Publications

Numerical methods for approximately solving partial differential equations (PDE) are at the core of scientific computing. Often, this requires high-resolution or adaptive discretization grids to capture relevant spatio-temporal features in the PDE solution, e.g., in applications like turbulence, combustion, and shock propagation. Numerical approximation also requires knowing the PDE in order to construct problem-specific discretizations. Systematically deriving such solution-adaptive discrete operators, however, is a current challenge. Here we present an artificial neural network architecture for data-driven learning of problemand resolution-specific local discretizations of nonlinear PDEs. Our proposed method achieves numerically stable discretization of the operators in an unknown nonlinear PDE by spatially and temporally adaptive parametric pooling on regular Cartesian grids, and by incorporating knowledge about discrete time integration. Knowing the actual PDE is not necessary, as solution data is sufficient to train the network to learn the discrete operators. A once-trained neural architecture model can be used to predict solutions of the PDE on larger spatial domains and for longer times than it was trained for, hence addressing the problem of PDE-constrained extrapolation from data. We present demonstrative examples on long-term forecasting of hard numerical problems including equation-free forecasting of non-linear dynamics of forced Burgers problem on coarse spatio-temporal grids.

We introduce c-lasso, a Python package that enables sparse and robust linear regression and classification with linear equality constraints. The underlying statistical forward model is assumed to be of the following form: [ y = X β+ σε subject to Cβ=0 ] Here, X ∈ℝ

mathematical illustrations by Alex H. Barnett: In Discriminating Data, Wendy Hui Kyong Chun reveals how polarization is a goal—not an error—within big data and machine learning. These methods, she argues, encode segregation, eugenics, and identity politics through their default assumptions and conditions. Correlation, which grounds big data's predictive potential, stems from twentieth-century eugenic attempts to “breed” a better future. Recommender systems foster angry clusters of sameness through homophily. Users are “trained” to become authentically predictable via a politics and technology of recognition. Machine learning and data analytics thus seek to disrupt the future by making disruption impossible.

Large surveys are providing a diversity of spectroscopic observations with Gaia alone set to deliver millions of Ca-triplet-region spectra across the Galaxy. We aim to understand the dimensionality of the chemical abundance information in the Gaia–Radial Velocity Spectrometer (RVS) data to inform galactic archeology pursuits. We fit a quadratic model of four primary sources of variability, described by labels of Teff, \mathrm{log}g, [Fe/H], and [α/Fe], to the normalized flux of 10,802 red-clump stars from the Gaia-RVS-like Abundances and Radial velocity Galactic Origins Survey (ARGOS). We examine the residuals between ARGOS spectra and the models and find that the models capture the flux variability across 85 percent of the wavelength region. The remaining residual variance is concentrated to the Ca-triplet features, at an amplitude up to 12 percent of the normalized flux. We use principal component analysis on the residuals and find orthogonal correlations in the Ca-triplet core and wings. This variability, not captured by our model, presumably marks departures from the completeness of the 1D LTE label description. To test the indication of low-dimensionality, we turn to abundance-space to infer how well we can predict measured [Si/H], [O/H], [Ca/H], [Ni/H], and [Al/H] abundances from the Gaia-RVS-like Radial Velocity Experiment survey with models of Teff, \mathrm{log}g, [Fe/H], and [Mg/Fe]. We find that we can nearly entirely predict these abundances. Using high-precision Apache Point Observatory Galactic Evolution Experiment abundances, we determine that a measurement uncertainty of <0.03 dex is required to capture additional information from these elements. This indicates that a four-label model sufficiently describes chemical abundance variance for an approximate signal-to-noise ratio <200 per pixel, in Gaia-RVS spectra.

Topological materials discovery has evolved at a rapid pace over the past 15 years following the identification of the first nonmagnetic topological insulators (TIs), topological crystalline insulators (TCIs), and 3D topological semimetals (TSMs). Most recently, through complete analyses of symmetry-allowed band structures - including the theory of Topological Quantum Chemistry (TQC) - researchers have determined crystal-symmetry-enhanced Wilson-loop and complete symmetry-based indicators for nonmagnetic topological phases, leading to the discovery of higher-order TCIs and TSMs. The recent application of TQC and related methods to high-throughput materials discovery has revealed that over half of all of the known stoichiometric, solid-state, nonmagnetic materials are topological at the Fermi level, over 85 percent of the known stoichiometric materials host energetically isolated topological bands, and that just under 2/3 of the energetically isolated bands in known materials carry the stable topology of a TI or TCI. In this Review, we survey topological electronic materials discovery in nonmagnetic crystalline solids from the prediction of the first 2D and 3D TIs to the recently introduced methods that have facilitated large-scale searches for topological materials. We also discuss future venues for the identification and manipulation of solid-state topological phases, including charge-density-wave compounds, magnetic materials, and 2D few-layer devices.

We develop a theory for manipulating the effective band structure of interacting helical edge states realized on the boundary of two-dimensional time-reversal symmetric topological insulators. For sufficiently strong interaction, an interacting edge band gap develops, spontaneously breaking time-reversal symmetry on the edge. The resulting spin texture, as well as the energy of the the time-reversal breaking gaps, can be tuned by an external moiré potential (i.e., a superlattice potential). Remarkably, we establish that by tuning the strength and period of the potential, the interacting gaps can be fully suppressed and interacting Dirac points re-emerge. In addition, nearly flat bands can be created by the moiré potential with a sufficiently long period. Our theory provides an unprecedented way to enhance the coherence length of interacting helical edges by suppressing the interacting gap. The implications of this finding for ongoing experiments on helical edge states is discussed.

We show that the onset of quantum chaos at infinite temperature in two many-body one-dimensional lattice models, the perturbed spin-1/2 XXZ and Anderson models, is characterized by universal behavior. Specifically, we show that the onset of quantum chaos is marked by maxima of the typical fidelity susceptibilities that scale with the square of the inverse average level spacing, saturating their upper bound, and that the strength of the integrability- or localization-breaking perturbation at these maxima decreases with increasing system size. We also show that the spectral function below the

We analyze a one-dimensional XXZ spin chain in a disordered magnetic field. As the main probes of the system's behavior we use the sensitivity of eigenstates to adiabatic transformations, as expressed through the fidelity susceptibility, in conjunction with the low frequency asymptotes of the spectral function. We identify a region of maximal chaos -- with exponentially enhanced susceptibility -- which separates the many-body localized phase from the diffusive ergodic phase. This regime is characterized by slow transport and we argue that the presence of such slow dynamics is incompatible with the localization transition in the thermodynamic limit. Instead of localizing, the system appears to enter a universal subdiffusive relaxation regime at moderate values of disorder, where the spectral function of the local longitudinal magnetization is inversely proportional to the frequency, corresponding to logarithmic in time relaxation of its auto-correlation function.

We investigate the interplay of electronic correlations and spin-orbit coupling (SOC) for a one-band and a two-band honeycomb lattice model. The main difference between the two models concerning SOC is that in the one-band case the SOC is a purely non-local term in the basis of the p

Physical systems obey strict symmetry principles. We expect that machine learning methods that intrinsically respect these symmetries should have higher prediction accuracy and better generalization in prediction of physical dynamics. In this work we implement a principled model based on invariant scalars, and release open-source code. We apply this Scalars method to a simple chaotic dynamical system, the springy double pendulum. We show that the Scalars method outperforms state-of-the-art approaches for learning the properties of physical systems with symmetries, both in terms of accuracy and speed. Because the method incorporates the fundamental symmetries, we expect it to generalize to different settings, such as changes in the force laws in the system.