In the context of science, the well-known adage “a picture is worth a thousand words” might well be “a model is worth a thousand datasets.” Scientific models, such as Newtonian physics or biological gene regulatory networks, are human-driven simplifications of complex phenomena that serve as surrogates for the countless experiments that validated the models. Recently, machine learning has been able to overcome the inaccuracies of approximate modeling by directly learning the entire set of nonlinear interactions from data. However, without any predetermined structure from the scientific basis behind the problem, machine learning approaches are flexible but data-expensive, requiring large databases of homogeneous labeled training data. A central challenge is reconciling data that is at odds with simplified models without requiring "big data". In this work demonstrate how a mathematical object, which we denote universal differential equations (UDEs), can be utilized as a theoretical underpinning to a diverse array of problems in scientific machine learning to yield efficient algorithms and generalized approaches. The UDE model augments scientific models with machine-learnable structures for scientifically-based learning. We show how UDEs can be utilized to discover previously unknown governing equations, accurately extrapolate beyond the original data, and accelerate model simulation, all in a time and data-efficient manner. This advance is coupled with open-source software that allows for training UDEs which incorporate physical constraints, delayed interactions, implicitly-defined events, and intrinsic stochasticity in the model. Our examples show how a diverse set of computationally-difficult modeling issues across scientific disciplines, from automatically discovering biological mechanisms to accelerating the training of physics-informed neural networks and large-eddy simulations, can all be transformed into UDE training problems that are efficiently solved by a single software methodology.

We numerically explore convection and general circulation of an ocean, encased in a spherical shell of uniform thickness, heated from below by a spatially uniform heat flux, and whose temperature at the upper surface is relaxed to the freezing point of water. The role of salt is not considered. We describe the phenomenology and equilibrium solutions across a broad range of two key non‐dimensional numbers: the natural Rossby number, a measure of the influence of rotation, and the ratio of inner to outer radius of the moon's ocean, a measure of the geometry of the moon's tangent cylinder. Two distinct regimes of circulation are identified, both dominated by Taylor columns aligned with the rotation axis—“plumes” and “rolls” which predominate inside and outside the tangent cylinder, respectively. Inside the tangent cylinder, convective plumes align with Taylor columns that extend from the bottom to the ice shell. The plumes energize geostrophic turbulence which in turn generates a general circulation consisting counter‐rotating zonal jets. Moreover, the plumes are efficient at transferring heat from the bottom to the surface, resulting in loss of heat from the ocean to the polar ice. If the plumes are suppressed rolls outside the tangent cylinder become the dominant mode of heat transfer resulting in equatorial cooling. We conclude that if moons such as Enceladus and Europa were to be predominantly heated from below, they will likely have a “Jovian‐like” circulation: an unstratified, turbulent, geostrophically controlled ocean with strong “Taylor column” behavior and a circulation dominated by counter‐rotating zonal jets.

Oceananigans.jl is a fast and friendly software package for the numerical simulation of incompressible, stratified, rotating fluid flows on CPUs and GPUs. Oceananigans.jl is fast and flexible enough for research yet simple enough for students and first-time programmers. Oceananigans.jl is being developed as part of the Climate Modeling Alliance project for the simulation of small-scale ocean physics at high-resolution that affect the evolution of Earth's climate.

ter understand their biases and uncertainties. • Parameterization parameter distributions, learned using high-resolution simulations, should be used as prior distributions for climate modeling studies.

Parameterizations of unresolved turbulent processes often compromise the fidelity of large-scale ocean models. In this work, we argue for a Bayesian approach to the refinement and evaluation of turbulence parameterizations. Using an ensemble of large eddy simulations of turbulent penetrative convection in the surface boundary layer, we demonstrate the method by estimating the uncertainty of parameters in the convective limit of the popular "K-Profile Parameterization." We uncover structural deficiencies and propose an alternative scaling that overcomes them. Plain Language SummaryClimate projections are often compromised by significant uncertainties which stem from the representation of physical processes that cannot be resolved-such as clouds in the atmosphere and turbulent swirls in the ocean-but which have to be parameterized. We propose a methodology for improving parameterizations in which they are tested against, and tuned to, high-resolution numerical simulations of subdomains that represent them more completely. A Bayesian methodology is used to calibrate the parameterizations against the highly resolved model, to assess their fidelity and identify shortcomings. Most importantly, the approach provides estimates of parameter uncertainty. While the method is illustrated for a particular parameterization of boundary layer mixing, it can be applied to any parameterization.

We explore how neural differential equations (NDEs) may be trained on highly resolved fluid-dynamical models of unresolved scales providing an ideal framework for data-driven parameterizations in climate models. NDEs overcome some of the limitations of traditional neural networks (NNs) in fluid dynamical applications in that they can readily incorporate conservation laws and boundary conditions and are stable when integrated over time. We advocate a method that employs a 'residual' approach, in which the NN is used to improve upon an existing parameterization through the representation of residual fluxes which are not captured by the base parameterization. This reduces the amount of training required and providing a method for capturing up-gradient and nonlocal fluxes. As an illustrative example, we consider the parameterization of free convection of the oceanic boundary layer triggered by buoyancy loss at the surface. We demonstrate that a simple parameterization of the process -convective adjustment -can be improved upon by training a NDE against highly resolved explicit models, to capture entrainment fluxes at the base of the well-mixed layer, fluxes that convective adjustment itself cannot represent. The augmented parameterization outperforms existing commonly used parameterizations such as the K-Profile Parameterization (KPP). We showcase that the NDE performs well independent of the time-stepper and that an online training approach using differentiable simulation via the Julia scientific machine learning software stack improves accuracy by an order-of-magnitude. We conclude that NDEs provide an exciting route forward to the development of representations of sub-grid-scale processes for climate science, opening up myriad new opportunities.

Between 5% and 25% of the total momentum transferred between the atmosphere and ocean is transmitted via the growth of long surface gravity waves called ''swell.'' In this paper, we use large-eddy simulations to show that swelltransmitted momentum excites near-inertial waves and drives turbulent mixing that deepens a rotating, stratified, turbulent ocean surface boundary layer. We find that swell-transmitted currents are less effective at producing turbulence and mixing the boundary layer than currents driven by an effective surface stress. Overall, however, the differences between swelldriven and surface-stress-driven boundary layers are relatively minor. In consequence, our results corroborate assumptions made in Earth system models that neglect the vertical structure of swell-transmitted momentum fluxes and instead parameterize all air-sea momentum transfer processes with an effective surface stress.

We infer circumpolar maps of stress imparted to the ocean by the wind, mediated by sea-ice, in and around the Seasonal Ice Zone (SIZ) of Antarctica. In the open ocean we compute the wind stress using surface winds from daily atmospheric reanalyses and applying bulk formulae. In the presence of sea ice, the stress imparted to the underlying ocean is computed from satellite observations of daily ice concentration and drift velocity assuming, first, that the ocean geostrophic currents beneath are negligible, and then including surface geostrophic ocean currents inferred from satellite altimetry. In this way maps of surface ocean stress in the SIZ are obtained. The maps are discussed and interpreted, and their importance in setting the circulation emphasised. Just as in parallel observational studies in the Arctic, we find that ocean currents significantly modify the stress field, the sense of the surface ageostrophic flow and thus pathways of exchange across the SIZ. Maps of Ekman pumping reveal broad patterns of upwelling within the SIZ enhanced near the sea ice edge, which are offset by strong narrow downwelling regions adjacent to the Antarctic continent.