Towards Interpretable Deep Learning Models for Knowledge Tracing

As an important technique for modeling the knowledge states of learners, the traditional knowledge tracing (KT) models have been widely used to support intelligent tutoring systems and MOOC platforms. Driven by the fast advancements of deep learning techniques, deep neural network has been recently adopted to design new KT models for achieving better prediction performance. However, the lack of interpretability of these models has painfully impeded their practical applications, as their outputs and working mechanisms suffer from the intransparent decision process and complex inner structures. We thus propose to adopt the post-hoc method to tackle the interpretability issue for deep learning based knowledge tracing (DLKT) models. Specifically, we focus on applying the layer-wise relevance propagation (LRP) method to interpret RNN-based DLKT model by backpropagating the relevance from the model's output layer to its input layer. The experiment results show the feasibility using the LRP method for interpreting the DLKT model's predictions, and partially validate the computed relevance scores from both question level and concept level. We believe it can be a solid step towards fully interpreting the DLKT models and promote their practical applications in the education domain

Cooling of young stars growing by disk accretion

In the initial formation stages young stars must acquire a significant fraction of their mass by accretion from a circumstellar disk that forms in the center of a collapsing protostellar cloud. Throughout this period mass accretion rates through the disk can reach 10^{-6}-10^{-5} M_Sun/yr leading to substantial energy release in the vicinity of stellar surface. We study the impact of irradiation of the stellar surface produced by the hot inner disk on properties of accreting fully convective low-mass stars, and also look at objects such as young brown dwarfs and giant planets. At high accretion rates irradiation raises the surface temperature of the equatorial region above the photospheric temperature T_0 that a star would have in the absence of accretion. The high-latitude (polar) parts of the stellar surface, where disk irradiation is weak, preserve their temperature at the level of T_0. In strongly irradiated regions an almost isothermal outer radiative zone forms on top of the fully convective interior, leading to the suppression of the local internal cooling flux derived from stellar contraction (similar suppression occurs in irradiated ``hot Jupiters''). Properties of this radiative zone likely determine the amount of thermal energy that gets advected into the convective interior of the star. Total intrinsic luminosity integrated over the whole stellar surface is reduced compared to the non-accreting case, by up to a factor of several in some systems (young brown dwarfs, stars in quasar disks, forming giants planets), potentially leading to the retardation of stellar contraction. Stars and brown dwarfs irradiated by their disks tend to lose energy predominantly through their cool polar regions while young giant planets accreting through the disk cool through their whole surface.Comment: 14 pages, 6 figures, submitted to Ap

Hydrogen Burning on Magnetar Surfaces

We compute the rate of diffusive nuclear burning for hydrogen on the surface of a "magnetar" (Soft Gamma-Ray Repeater or Anomalous X-Ray Pulsar). We find that hydrogen at the photosphere will be burned on an extremely rapid timescale of hours to years, depending on composition of the underlying material. Improving on our previous studies, we explore the effect of a maximally thick "inert" helium layer, previously thought to slow down the burning rate. Since hydrogen diffuses faster in helium than through heavier elements, we find this helium buffer actually increases the burning rate for magnetars. We compute simple analytic scalings of the burning rate with temperature and magnetic field for a range of core temperature. We conclude that magnetar photospheres are very unlikely to contain hydrogen. This motivates theoretical work on heavy element atmospheres that are needed to measure effective temperature from the observed thermal emission and constrains models of AXPs that rely on magnetar cooling through thick light element envelopes.Comment: 4 pages, 2 figures, To be published in ApJ Letter

R-modes in Neutron Stars with Crusts: Turbulent Saturation, Spin-down, and Crust Melting

Rossby waves (r-modes) have been suggested as a means to regulate the spin periods of young or accreting neutron stars, and also to produce observable gravitational wave radiation. R-modes involve primarily transverse, incompressive motions of the star's fluid core. However, neutron stars gain crusts early in their lives: therefore, r-modes also imply shear in the fluid beneath the crust. We examine the criterion for this shear layer to become turbulent, and derive the rate of dissipation in the turbulent regime. Unlike dissipation from a viscous boundary layer, turbulent energy loss is nonlinear in mode energy and can therefore cause the mode to saturate at amplitudes typically much less than unity. This energy loss also reappears as heat below the crust. We study the possibility of crust melting as well as its implications for the spin evolution of low-mass X-ray binaries. Lastly, we identify some universal features of the spin evolution that may have observational consequences.Comment: 12 pages, 4 figures, submitted to Ap

Seismology of the Accreting White Dwarf in GW Lib

We present a first analysis of the g-mode oscillation spectrum for the white dwarf (WD) primary of GW Lib, a faint cataclysmic variable (CV). Stable periodicities have been observed from this WD for a number of years, but their interpretation as stellar pulsations has been hampered by a lack of theoretical models appropriate to an accreting WD. Using the results of Townsley and Bildsten, we construct accreting models for the observed effective temperature and approximate mass of the WD in GW Lib. We compute g-mode frequencies for a range of accreted layer masses, Macc, and long term accretion rates, . If we assume that the observed oscillations are from l=1 g-modes, then the observed periods are matched when M ~= 1.02 Msun, Macc ~= 0.31 x 10^-4 Msun and ~= 7.3 x 10^-11 Msun/yr. Much more sensitive observations are needed to discover more modes, after which we will be able to more accurately measure these parameters and constrain or measure the WD's rotation rate.Comment: 4 pages, 3 figures; uses emulateapj; Accepted by the Astrophysical Journal Letter

Differential rotation of nonlinear r-modes

Differential rotation of r-modes is investigated within the nonlinear theory up to second order in the mode amplitude in the case of a slowly-rotating, Newtonian, barotropic, perfect-fluid star. We find a nonlinear extension of the linear r-mode, which represents differential rotation that produces large scale drifts of fluid elements along stellar latitudes. This solution includes a piece induced by first-order quantities and another one which is a pure second-order effect. Since the latter is stratified on cylinders, it cannot cancel differential rotation induced by first-order quantities, which is not stratified on cylinders. It is shown that, unlikely the situation in the linearized theory, r-modes do not preserve vorticity of fluid elements at second-order. It is also shown that the physical angular momentum and energy of the perturbation are, in general, different from the corresponding canonical quantities.Comment: 9 pages, revtex4; section III revised, comments added in Introduction and Conclusions, references updated; to appear in Phys. Rev.

Reionization Constraints on the Contribution of Primordial Compact Objects to Dark Matter

Many lines of evidence suggest that nonbaryonic dark matter constitutes roughly 30% of the critical closure density, but the composition of this dark matter is unknown. One class of candidates for the dark matter is compact objects formed in the early universe, with typical masses M between 0.1 and 1 solar masses to correspond to the mass scale of objects found with microlensing observing projects. Specific candidates of this type include black holes formed at the epoch of the QCD phase transition, quark stars, and boson stars. Here we show that accretion onto these objects produces substantial ionization in the early universe, with an optical depth to Thomson scattering out to z=1100 of approximately tau=2-4 [f_CO\epsilon_{-1}(M/Msun)]^{1/2} (H_0/65)^{-1}, where \epsilon_{-1} is the accretion efficiency \epsilon\equiv L/{\dot M}c^2 divided by 0.1 and f_CO is the fraction of matter in the compact objects. The current upper limit to the scattering optical depth, based on the anisotropy of the microwave background, is approximately 0.4. Therefore, if accretion onto these objects is relatively efficient, they cannot be the main component of nonbaryonic dark matter.Comment: 12 pages including one figure, uses aaspp4, submitted to Ap

Bulk viscosity in the nonlinear and anharmonic regime of strange quark matter

The bulk viscosity of cold, dense three-flavor quark matter is studied as a function of temperature and the amplitude of density oscillations. The study is also extended to the case of two different types of anharmonic oscillations of density. We point several qualitative effects due to the anharmonicity, although quantitatively they appear to be relatively small. We also find that, in most regions of the parameter space, with the exception of the case of a very large amplitude of density oscillations (i.e. 10% and above), nonlinear effects and anharmonicity have a small effect on the interplay of the nonleptonic and semileptonic processes in the bulk viscosity.Comment: 14 pages, 6 figures; v2: Appendix B is omitted, a few new discussions added and some new references adde

Developing a catalogue of explainability methods to support expert and non-expert users.

Organisations face growing legal requirements and ethical responsibilities to ensure that decisions made by their intelligent systems are explainable. However, provisioning of an explanation is often application dependent, causing an extended design phase and delayed deployment. In this paper we present an explainability framework formed of a catalogue of explanation methods, allowing integration to a range of projects within a telecommunications organisation. These methods are split into low-level explanations, high-level explanations and co-created explanations. We motivate and evaluate this framework using the specific case-study of explaining the conclusions of field engineering experts to non-technical planning staff. Feedback from an iterative co-creation process and a qualitative evaluation is indicative that this is a valuable development tool for use in future company projects

Ambipolar diffusion in superfluid neutron stars

In this paper we reconsider the problem of magnetic field diffusion in neutron star cores. We model the star as consisting of a mixture of neutrons, protons and electrons, and allow for particle reactions and binary collisions between species. Our analysis is in much the same spirit as that of Goldreich & Reisenegger (1992), and we content ourselves with rough estimates of magnetic diffusion timescales, rather than solving accurately for some particular field geometry. However, our work improves upon previous treatments in one crucial respect: we allow for superfluidity in the neutron star matter. We find that the consequent mutual friction force, coupling the neutrons and charged particles, together with the suppression of particles collisions and reactions, drastically affect the ambipolar magnetic field diffusion timescale. In particular, the addition of superfluidity means that it is unlikely that there is ambipolar diffusion in magnetar cores on the timescale of the lifetimes of these objects, contradicting an assumption often made in the modelling of the flaring activity commonly observed in magnetars. Our work suggests that if a decaying magnetic field is indeed the cause of magnetar activity, the field evolution is likely to take place outside of the core, and might represent Hall/Ohmic diffusion in the stellar crust, or else that a mechanism other than standard ambipolar diffusion is active, e.g. flux expulsion due to the interaction between neutron vortices and magnetic fluxtubes.Comment: Paper changed to incorporate comments from referee. To appear in MNRA