Physics Informed Neural Networks for Simulating Radiative Transfer

We propose a novel machine learning algorithm for simulating radiative transfer. Our algorithm is based on physics informed neural networks (PINNs), which are trained by minimizing the residual of the underlying radiative tranfer equations. We present extensive experiments and theoretical error estimates to demonstrate that PINNs provide a very easy to implement, fast, robust and accurate method for simulating radiative transfer. We also present a PINN based algorithm for simulating inverse problems for radiative transfer efficiently

A Multi-level procedure for enhancing accuracy of machine learning algorithms

We propose a multi-level method to increase the accuracy of machine learning algorithms for approximating observables in scientific computing, particularly those that arise in systems modeled by differential equations. The algorithm relies on judiciously combining a large number of computationally cheap training data on coarse resolutions with a few expensive training samples on fine grid resolutions. Theoretical arguments for lowering the generalization error, based on reducing the variance of the underlying maps, are provided and numerical evidence, indicating significant gains over underlying single-level machine learning algorithms, are presented. Moreover, we also apply the multi-level algorithm in the context of forward uncertainty quantification and observe a considerable speed-up over competing algorithms

Neural Inverse Operators for Solving PDE Inverse Problems

A large class of inverse problems for PDEs are only well-defined as mappings from operators to functions. Existing operator learning frameworks map functions to functions and need to be modified to learn inverse maps from data. We propose a novel architecture termed Neural Inverse Operators (NIOs) to solve these PDE inverse problems. Motivated by the underlying mathematical structure, NIO is based on a suitable composition of DeepONets and FNOs to approximate mappings from operators to functions. A variety of experiments are presented to demonstrate that NIOs significantly outperform baselines and solve PDE inverse problems robustly, accurately and are several orders of magnitude faster than existing direct and PDE-constrained optimization methods

Representation Equivalent Neural Operators: a Framework for Alias-free Operator Learning

Recently, operator learning, or learning mappings between infinite-dimensional function spaces, has garnered significant attention, notably in relation to learning partial differential equations from data. Conceptually clear when outlined on paper, neural operators necessitate discretization in the transition to computer implementations. This step can compromise their integrity, often causing them to deviate from the underlying operators. This research offers a fresh take on neural operators with a framework Representation equivalent Neural Operators (ReNO) designed to address these issues. At its core is the concept of operator aliasing, which measures inconsistency between neural operators and their discrete representations. We explore this for widely-used operator learning techniques. Our findings detail how aliasing introduces errors when handling different discretizations and grids and loss of crucial continuous structures. More generally, this framework not only sheds light on existing challenges but, given its constructive and broad nature, also potentially offers tools for developing new neural operators.Comment: 28 page

Gaia's Cepheids and RR Lyrae Stars and Luminosity Calibrations Based on Tycho-Gaia Astrometric Solution

Gaia Data Release 1 contains parallaxes for more than 700 Galactic Cepheids and RR Lyrae stars, computed as part of the Tycho-Gaia Astrometric Solution (TGAS). We have used TGAS parallaxes, along with literature ($V, I, J, {K_\mathrm{s}}, W_1$) photometry and spectroscopy, to calibrate the zero point of the Period-Luminosity and Period-Wesenheit relations of classical and type II Cepheids, and the near-infrared Period-Luminosity, Period-Luminosity-Metallicity and optical Luminosity-Metallicity relations of RR Lyrae stars. In this contribution we briefly summarise results obtained by fitting these basic relations adopting different techniques that operate either in parallax or distance (absolute magnitude) space.Comment: 5 pages, 4 figures, proceedings for the 22nd Los Alamos Stellar Pulsation Conference Series Meeting "Wide field variability surveys: a 21st-century perspective", held in San Pedro de Atacama, Chile, Nov. 28 - Dec. 2, 201

Updated theoretical Period-Age and Period-Age-Color relations for Galactic Classical Cepheids: an application to the Gaia DR2 sample

Updated evolutionary and pulsational model predictions are combined in order to interpret the properties of Galactic Classical Cepheids in the Gaia Data Release 2. In particular, the location of the instability strip boundaries and the analytical relations connecting pulsation periods to the intrinsic stellar parameters are combined with evolutionary tracks to derive reliable and accurate period-age, and the first theoretical period-age-color relations in the Gaia bands for a solar chemical abundance pattern ($Z$=$0.02$, $Y$=$0.28$). The adopted theoretical framework takes into account possible variations in the mass-luminosity relation for the core helium-burning stage as due to changes in the core convective overshooting and/or mass loss efficiency, as well as the impact on the instability strip boundaries due to different assumptions for superadiabatic convection efficiency. The inferred period-age and period-age-color relations are applied to a selected sample of both fundamental and first overtone Gaia Cepheids, and individual ages for the various adopted theoretical scenarios are derived. The retrieved age distributions confirm that a variation in the efficiency of superadiabatic convection in the pulsational model computations has a negligible effect, whereas a brighter Mass-Luminosity relation, as produced by mild overshooting, rotation or mass loss, implies significantly older age predictions. Moreover, older Cepheids are found at larger Galactocentric distances, while first overtone Cepheids are found to be systematically older than the fundamental ones. The comparison with independent age distribution analysis in literature supports the predictive capability of current theoretical framework.Comment: 14 pages, 11 figures, 8 tables, accepted for publication in MNRA