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Regularization is one of the most fundamental topics in optimization, statistics and machine learning. To get sparsity in estimating a parameter $u\in\mathbb{R}^d$, an $\ell_q$ penalty term, $\Vert u\Vert_q$, is usually added to the objective function. What is the probabilistic distribution corresponding to such $\ell_q$ penalty? What is the correct stochastic process corresponding to $\Vert u\Vert_q$ when we model functions $u\in L^q$? This is important for statistically modeling large dimensional objects, e.g. images, with penalty to preserve certainty properties, e.g. edges in the image. In this work, we generalize the $q$-exponential distribution (with density proportional to) $\exp{(- \frac{1}{2}|u|^q)}$ to a stochastic process named $Q$-exponential (Q-EP) process that corresponds to the $L_q$ regularization of functions. The key step is to specify consistent multivariate $q$-exponential distributions by choosing from a large family of elliptic contour distributions. The work is closely related to Besov process which is usually defined by the expanded series. Q-EP can be regarded as a definition of Besov process with explicit probabilistic formulation and direct control on the correlation length. From the Bayesian perspective, Q-EP provides a flexible prior on functions with sharper penalty ($q<2$) than the commonly used Gaussian process (GP). We compare GP, Besov and Q-EP in modeling functional data, reconstructing images, and solving inverse problems and demonstrate the advantage of our proposed methodology., Comment: 21 pages, 15 figures

We present OrbNet Denali, a machine learning model for electronic structure that is designed as a drop-in replacement for ground-state density functional theory (DFT) energy calculations. The model is a message-passing neural network that uses symmetry-adapted atomic orbital features from a low-cost quantum calculation to predict the energy of a molecule. OrbNet Denali is trained on a vast dataset of 2.3 million DFT calculations on molecules and geometries. This dataset covers the most common elements in bio- and organic chemistry (H, Li, B, C, N, O, F, Na, Mg, Si, P, S, Cl, K, Ca, Br, I) as well as charged molecules. OrbNet Denali is demonstrated on several well-established benchmark datasets, and we find that it provides accuracy that is on par with modern DFT methods while offering a speedup of up to three orders of magnitude. For the GMTKN55 benchmark set, OrbNet Denali achieves WTMAD-1 and WTMAD-2 scores of 7.19 and 9.84, on par with modern DFT functionals. For several GMTKN55 subsets, which contain chemical problems that are not present in the training set, OrbNet Denali produces a mean absolute error comparable to those of DFT methods. For the Hutchison conformers benchmark set, OrbNet Denali has a median correlation coefficient of R^2=0.90 compared to the reference DLPNO-CCSD(T) calculation, and R^2=0.97 compared to the method used to generate the training data (wB97X-D3/def2-TZVP), exceeding the performance of any other method with a similar cost. Similarly, the model reaches chemical accuracy for non-covalent interactions in the S66x10 dataset. For torsional profiles, OrbNet Denali reproduces the torsion profiles of wB97X-D3/def2-TZVP with an average MAE of 0.12 kcal/mol for the potential energy surfaces of the diverse fragments in the TorsionNet500 dataset.

The balanced development of cognitive and noncognitive skills is crucial not only for well-rounded child development in schools but also for career and life success. Unlike academic learning standards, however, there are rarely comparable "socioemotional learning" (SEL) standards. Under the pressure of high-stakes testing policies ("educational accountability 1.0") during the No Child Left Behind Act (NCLB) era, public schools paid more attention to academic performance in core subjects as measured by standardized test scores at the expense of subjects that are not assessed. The Every Student Succeeds Act (ESSA), which replaced NCLB in 2015, gives state policymakers opportunities to redesign educational accountability systems and incorporate nonacademic measures. While the Common Core State Standards (CCSS) introduced more rigorous curriculum standards and assessments, the full and accurate measurement of academic progress was hindered by a test opt-out movement. This study applies mixed methods to assess states' standards and measures for children's socioemotional learning. Drawing on state policy documents, survey measures, and the National Survey of Children's Health (NSCH) data, the study explores the SEL standards policy movement, and the standards' potential impact and challenges. In light of broad-based educational mission and policy shifts under the ESSA, this study helps fill the knowledge gap in school accountability policy. The premise of this study is that systemic improvement of a child's social ecology -- including family, school, and neighborhood environment -- should work in tandem with SEL standards and measures, particularly for high-risk children who are vulnerable due to multiple environmental disadvantages.

The ability to understand and engineer molecular structures relies on having accurate descriptions of the energy as a function of atomic coordinates. Here we outline a new paradigm for deriving energy functions of hyperdimensional molecular systems, which involves generating data for low-dimensional systems in virtual reality (VR) to then efficiently train atomic neural networks (ANNs). This generates high quality data for specific areas of interest within the hyperdimensional space that characterizes a molecule's potential energy surface (PES). We demonstrate the utility of this approach by gathering data within VR to train ANNs on chemical reactions involving fewer than 8 heavy atoms. This strategy enables us to predict the energies of much higher-dimensional systems, e.g. containing nearly 100 atoms. Training on datasets containing only 15K geometries, this approach generates mean absolute errors around 2 kcal/mol. This represents one of the first times that an ANN-PES for a large reactive radical has been generated using such a small dataset. Our results suggest VR enables the intelligent curation of high-quality data, which accelerates the learning process.

Simulating drug binding and unbinding is a challenge, as the rugged energy landscapes that separate bound and unbound states require extensive sampling that consumes significant computational resources. Here, we describe the use of interactive molecular dynamics in virtual reality (iMD-VR) as an accurate low-cost strategy for flexible protein-ligand docking. We outline an experimental protocol which enables expert iMD-VR users to guide ligands into and out of the binding pockets of trypsin, neuraminidase, and HIV-1 protease, and recreate their respective crystallographic protein-ligand binding poses within 5 - 10 minutes. Following a brief training phase, our studies shown that iMD-VR novices were able to generate unbinding and rebinding pathways on similar timescales as iMD-VR experts, with the majority able to recover binding poses within 2.15 Angstrom RMSD of the crystallographic binding pose. These results indicate that iMD-VR affords sufficient control for users to carry out the detailed atomic manipulations required to dock flexible ligands into dynamic enzyme active sites and recover crystallographic poses, offering an interesting new approach for simulating drug docking and generating binding hypotheses., Comment: PLOS ONE

As molecular scientists have made progress in their ability to engineer nano-scale molecular structure, we are facing new challenges in our ability to engineer molecular dynamics (MD) and flexibility. Dynamics at the molecular scale differs from the familiar mechanics of everyday objects, because it involves a complicated, highly correlated, and three-dimensional many-body dynamical choreography which is often non-intuitive even for highly trained researchers. We recently described how interactive molecular dynamics in virtual reality (iMD-VR) can help to meet this challenge, enabling researchers to manipulate real-time MD simulations of flexible structures in 3D. In this article, we outline efforts to extend immersive technologies to the molecular sciences, and we introduce 'Narupa', a flexible, open-source, multi-person iMD-VR software framework which enables groups of researchers to simultaneously cohabit real-time simulation environments to interactively visualize and manipulate the dynamics of molecular structures with atomic-level precision. We outline several application domains where iMD-VR is facilitating research, communication, and creative approaches within the molecular sciences, including training machines to learn reactive potential energy surfaces (PESs), biomolecular conformational sampling, protein-ligand binding, reaction discovery using 'on-the-fly' quantum chemistry, and transport dynamics in materials. We touch on iMD-VR's various cognitive and perceptual affordances, and how these provide research insight for molecular systems. By synergistically combining human spatial reasoning and design insight with computational automation, technologies like iMD-VR have the potential to improve our ability to understand, engineer, and communicate microscopic dynamical behavior, offering the potential to usher in a new paradigm for engineering molecules and nano-architectures.