Deep generative chemistry models emerge as powerful tools to expedite drug discovery. However, the immense size and complexity of the structural space of all possible drug-like molecules pose significant obstacles, which could be overcome with hybrid architectures combining quantum computers with deep classical networks. As the first step toward this goal, we built a compact discrete variational autoencoder (DVAE) with a Restricted Boltzmann Machine (RBM) of reduced size in its latent layer. The size of the proposed model was small enough to fit on a state-of-the-art D-Wave quantum annealer and allowed training on a subset of the ChEMBL dataset of biologically active compounds. Finally, we generated 2331 novel chemical structures with medicinal chemistry and synthetic accessibility properties in the ranges typical for molecules from ChEMBL. The presented results demonstrate the feasibility of using already existing or soon-to-be-available quantum computing devices as testbeds for future drug discovery applications.
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This paper develops and benchmarks an immersed peridynamics method to simulate the deformation, damage, and failure of hyperelastic materials within a fluid-structure interaction framework. The immersed peridynamics method describes an incompressible structure immersed in a viscous incompressible fluid. It expresses the momentum equation and incompressibility constraint in Eulerian form, and it describes the structural motion and resultant forces in Lagrangian form. Coupling between Eulerian and Lagrangian variables is achieved by integral transforms with Dirac delta function kernels, as in standard immersed boundary methods. The major difference between our approach and conventional immersed boundary methods is that we use peridynamics, instead of classical continuum mechanics, to determine the structural forces. We focus on non-ordinary state-based peridynamic material descriptions that allow us to use a constitutive correspondence framework that can leverage well characterized nonlinear constitutive models of soft materials. The convergence and accuracy of our approach are compared to both conventional and immersed finite element methods using widely used benchmark problems of nonlinear incompressible elasticity. We demonstrate that the immersed peridynamics method yields comparable accuracy with similar numbers of structural degrees of freedom for several choices of the size of the peridynamic horizon. We also demonstrate that the method can generate grid-converged simulations of fluid-driven material damage growth, crack formation and propagation, and rupture under large deformations.
The proximal Galerkin finite element method is a high-order, low iteration complexity, nonlinear numerical method that preserves the geometric and algebraic structure of bound constraints in infinite-dimensional function spaces. This paper introduces the proximal Galerkin method and applies it to solve free boundary problems, enforce discrete maximum principles, and develop scalable, mesh-independent algorithms for optimal design. The paper leads to a derivation of the latent variable proximal point (LVPP) algorithm: an unconditionally stable alternative to the interior point method. LVPP is an infinite-dimensional optimization algorithm that may be viewed as having an adaptive barrier function that is updated with a new informative prior at each (outer loop) optimization iteration. One of the main benefits of this algorithm is witnessed when analyzing the classical obstacle problem. Therein, we find that the original variational inequality can be replaced by a sequence of semilinear partial differential equations (PDEs) that are readily discretized and solved with, e.g., high-order finite elements. Throughout this work, we arrive at several unexpected contributions that may be of independent interest. These include (1) a semilinear PDE we refer to as the entropic Poisson equation; (2) an algebraic/geometric connection between high-order positivity-preserving discretizations and certain infinite-dimensional Lie groups; and (3) a gradient-based, bound-preserving algorithm for two-field density-based topology optimization. The complete latent variable proximal Galerkin methodology combines ideas from nonlinear programming, functional analysis, tropical algebra, and differential geometry and can potentially lead to new synergies among these areas as well as within variational and numerical analysis.
Using diffusion models to solve inverse problems is a growing field of research. Current methods assume the degradation to be known and provide impressive results in terms of restoration quality and diversity. In this work, we leverage the efficiency of those models to jointly estimate the restored image and unknown parameters of the degradation model. In particular, we designed an algorithm based on the well-known Expectation-Minimization (EM) estimation method and diffusion models. Our method alternates between approximating the expected log-likelihood of the inverse problem using samples drawn from a diffusion model and a maximization step to estimate unknown model parameters. For the maximization step, we also introduce a novel blur kernel regularization based on a Plug \& Play denoiser. Diffusion models are long to run, thus we provide a fast version of our algorithm. Extensive experiments on blind image deblurring demonstrate the effectiveness of our method when compared to other state-of-the-art approaches.
Hawkes processes are often applied to model dependence and interaction phenomena in multivariate event data sets, such as neuronal spike trains, social interactions, and financial transactions. In the nonparametric setting, learning the temporal dependence structure of Hawkes processes is generally a computationally expensive task, all the more with Bayesian estimation methods. In particular, for generalised nonlinear Hawkes processes, Monte-Carlo Markov Chain methods applied to compute the doubly intractable posterior distribution are not scalable to high-dimensional processes in practice. Recently, efficient algorithms targeting a mean-field variational approximation of the posterior distribution have been proposed. In this work, we first unify existing variational Bayes approaches under a general nonparametric inference framework, and analyse the asymptotic properties of these methods under easily verifiable conditions on the prior, the variational class, and the nonlinear model. Secondly, we propose a novel sparsity-inducing procedure, and derive an adaptive mean-field variational algorithm for the popular sigmoid Hawkes processes. Our algorithm is parallelisable and therefore computationally efficient in high-dimensional setting. Through an extensive set of numerical simulations, we also demonstrate that our procedure is able to adapt to the dimensionality of the parameter of the Hawkes process, and is partially robust to some type of model mis-specification.
This study focuses on the presence of (multi)fractal structures in confined hadronic matter through the momentum distributions of mesons produced in proton-proton collisions between 23 GeV and 63 GeV. The analysis demonstrates that the $q$-exponential behaviour of the particle momentum distributions is consistent with fractal characteristics, exhibiting fractal structures in confined hadronic matter with features similar to those observed in the deconfined quark-gluon plasma (QGP) regime. Furthermore, the systematic analysis of meson production in hadronic collisions at energies below 1 TeV suggests that specific fractal parameters are universal, independently of confinement or deconfinement, while others may be influenced by the quark content of the produced meson. These results pave the way for further research exploring the implications of fractal structures on various physical distributions and offer insights into the nature of the phase transition between confined and deconfined regimes.
This paper presents the error analysis of numerical methods on graded meshes for stochastic Volterra equations with weakly singular kernels. We first prove a novel regularity estimate for the exact solution via analyzing the associated convolution structure. This reveals that the exact solution exhibits an initial singularity in the sense that its H\"older continuous exponent on any neighborhood of $t=0$ is lower than that on every compact subset of $(0,T]$. Motivated by the initial singularity, we then construct the Euler--Maruyama method, fast Euler--Maruyama method, and Milstein method based on graded meshes. By establishing their pointwise-in-time error estimates, we give the grading exponents of meshes to attain the optimal uniform-in-time convergence orders, where the convergence orders improve those of the uniform mesh case. Numerical experiments are finally reported to confirm the sharpness of theoretical findings.
We present a multigrid algorithm to solve efficiently the large saddle-point systems of equations that typically arise in PDE-constrained optimization under uncertainty. The algorithm is based on a collective smoother that at each iteration sweeps over the nodes of the computational mesh, and solves a reduced saddle-point system whose size depends on the number $N$ of samples used to discretized the probability space. We show that this reduced system can be solved with optimal $O(N)$ complexity. We test the multigrid method on three problems: a linear-quadratic problem for which the multigrid method is used to solve directly the linear optimality system; a nonsmooth problem with box constraints and $L^1$-norm penalization on the control, in which the multigrid scheme is used within a semismooth Newton iteration; a risk-adverse problem with the smoothed CVaR risk measure where the multigrid method is called within a preconditioned Newton iteration. In all cases, the multigrid algorithm exhibits very good performances and robustness with respect to all parameters of interest.
In genetic studies, haplotype data provide more refined information than data about separate genetic markers. However, large-scale studies that genotype hundreds to thousands of individuals may only provide results of pooled data, where only the total allele counts of each marker in each pool are reported. Methods for inferring haplotype frequencies from pooled genetic data that scale well with pool size rely on a normal approximation, which we observe to produce unreliable inference when applied to real data. We illustrate cases where the approximation breaks down, due to the normal covariance matrix being near-singular. As an alternative to approximate methods, in this paper we propose exact methods to infer haplotype frequencies from pooled genetic data based on a latent multinomial model, where the observed allele counts are considered integer combinations of latent, unobserved haplotype counts. One of our methods, latent count sampling via Markov bases, achieves approximately linear runtime with respect to pool size. Our exact methods produce more accurate inference over existing approximate methods for synthetic data and for data based on haplotype information from the 1000 Genomes Project. We also demonstrate how our methods can be applied to time-series of pooled genetic data, as a proof of concept of how our methods are relevant to more complex hierarchical settings, such as spatiotemporal models.
In spatial regression models, spatial heterogeneity may be considered with either continuous or discrete specifications. The latter is related to delineation of spatially connected regions with homogeneous relationships between variables (spatial regimes). Although various regionalization algorithms have been proposed and studied in the field of spatial analytics, methods to optimize spatial regimes have been largely unexplored. In this paper, we propose two new algorithms for spatial regime delineation, two-stage K-Models and Regional-K-Models. We also extend the classic Automatic Zoning Procedure to spatial regression context. The proposed algorithms are applied to a series of synthetic datasets and two real-world datasets. Results indicate that all three algorithms achieve superior or comparable performance to existing approaches, while the two-stage K-Models algorithm largely outperforms existing approaches on model fitting, region reconstruction, and coefficient estimation. Our work enriches the spatial analytics toolbox to explore spatial heterogeneous processes.
We investigate a class of parametric elliptic eigenvalue problems with homogeneous essential boundary conditions where the coefficients (and hence the solution $u$) may depend on a parameter $y$. For the efficient approximate evaluation of parameter sensitivities of the first eigenpairs on the entire parameter space we propose and analyse Gevrey class and analytic regularity of the solution with respect to the parameters. This is made possible by a novel proof technique which we introduce and demonstrate in this paper. Our regularity result has immediate implications for convergence of various numerical schemes for parametric elliptic eigenvalue problems, in particular, for elliptic eigenvalue problems with infinitely many parameters arising from elliptic differential operators with random coefficients.
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