In this paper, we present a new methodology to develop arbitrary high-order structure-preserving methods for solving the quantum Zakharov system. The key ingredients of our method are: (i) the original Hamiltonian energy is reformulated into a quadratic form by introducing a new quadratic auxiliary variable; (ii) based on the energy variational principle, the original system is then rewritten into a new equivalent system which inherits the mass conservation law and a quadratic energy; (iii) the resulting system is discretized by symplectic Runge-Kutta method in time combining with the Fourier pseudo-spectral method in space. The proposed method achieves arbitrary high-order accurate in time and can preserve the discrete mass and original Hamiltonian energy exactly. Moreover, an efficient iterative solver is presented to solve the resulting discrete nonlinear equations. Finally, ample numerical examples are presented to demonstrate the theoretical claims and illustrate the efficiency of our methods.
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Linear-quadratic regulator (LQR) is a landmark problem in the field of optimal control, which is the concern of this paper. Generally, LQR is classified into state-feedback LQR (SLQR) and output-feedback LQR (OLQR) based on whether the full state is obtained. It has been suggested in existing literature that both the SLQR and the OLQR could be viewed as \textit{constrained nonconvex matrix optimization} problems in which the only variable to be optimized is the feedback gain matrix. In this paper, we introduce a first-order accelerated optimization framework of handling the LQR problem, and give its convergence analysis for the cases of SLQR and OLQR, respectively. Specifically, a Lipschiz Hessian property of LQR performance criterion is presented, which turns out to be a crucial property for the application of modern optimization techniques. For the SLQR problem, a continuous-time hybrid dynamic system is introduced, whose solution trajectory is shown to converge exponentially to the optimal feedback gain with Nesterov-optimal order $1-\frac{1}{\sqrt{\kappa}}$ ($\kappa$ the condition number). Then, the symplectic Euler scheme is utilized to discretize the hybrid dynamic system, and a Nesterov-type method with a restarting rule is proposed that preserves the continuous-time convergence rate, i.e., the discretized algorithm admits the Nesterov-optimal convergence order. For the OLQR problem, a Hessian-free accelerated framework is proposed, which is a two-procedure method consisting of semiconvex function optimization and negative curvature exploitation. In a time $\mathcal{O}(\epsilon^{-7/4}\log(1/\epsilon))$, the method can find an $\epsilon$-stationary point of the performance criterion; this entails that the method improves upon the $\mathcal{O}(\epsilon^{-2})$ complexity of vanilla gradient descent. Moreover, our method provides the second-order guarantee of stationary point.
Recent developments in deep learning have made remarkable progress in speeding up the prediction of quantum chemical (QC) properties by removing the need for expensive electronic structure calculations like density functional theory. However, previous methods learned from 1D SMILES sequences or 2D molecular graphs failed to achieve high accuracy as QC properties primarily depend on the 3D equilibrium conformations optimized by electronic structure methods, far different from the sequence-type and graph-type data. In this paper, we propose a novel approach called Uni-Mol+ to tackle this challenge. Uni-Mol+ first generates a raw 3D molecule conformation from inexpensive methods such as RDKit. Then, the raw conformation is iteratively updated to its target DFT equilibrium conformation using neural networks, and the learned conformation will be used to predict the QC properties. To effectively learn this update process towards the equilibrium conformation, we introduce a two-track Transformer model backbone and train it with the QC property prediction task. We also design a novel approach to guide the model's training process. Our extensive benchmarking results demonstrate that the proposed Uni-Mol+ significantly improves the accuracy of QC property prediction in various datasets. We have made the code and model publicly available at \url{//github.com/dptech-corp/Uni-Mol}.
Many stochastic processes in the physical and biological sciences can be modelled using Brownian dynamics with multiplicative noise. However, numerical integrators for these processes can lose accuracy or even fail to converge when the diffusion term is configuration-dependent. One remedy is to construct a transform to a constant-diffusion process and sample the transformed process instead. In this work, we explain how coordinate-based and time-rescaling-based transforms can be used either individually or in combination to map a general class of variable-diffusion Brownian motion processes into constant-diffusion ones. The transforms are invertible, thus allowing recovery of the original dynamics. We motivate our methodology using examples in one dimension before then considering multivariate diffusion processes. We illustrate the benefits of the transforms through numerical simulations, demonstrating how the right combination of integrator and transform can improve computational efficiency and the order of convergence to the invariant distribution. Notably, the transforms that we derive are applicable to a class of multibody, anisotropic Stokes-Einstein diffusion that has applications in biophysical modelling.
Limited by imaging systems, the reconstruction of Magnetic Resonance Imaging (MRI) images from partial measurement is essential to medical imaging research. Benefiting from the diverse and complementary information of multi-contrast MR images in different imaging modalities, multi-contrast Super-Resolution (SR) reconstruction is promising to yield SR images with higher quality. In the medical scenario, to fully visualize the lesion, radiologists are accustomed to zooming the MR images at arbitrary scales rather than using a fixed scale, as used by most MRI SR methods. In addition, existing multi-contrast MRI SR methods often require a fixed resolution for the reference image, which makes acquiring reference images difficult and imposes limitations on arbitrary scale SR tasks. To address these issues, we proposed an implicit neural representations based dual-arbitrary multi-contrast MRI super-resolution method, called Dual-ArbNet. First, we decouple the resolution of the target and reference images by a feature encoder, enabling the network to input target and reference images at arbitrary scales. Then, an implicit fusion decoder fuses the multi-contrast features and uses an Implicit Decoding Function~(IDF) to obtain the final MRI SR results. Furthermore, we introduce a curriculum learning strategy to train our network, which improves the generalization and performance of our Dual-ArbNet. Extensive experiments in two public MRI datasets demonstrate that our method outperforms state-of-the-art approaches under different scale factors and has great potential in clinical practice.
This paper examines the approximation of log-determinant for large-scale symmetric positive definite matrices. Inspired by the variance reduction technique, we split the approximation of $\log\det(A)$ into two parts. The first to compute is the trace of the projection of $\log(A)$ onto a suboptimal subspace, while the second is the trace of the projection on the corresponding orthogonal complementary space. For these two approximations, the stochastic Lanczos quadrature method is used. Furthermore, in the construction of the suboptimal subspace, we utilize a projection-cost-preserving sketch to bound the size of the Gaussian random matrix and the dimension of the suboptimal subspace. We provide a rigorous error analysis for our proposed method and explicit lower bounds for its design parameters, offering guidance for practitioners. We conduct numerical experiments to demonstrate our method's effectiveness and illustrate the quality of the derived bounds.
The Independent Cutset problem asks whether there is a set of vertices in a given graph that is both independent and a cutset. Such a problem is $\textsf{NP}$-complete even when the input graph is planar and has maximum degree five. In this paper, we first present a $\mathcal{O}^*(1.4423^{n})$-time algorithm for the problem. We also show how to compute a minimum independent cutset (if any) in the same running time. Since the property of having an independent cutset is MSO$_1$-expressible, our main results are concerned with structural parameterizations for the problem considering parameters that are not bounded by a function of the clique-width of the input. We present $\textsf{FPT}$-time algorithms for the problem considering the following parameters: the dual of the maximum degree, the dual of the solution size, the size of a dominating set (where a dominating set is given as an additional input), the size of an odd cycle transversal, the distance to chordal graphs, and the distance to $P_5$-free graphs. We close by introducing the notion of $\alpha$-domination, which allows us to identify more fixed-parameter tractable and polynomial-time solvable cases.
Maximum weight independent set (MWIS) admits a $\frac1k$-approximation in inductively $k$-independent graphs and a $\frac{1}{2k}$-approximation in $k$-perfectly orientable graphs. These are a a parameterized class of graphs that generalize $k$-degenerate graphs, chordal graphs, and intersection graphs of various geometric shapes such as intervals, pseudo-disks, and several others. We consider a generalization of MWIS to a submodular objective. Given a graph $G=(V,E)$ and a non-negative submodular function $f: 2^V \rightarrow \mathbb{R}_+$, the goal is to approximately solve $\max_{S \in \mathcal{I}_G} f(S)$ where $\mathcal{I}_G$ is the set of independent sets of $G$. We obtain an $\Omega(\frac1k)$-approximation for this problem in the two mentioned graph classes. The first approach is via the multilinear relaxation framework and a simple contention resolution scheme, and this results in a randomized algorithm with approximation ratio at least $\frac{1}{e(k+1)}$. This approach also yields parallel (or low-adaptivity) approximations. Motivated by the goal of designing efficient and deterministic algorithms, we describe two other algorithms for inductively $k$-independent graphs that are inspired by work on streaming algorithms: a preemptive greedy algorithm and a primal-dual algorithm. In addition to being simpler and faster, these algorithms, in the monotone submodular case, yield the first deterministic constant factor approximations for various special cases that have been previously considered such as intersection graphs of intervals, disks and pseudo-disks.
In this paper, we propose a new formulation and a suitable finite element method for the steady coupling of viscous flow in deformable porous media using divergence-conforming filtration fluxes. The proposed method is based on the use of parameter-weighted spaces, which allows for a more accurate and robust analysis of the continuous and discrete problems. Furthermore, we conduct a solvability analysis of the proposed method and derive optimal error estimates in appropriate norms. These error estimates are shown to be robust in the case of large Lam\'e parameters and small permeability and storativity coefficients. To illustrate the effectiveness of the proposed method, we provide a few representative numerical examples, including convergence verification, poroelastic channel flow simulation, and test the robustness of block-diagonal preconditioners with respect to model parameters.
Given a target distribution $\pi$ and an arbitrary Markov infinitesimal generator $L$ on a finite state space $\mathcal{X}$, we develop three structured and inter-related approaches to generate new reversiblizations from $L$. The first approach hinges on a geometric perspective, in which we view reversiblizations as projections onto the space of $\pi$-reversible generators under suitable information divergences such as $f$-divergences. With different choices of functions $f$, we not only recover nearly all established reversiblizations but also unravel and generate new reversiblizations. Along the way, we unveil interesting geometric results such as bisection properties, Pythagorean identities, parallelogram laws and a Markov chain counterpart of the arithmetic-geometric-harmonic mean inequality governing these reversiblizations. This further serves as motivation for introducing the notion of information centroids of a sequence of Markov chains and to give conditions for their existence and uniqueness. Building upon the first approach, we view reversiblizations as generalized means. In this second approach, we construct new reversiblizations via different natural notions of generalized means such as the Cauchy mean or the dual mean. In the third approach, we combine the recently introduced locally-balanced Markov processes framework and the notion of convex $*$-conjugate in the study of $f$-divergence. The latter offers a rich source of balancing functions to generate new reversiblizations.
Recommender system is one of the most important information services on today's Internet. Recently, graph neural networks have become the new state-of-the-art approach of recommender systems. In this survey, we conduct a comprehensive review of the literature in graph neural network-based recommender systems. We first introduce the background and the history of the development of both recommender systems and graph neural networks. For recommender systems, in general, there are four aspects for categorizing existing works: stage, scenario, objective, and application. For graph neural networks, the existing methods consist of two categories, spectral models and spatial ones. We then discuss the motivation of applying graph neural networks into recommender systems, mainly consisting of the high-order connectivity, the structural property of data, and the enhanced supervision signal. We then systematically analyze the challenges in graph construction, embedding propagation/aggregation, model optimization, and computation efficiency. Afterward and primarily, we provide a comprehensive overview of a multitude of existing works of graph neural network-based recommender systems, following the taxonomy above. Finally, we raise discussions on the open problems and promising future directions of this area. We summarize the representative papers along with their codes repositories in //github.com/tsinghua-fib-lab/GNN-Recommender-Systems.
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