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Data-driven surrogate models have shown immense potential as quick, inexpensive approximations to complex numerical and experimental modelling tasks. However, most surrogate models characterising physical systems do not quantify their uncertainty, rendering their predictions unreliable, and needing further validation. Though Bayesian approximations offer some solace in estimating the error associated with these models, they cannot provide they cannot provide guarantees, and the quality of their inferences depends on the availability of prior information and good approximations to posteriors for complex problems. This is particularly pertinent to multi-variable or spatio-temporal problems. Our work constructs and formalises a conformal prediction framework that satisfies marginal coverage for spatio-temporal predictions in a model-agnostic manner, requiring near-zero computational costs. The paper provides an extensive empirical study of the application of the framework to ascertain valid error bars that provide guaranteed coverage across the surrogate model's domain of operation. The application scope of our work extends across a large range of spatio-temporal models, ranging from solving partial differential equations to weather forecasting. Through the applications, the paper looks at providing statistically valid error bars for deterministic models, as well as crafting guarantees to the error bars of probabilistic models. The paper concludes with a viable conformal prediction formalisation that provides guaranteed coverage of the surrogate model, regardless of model architecture, and its training regime and is unbothered by the curse of dimensionality.

Neural weather models have shown immense potential as inexpensive and accurate alternatives to physics-based models. However, most models trained to perform weather forecasting do not quantify the uncertainty associated with their forecasts. This limits the trust in the model and the usefulness of the forecasts. In this work we construct and formalise a conformal prediction framework as a post-processing method for estimating this uncertainty. The method is model-agnostic and gives calibrated error bounds for all variables, lead times and spatial locations. No modifications are required to the model and the computational cost is negligible compared to model training. We demonstrate the usefulness of the conformal prediction framework on a limited area neural weather model for the Nordic region. We further explore the advantages of the framework for deterministic and probabilistic models.

Predicting plasma evolution within a Tokamak reactor is crucial to realizing the goal of sustainable fusion. Capabilities in forecasting the spatio-temporal evolution of plasma rapidly and accurately allow us to quickly iterate over design and control strategies on current Tokamak devices and future reactors. Modelling plasma evolution using numerical solvers is often expensive, consuming many hours on supercomputers, and hence, we need alternative inexpensive surrogate models. We demonstrate accurate predictions of plasma evolution both in simulation and experimental domains using deep learning-based surrogate modelling tools, viz., Fourier Neural Operators (FNO). We show that FNO has a speedup of six orders of magnitude over traditional solvers in predicting the plasma dynamics simulated from magnetohydrodynamic models, while maintaining a high accuracy (MSE in the normalised domain $\approx$ $10^{-5}$). Our modified version of the FNO is capable of solving multi-variable Partial Differential Equations (PDE), and can capture the dependence among the different variables in a single model. FNOs can also predict plasma evolution on real-world experimental data observed by the cameras positioned within the MAST Tokamak, i.e., cameras looking across the central solenoid and the divertor in the Tokamak. We show that FNOs are able to accurately forecast the evolution of plasma and have the potential to be deployed for real-time monitoring. We also illustrate their capability in forecasting the plasma shape, the locations of interactions of the plasma with the central solenoid and the divertor for the full (available) duration of the plasma shot within MAST. The FNO offers a viable alternative for surrogate modelling as it is quick to train and infer, and requires fewer data points, while being able to do zero-shot super-resolution and getting high-fidelity solutions.

Predicting plasma evolution within a Tokamak is crucial to building a sustainable fusion reactor. Whether in the simulation space or within the experimental domain, the capability to forecast the spatio-temporal evolution of plasma field variables rapidly and accurately could improve active control methods on current tokamak devices and future fusion reactors. In this work, we demonstrate the utility of using Fourier Neural Operator (FNO) to model the plasma evolution in simulations and experiments. Our work shows that the FNO is capable of predicting magnetohydrodynamic models governing the plasma dynamics, 6 orders of magnitude faster than the traditional numerical solver, while maintaining considerable accuracy (NMSE $\sim 10^{-5})$. Our work also benchmarks the performance of the FNO against other standard surrogate models such as Conv-LSTM and U-Net and demonstrate that the FNO takes significantly less time to train, requires less parameters and outperforms other models. We extend the FNO approach to model the plasma evolution observed by the cameras positioned within the MAST spherical tokamak. We illustrate its capability in forecasting the formation of filaments within the plasma as well as the heat deposits. The FNO deployed to model the camera is capable of forecasting the full length of the plasma shot within half the time of the shot duration.

Classical sequential models employed in time-series prediction rely on learning the mappings from the past to the future instances by way of a hidden state. The Hidden states characterise the historical information and encode the required temporal dependencies. However, most existing sequential models operate within finite-dimensional Euclidean spaces which offer limited functionality when employed in modelling physics relevant data. Alternatively recent work with neural operator learning within the Fourier space has shown efficient strategies for parameterising Partial Differential Equations (PDE). In this work, we propose a novel sequential model, built to handle Physics relevant data by way of amalgamating the conventional RNN architecture with that of the Fourier Neural Operators (FNO). The Fourier-RNN allows for learning the mappings from the input to the output as well as to the hidden state within the Fourier space associated with the temporal data. While the Fourier-RNN performs identical to the FNO when handling PDE data, it outperforms the FNO and the conventional RNN when deployed in modelling noisy, non-Markovian data.

The relative roles of mergers and star formation in regulating galaxy growth are still a matter of intense debate. We here present our DECODE, a new Discrete statistical sEmi-empiriCal mODEl specifically designed to predict rapidly and efficiently, in a full cosmological context, galaxy assembly and merger histories for any given input stellar mass-halo mass (SMHM) relation. DECODE generates object-by-object dark matter merger trees (hence discrete) from accurate subhalo mass and infall redshift probability functions (hence statistical) for all subhaloes, including those residing within other subhaloes, with virtually no resolution limits on mass or volume. Merger trees are then converted into galaxy assembly histories via an input, redshift dependent SMHM relation, which is highly sensitive to the significant systematics in the galaxy stellar mass function and on its evolution with cosmic time. DECODE can accurately reproduce the predicted mean galaxy merger rates and assembly histories of hydrodynamic simulations and semi-analytic models, when adopting in input their SMHM relations. In the present work we use DECODE to prove that only SMHM relations implied by stellar mass functions characterized by large abundances of massive galaxies and significant redshift evolution, at least at $M_\star \gtrsim 10^{11} \, M_\odot$, can simultaneously reproduce the local abundances of satellite galaxies, the galaxy (major merger) pairs since $z \sim 3$, and the growth of Brightest Cluster Galaxies. The same models can also reproduce the local fraction of elliptical galaxies, on the assumption that these are strictly formed by major mergers, but not the full bulge-to-disc ratio distributions, which require additional processes., Comment: MNRAS, accepted, 29 pages, 25 figures

In previous work we showed that a semi-empirical model in which galaxies in host dark matter haloes are assigned stellar masses via a stellar mass-halo mass (SMHM) relation and sizes ($R_e$) via a linear and tight $R_e-R_h$ relation, can faithfully reproduce the size function of local SDSS central galaxies and the strong size evolution of massive galaxies (MGs, $M_{\rm star}>10^{11.2}M_\odot$). In this third paper of the series, we focus on the population of satellite MGs. We find that without any additional calibration and irrespective of the exact SMHM relation, fraction of quenched galaxies or level of stellar stripping, the same model is able to reproduce the local size function of quiescent satellite MGs in SDSS. In addition, the same model can reproduce the puzzling weak dependence of mean size on host halo mass for both central and satellite galaxies. The model also matches the size function of starforming satellite MGs, after assuming that some of them transform into massive lenticulars in a few Gyr after infalling in the group/cluster environment. However, the vast majority of satellite lenticulars is predicted to form before infall. The $R_e-R_h$ appears to be fundamental to connect galaxies and their host haloes., Comment: Accepted for publication in MNRAS. 5 pages, 3 figures. Comments and suggestions still welcome

Hydrodynamical simulations of galaxy formation and evolution attempt to fully model the physics that shapes galaxies. The agreement between the morphology of simulated and real galaxies, and the way the morphological types are distributed across galaxy scaling relations are important probes of our knowledge of galaxy formation physics. Here we propose an unsupervised deep learning approach to perform a stringent test of the fine morphological structure of galaxies coming from the Illustris and IllustrisTNG (TNG100 and TNG50) simulations against observations from a subsample of the Sloan Digital Sky Survey. Our framework is based on PixelCNN, an autoregressive model for image generation with an explicit likelihood. We adopt a strategy that combines the output of two PixelCNN networks in a metric that isolates the fine morphological details of galaxies from the sky background. We are able to \emph{quantitatively} identify the improvements of IllustrisTNG, particularly in the high-resolution TNG50 run, over the original Illustris. However, we find that the fine details of galaxy structure are still different between observed and simulated galaxies. This difference is driven by small, more spheroidal, and quenched galaxies which are globally less accurate regardless of resolution and which have experienced little improvement between the three simulations explored. We speculate that this disagreement, that is less severe for quenched disky galaxies, may stem from a still too coarse numerical resolution, which struggles to properly capture the inner, dense regions of quenched spheroidal galaxies., Comment: Accepted for publication in MNRAS. Comments welcome

The masses of supermassive black holes at the centres of local galaxies appear to be tightly correlated with the mass and velocity dispersions of their galactic hosts. However, the local Mbh-Mstar relation inferred from dynamically measured inactive black holes is up to an order-of-magnitude higher than some estimates from active black holes, and recent work suggests that this discrepancy arises from selection bias on the sample of dynamical black hole mass measurements. In this work we combine X-ray measurements of the mean black hole accretion luminosity as a function of stellar mass and redshift with empirical models of galaxy stellar mass growth, integrating over time to predict the evolving Mbh-Mstar relation. The implied relation is nearly independent of redshift, indicating that stellar and black hole masses grow, on average, at similar rates. Matching the de-biased local Mbh-Mstar relation requires a mean radiative efficiency ~0.15, in line with theoretical expectations for accretion onto spinning black holes. However, matching the "raw" observed relation for inactive black holes requires a mean radiative efficiency around 0.02, far below theoretical expectations. This result provides independent evidence for selection bias in dynamically estimated black hole masses, a conclusion that is robust to uncertainties in bolometric corrections, obscured active black hole fractions, and kinetic accretion efficiency. For our fiducial assumptions, they favour moderate-to-rapid spins of typical supermassive black holes, to achieve a mean radiative efficiency ~0.12-0.20. Our approach has similarities to the classic Soltan analysis, but by using galaxy-based data instead of integrated quantities we are able to focus on regimes where observational uncertainties are minimized., Comment: 12 pages, 8 Figures. MNRAS, accepted. A discussion around the points raised by arXiv:1909.10821 is included in the Appendix

The mass and structural assembly of galaxies is a matter of intense debate. Current theoretical models predict the existence of a linear relationship between galaxy size ($R_e$) and the host dark matter halo virial radius ($R_h$).\\ By making use of semi-empirical models compared to the size distributions of central galaxies from the Sloan Digital Sky Survey, we provide robust constraints on the normalization and scatter of the $R_e-R_h$ relation. We explore the parameter space of models in which the $R_e-R_h$ relation is mediated by either the spin parameter or the concentration of the host halo, or a simple constant the nature of which is in principle unknown. We find that the data require extremely tight relations for both early-type and late-type galaxies (ETGs,LTGs), especially for more massive galaxies. These constraints challenge models based solely on angular momentum conservation, which predict significantly wider distributions of galaxy sizes and no trend with stellar mass, if taken at face value. We discuss physically-motivated alterations to the original models that bring the predictions into better agreement with the data. We argue that the measured tight size distributions of SDSS disk galaxies can be reproduced by semi-empirical models in which the $R_e-R_h$ connection is mediated by the \emph{stellar} specific angular momenta $j_{star}.$ We find that current cosmological models of galaxy formation broadly agree with our constraints for LTGs, and justify the strong link between $R_e$ and $j_{star}$ that we propose, however the tightness of the $R_e-R_h$ relation found in such ab-initio theoretical models for ETGs is in tension with our semi-empirical findings., Comment: 22 pages, 14 figures

A supermassive black hole has been found at the centre of nearly every galaxy observed with sufficient sensitivity. The masses of these black holes are observed to increase with either the total mass or the mean (random) velocity of the stars in their host galaxies. The origin of these correlations remains elusive. Observational systematics and biases severely limit our knowledge of the local demography of supermassive black holes thus preventing accurate model comparisons and progress in this field. Here we show that the large-scale spatial distribution of local active galactic nuclei (AGN), believed to be accreting supermassive black holes, can constrain the shape and normalization of the black hole-stellar mass relation thus bypassing resolution-related observational biases. In turn, our results can set more stringent constraints on the so-called "radiative efficiency", a fundamental parameter describing the inner physics of supermassive black holes that is closely linked to their spin, geometry, and ability to release energy. The mean value of the radiative efficiency can be estimated by comparing the average total luminous output of AGN with the relic mass density locked up in quiescent supermassive black holes at galaxy centres today. For currently accepted values of the AGN obscured fractions and bolometric corrections, our newest estimates of the local supermassive black hole mass density favour mean radiative efficiencies of ~10-20\%, suggesting that the vast majority of supermassive black holes are spinning moderately to rapidly. With large-scale AGN surveys coming online, our novel methodology will enable even tighter constraints on the fundamental parameters that regulate the growth of supermassive black holes., Comment: 31 pages, 7 Figures. Accepted

We carry out a systematic investigation of the total mass density profile of massive (Mstar~3e11 Msun) early-type galaxies and its dependence on redshift, specifically in the range 0