnetospheres with a Bayesian multiple linear regression model to predict the draped field as a function of the upstream interplanetary magnetic field (IMF), remanent crustal fields, and a previously underestimated induced effect. From our model we conclude that unexpected twists in high altitude dayside draping (>800 km) are a result of the IMF component in the +/- X MSO direction. We propose that this is a natural outcome of current theories of induced magnetospheres but has been underestimated due to approximations of the IMF as solely +/- Y directed. We additionally estimate that distortions in low altitude (<800 km) dayside draping along Z are directly related to remanent crustal fields. We show dayside draping traces down tail and previously reported inner magnetotail twists are likely caused by the crustal field of Mars, while the outer tail morphology is governed by an induced response to the IMF direction. We conclude with an updated understanding of induced magnetospheres which details dayside draping for multiple directions of the incoming IMF and discuss the repercussions of this draping for magnetotail morphology.

Supplemental material (SM; also known as supplementary information) comes with its associated research article and provides study details such as metadata, additional figures and text, multimedia, and code. Well-designed SM helps readers fully understand the underlying scientific analysis, reproduce the work, and even reuse the workflows for exploratory ideas. Thus, the concept of FAIR (Findable, Accessible, Interoperable, and Reusable), which is originally designed for data sharing guidelines, also matches these core qualities for SM. We evaluate different SM-preparation practices that are commonly found in Earth Science journal articles. These practices are classified into five tiers based on the FAIR principles and the narrative structure. We show that Jupyter Book-based SM belongs to the top tier and outperforms the other practices, despite being not as popular as the other SM-preparation practices as of 2022. We identify the advantages of the Jupyter Book-based SM as follows. Jupyter Book uses a narrative structure to combine different elements of SM into a single scholarly object, increasing readability. Jupyter Book’s direct support of HTML publishing allows users to web host the SM using services such as Github Pages, improving the web indexing ranks and resulting in higher exposure of both the research article and the SM. The entire SM is also eligible to be archived in a data repository and receive a Digital Object Identifier (DOI) that can be used for citations. In addition, Jupyter Book-based SM lowers the threshold of reproducing and reusing the work by accessing an interactive cloud computing service (e.g., MyBinder.org) with all data and code imported if the content is available on a code-hosting platform (e.g., Github). These features summarize the core values of SM from the perspective of open science. We encourage researchers to use these good practices and urge journal publishers to be open to receiving such supplements for maximum effectiveness.

Inversion methods play an important role in glacier models, both to calibrate and estimate parameters of interest (e.g. Glen's coefficients). However, inversions are usually made for each glacier individually, without using any global information, i.e. without deriving general laws governing the spatiotemporal variability of those parameters. The reason behind this limitation is twofold: the statistical challenge of making constrained inferences with multiple glaciers, and the computational limitation of processing massive glacier datasets. Machine learning powered with differential programming is a tool that can address both limitations. We introduce a statistical framework for functional inversion of physical processes governing global-scale glacier changes. We apply this framework to invert a prescribed function describing the spatial variability of Glen’s coefficient (A). Instead of estimating a single parameter per glacier, we learn the parameters of a regressor (i.e. a neural network) that encodes information related to each glacier (i.e. long-term air temperature) to the parameter of interest. The inversion is done by embedding a neural network inside the Shallow Ice Approximation PDE - resulting in a Universal Differential Equation - with the goal of minimizing the error on the simulated ice surface velocities. We previously had shown that this hybrid model training is possible thanks to the use of differential programming, enabling differentiation of a PDE, a numerical solver and a neural network simultaneously. In this work we upscale this approach to include larger datasets and with the goal of learning real empirical laws from observations. This framework is built inside ODINN.jl, an open-source package in the Julia programming language for global glacier evolution modelling using Universal Differential Equations. ODINN exploits the latest generation of ice surface velocities and geodetic mass balance remote sensing products, as well as many preprocessing tools from the Open Global Glacier Model (OGGM).

Our understanding of paleogeography through Earth history relies heavily on apparent polar wander paths (APWPs), which represent the time-dependent position of Earth’s spin axis relative to a given tectonic plate. However, there are a number of limitations associated with conventional approaches to APWP construction. First, the paleomagnetic record contains significant uncertainty in individual pole positions that is not propagated into APWPs. This traditional approach makes it difficult to incorporate age and positional uncertainty into synthesized paths and assigns equal weight to paleomagnetic poles with vastly different numbers of underlying sites. Second, the effective propagation of site-level uncertainties into the APWP requires a transformation that renders traditional parametric assumptions (i.e., Fisher statistics) on the pole level ineffective. Here, we overcome these limitations with a bottom-up Monte Carlo uncertainty propagation scheme that operates on site-level paleomagnetic data. To demonstrate our methodology, we present a large compilation of site-level Cenozoic paleomagnetic data from North America, which we use to generate a high-resolution APWP. We show that even in the presence of significant noise, polar wandering can be assessed with unprecedented temporal and spatial resolution.

Geoscientific models are facing increasing challenges to exploit growing datasets coming from remote sensing. Universal differential equations (UDEs), aided by differentiable programming, provide a new scientific modelling paradigm enabling both complex functional inversions to potentially discover new physical laws and data assimilation from heterogeneous and sparse observations. We demonstrate an application of UDEs as a proof of concept to learn the creep component of ice flow, i.e. a nonlinear diffusivity differential equation, of a glacier evolution model. By combining a mechanistic model based on a two-dimensional shallow-ice approximation partial differential equation with an embedded neural network, i.e. a UDE, we can learn parts of an equation as nonlinear functions that then can be translated into mathematical expressions. We implemented this modelling framework as ODINN.jl, a package in the Julia programming language, providing high performance, source-to-source automatic differentiation (AD) and seamless integration with tools and global datasets from the Open Global Glacier Model in Python. We demonstrate this concept for 17 different glaciers around the world, for which we successfully recover a prescribed artificial law describing ice creep variability by solving ∼ 500 000 ordinary differential equations in parallel. Furthermore, we investigate which are the best tools in the scientific machine learning ecosystem in Julia to differentiate and optimize large nonlinear diffusivity UDEs. This study represents a proof of concept for a new modelling framework aiming at discovering empirical laws for large-scale glacier processes, such as the variability in ice creep and basal sliding for ice flow, and new hybrid surface mass balance models.

We study the selection of covariate adjustment sets for estimating the value of point exposure dynamic policies, also known as dynamic treatment regimes, assuming a non-parametric causal graphical model with hidden variables, in which at least one adjustment set is fully observable. We show that recently developed criteria, for graphs without hidden variables, to compare the asymptotic variance of non-parametric estimators of static policy values that control for certain adjustment sets, are also valid under dynamic policies and graphs with hidden variables. We show that there exist adjustment sets that are optimal minimal (minimum), in the sense of yielding estimators with the smallest variance among those that control for adjustment sets that are minimal (of minimum cardinality). Moreover, we show that if either no variables are hidden or if all the observable variables are ancestors of either treatment, outcome, or the variables that are used to decide treatment, a globally optimal adjustment set exists. We provide polynomial time algorithms to compute the globally optimal (when it exists), optimal minimal, and optimal minimum adjustment sets. Our results are based on the construction of an undirected graph in which vertex cuts between the treatment and outcome variables correspond to adjustment sets. In this undirected graph, a partial order between minimal vertex cuts can be defined that makes the set of minimal cuts a lattice. This partial order corresponds directly to the ordering of the asymptotic variances of the corresponding non-parametrically adjusted estimators.

The last few years have been marked by a number of motivating novel ideas and methodological advancements in paleomagnetic analysis (e.g. trans-hierarchical uncertainty propagation), observational and theoretical geodynamics, and paleogeographical modeling (e.g. optimisation and Bayesian approaches). Many of these developments offer new insights on, and/or approaches to estimating, the past motions of tectonic plates—but so far these developments have largely unfolded in isolation of one another. In November 2021 an international group of 15 young scientists with highly complementary backgrounds (spanning the aforementioned fields) gathered to explore and discuss these exciting new developments and to brainstorm strategies that may enable their integration. We anticipate that the integration of these diverse new ideas and methods will open new frontiers in plate tectonic research, and notably lead to much better-constrained paleogeographic models. In this presentation, we will share some of the insights and strategies that emerged from the workshop, including the advantages of conducting paleomagnetic analysis at the site-level, the application of emerging paleomagnetic Euler pole analysis frameworks, and the use of insights extracted from Earth-like geodynamic models (which self-generate plate tectonic behavior) to further constrain the results of these paleomagnetic methods. We also present some preliminary results of early experiments putting these strategies into practice on a paleomagnetic dataset from North America.

Owing to the axial symmetry of the Earth's magnetic field, paleomagnetic data only directly record the latitudinal and azimuthal positions of crustal blocks in the past, and paleolongitude cannot be constrained. An ability to overcome this obstacle is thus of fundamental importance to paleogeographic reconstruction. Paleomagnetic Euler pole (PEP) analysis presents a unique means to recover such information, but prior implementations of the PEP method have incorporated subjective decisions into its execution, undercutting its fidelity and rigor. Here we present an optimization approach to PEP analysis that addresses some of these deficiencies-namely the objective identification of change-points and small-circle arcs that together approximate an apparent polar wander path. We elaborate on our novel methodology and conduct some experiments with synthetic data to demonstrate its performance. We furthermore present implementations of our methods both as adaptable, stand-alone scripts in Python and as a streamlined interactive workflow that can be operated through a web browser.

Consider an i.i.d. sample from an unknown density function supported on an unknown manifold embedded in a high dimensional Euclidean space. We tackle the problem of learning a distance between points, able to capture both the geometry of the manifold and the underlying density. We prove the convergence of this microscopic distance, as the sample size goes to infinity, to a macroscopic one that we call Fermat distance as it minimizes a path functional, resembling Fermat principle in optics. The proof boils down to the study of geodesics in Euclidean first-passage percolation for nonhomogeneous Poisson point processes.

Owing to the inherent axial symmetry of the Earth’s magnetic field, paleomagnetic data only directly record the latitudinal and azimuthal positions of crustal blocks in the past, and paleolongitude cannot be constrained. An ability to overcome this obstacle is thus of fundamental importance to paleogeographic reconstruction. Paleomagnetic Euler pole (PEP) analysis presents a unique means to recover such information, but prior implementations of the PEP method have incorporated subjective decisions into its execution, undercutting its fidelity and rigor. Here we introduce an optimization approach to PEP analysis that addresses some of these deficiencies---namely the objective identification of change-points and small-circle arcs that together approximate an apparent polar wander path. Equipped with a new analytical approach to PEP analysis we turn to consider its feasibility in terms of its potential in the context of the theoretical and practical limitations of paleomagnetic data. In this presentation, we will share some key insights that emerged through experimentation with randomly-drawn plate kinematic models.

We introduce heat engines working in the nanoregime that allow one to extract a finite amount of deterministic work. Using the resource theory approach to themodynamics, we show that the efficiency of these cycles is strictly smaller than Carnot's, and we associate this difference with a fundamental irreversibility that is present in single-shot transformations. When fluctuations in the extracted work are allowed there is a trade-off between their size and the efficiency. As the size of fluctuations increases so does the efficiency and optimal efficiency is attained for unbounded fluctuations, while a certain amount of deterministic work is drawn from the cycle. Finally, we show that when the working medium is composed of many particles, by creating an amount of correlations between the subsystems that scale logarithmically with their number, Carnot's efficiency can also be approached in the asymptotic limit along with deterministic work extraction.