arXiv subjects

Jianfeng Lu

Publications and source records attributed to Jianfeng Lu.

At least 235 records · Page 13Linked to original sources

arXiv2018

Learning interacting particle systems: diffusion parameter estimation for aggregation equations

In this article, we study the parameter estimation of interacting particle systems subject to the Newtonian aggregation and Brownian diffusion. Specifically, we construct an estimator $\widehatν$ with partial observed data to approximate the diffusion parameter $ν$, and the estimation error is achieved. Furthermore, we extend this result to general aggregation equations with a bounded Lipschitz interaction field.

math.AP↗

arXiv2018

Analysis of multiscale integrators for multiple attractors and irreversible Langevin samplers

We study multiscale integrator numerical schemes for a class of stiff stochastic differential equations (SDEs). We consider multiscale SDEs with potentially multiple attractors that behave as diffusions on graphs as the stiffness parameter goes to its limit. Classical numerical discretization schemes, such as the Euler-Maruyama scheme, become unstable as the stiffness parameter converges to its limit and appropriate multiscale integrators can correct for this. We rigorously establish the convergence of the numerical method to the related diffusion on graph, identifying the appropriate choice of discretization parameters. Theoretical results are supplemented by numerical studies on the problem of the recently developing area of introducing irreversibility in Langevin samplers in order to accelerate convergence to equilibrium.

math.NA↗

arXiv2018

Stop memorizing: A data-dependent regularization framework for intrinsic pattern learning

Deep neural networks (DNNs) typically have enough capacity to fit random data by brute force even when conventional data-dependent regularizations focusing on the geometry of the features are imposed. We find out that the reason for this is the inconsistency between the enforced geometry and the standard softmax cross entropy loss. To resolve this, we propose a new framework for data-dependent DNN regularization, the Geometrically-Regularized-Self-Validating neural Networks (GRSVNet). During training, the geometry enforced on one batch of features is simultaneously validated on a separate batch using a validation loss consistent with the geometry. We study a particular case of GRSVNet, the Orthogonal-Low-rank Embedding (OLE)-GRSVNet, which is capable of producing highly discriminative features residing in orthogonal low-rank subspaces. Numerical experiments show that OLE-GRSVNet outperforms DNNs with conventional regularization when trained on real data. More importantly, unlike conventional DNNs, OLE-GRSVNet refuses to memorize random data or random labels, suggesting it only learns intrinsic patterns by reducing the memorizing capacity of the baseline DNN.

cs.CV↗

arXiv2018

Double Path Networks for Sequence to Sequence Learning

Encoder-decoder based Sequence to Sequence learning (S2S) has made remarkable progress in recent years. Different network architectures have been used in the encoder/decoder. Among them, Convolutional Neural Networks (CNN) and Self Attention Networks (SAN) are the prominent ones. The two architectures achieve similar performances but use very different ways to encode and decode context: CNN use convolutional layers to focus on the local connectivity of the sequence, while SAN uses self-attention layers to focus on global semantics. In this work we propose Double Path Networks for Sequence to Sequence learning (DPN-S2S), which leverage the advantages of both models by using double path information fusion. During the encoding step, we develop a double path architecture to maintain the information coming from different paths with convolutional layers and self-attention layers separately. To effectively use the encoded context, we develop a cross attention module with gating and use it to automatically pick up the information needed during the decoding step. By deeply integrating the two paths with cross attention, both types of information are combined and well exploited. Experiments show that our proposed method can significantly improve the performance of sequence to sequence learning over state-of-the-art systems.

cs.CL↗

arXiv2018

The full configuration interaction quantum Monte Carlo method in the lens of inexact power iteration

In this paper, we propose a general analysis framework for inexact power iteration, which can be used to efficiently solve high dimensional eigenvalue problems arising from quantum many-body problems. Under the proposed framework, we establish the convergence theorems for several recently proposed randomized algorithms, including the full configuration interaction quantum Monte Carlo (FCIQMC) and the fast randomized iteration (FRI). The analysis is consistent with numerical experiments for physical systems such as Hubbard model and small chemical molecules. We also compare the algorithms both in convergence analysis and numerical results.

math.NA↗

arXiv2018

Stochastic dynamical low-rank approximation method

In this paper, we extend the dynamical low-rank approximation method to the space of finite signed measures. Under this framework, we derive stochastic low-rank dynamics for stochastic differential equations (SDEs) coming from classical stochastic dynamics or unraveling of Lindblad quantum master equations. We justify the proposed method by error analysis and also numerical examples for applications in solving high-dimensional SDE, stochastic Burgers' equation, and high-dimensional Lindblad equation.

math.NA↗

arXiv2018

Integrated Tempering Enhanced Sampling Method as the Infinite Switching Limit of Simulated Tempering

Fast and accurate sampling method is in high demand, in order to bridge the large gaps between molecular dynamic simulations and experimental observations. Recently, integrated tempering enhanced sampling method (ITS) has been proposed and successfully applied to various biophysical examples, significantly accelerating conformational sampling. The mathematical validation for its effectiveness has not been elucidated yet. Here we show that the integrated tempering enhanced sampling method can be viewed as a reformulation of the infinite switching limit of simulated tempering method over a mixed potential. Moreover, we demonstrate that the efficiency of simulated tempering molecular dynamics (STMD) improves as the frequency of switching between the temperatures is increased, based on the large deviation principle of empirical distributions. Our theory provides the theoretical justification of the advantage of ITS. Finally, we illustrate the utility of the infinite switching simulated tempering method through several numerical examples.

math.NA↗

arXiv2018

Solving parametric PDE problems with artificial neural networks

The curse of dimensionality is commonly encountered in numerical partial differential equations (PDE), especially when uncertainties have to be modeled into the equations as random coefficients. However, very often the variability of physical quantities derived from a PDE can be captured by a few features on the space of the coefficient fields. Based on such an observation, we propose using a neural-network (NN) based method to parameterize the physical quantity of interest as a function of input coefficients. The representability of such quantity using a neural-network can be justified by viewing the neural-network as performing time evolution to find the solutions to the PDE. We further demonstrate the simplicity and accuracy of the approach through notable examples of PDEs in engineering and physics.

math.NA↗

arXiv2018

Phase Space Sketching for Crystal Image Analysis based on Synchrosqueezed Transforms

Recent developments of imaging techniques enable researchers to visualize materials at the atomic resolution to better understand the microscopic structures of materials. This paper aims at automatic and quantitative characterization of potentially complicated microscopic crystal images, providing feedback to tweak theories and improve synthesis in materials science. As such, an efficient phase-space sketching method is proposed to encode microscopic crystal images in a translation, rotation, illumination, and scale invariant representation, which is also stable with respect to small deformations. Based on the phase-space sketching, we generalize our previous analysis framework for crystal images with simple structures to those with complicated geometry.

cond-mat.mtrl-sci↗

arXiv2018

Optimal artificial boundary condition for random elliptic media

We are given a uniformly elliptic coefficient field that we regard as a realization of a stationary and finite-range (say, range unity) ensemble of coefficient fields. Given a (deterministic) right-hand-side supported in a ball of size $\ell\gg 1$ and of vanishing average, we are interested in an algorithm to compute the (gradient of the) solution near the origin, just using the knowledge of the (given realization of the) coefficient field in some large box of size $L\gg\ell$. More precisely, we are interested in the most seamless (artificial) boundary condition on the boundary of the computational domain of size $L$. Motivated by the recently introduced multipole expansion in random media, we propose an algorithm. We rigorously establish an error estimate (on the level of the gradient) in terms of $L\gg\ell\gg 1$, using recent results in quantitative stochastic homogenization. More precisely, our error estimate has an a priori and an a posteriori aspect: With a priori overwhelming probability, the (random) prefactor can be bounded by a constant that is computable without much further effort, on the basis of the given realization in the box of size $L$. We also rigorously establish that the order of the error estimate in both $L$ and $\ell$ is optimal, where in this paper we focus on the case of $d=2$. This amounts to a lower bound on the variance of the quantity of interest when conditioned on the coefficients inside the computational domain, and relies on the deterministic insight that a sensitivity analysis wrt a defect commutes with (stochastic) homogenization. Finally, we carry out numerical experiments that show that this optimal convergence rate already sets in at only moderately large $L$, and that more naive boundary conditions perform worse both in terms of rate and prefactor.

math.NA↗

arXiv2018

Detecting localized eigenstates of linear operators

We describe a way of detecting the location of localized eigenvectors of a linear system $Ax = λx$ for eigenvalues $λ$ with $|λ|$ comparatively large. We define the family of functions $f_α: \left\{1.2. \dots, n\right\} \rightarrow \mathbb{R}_{}$ $$ f_α(k) = \log \left( \| A^α e_k \|_{\ell^2} \right),$$ where $α\geq 0$ is a parameter and $e_k = (0,0,\dots, 0,1,0, \dots, 0)$ is the $k-$th standard basis vector. We prove that eigenvectors associated to eigenvalues with large absolute value localize around local maxima of $f_α$: the metastable states in the power iteration method (slowing down its convergence) can be used to predict localization. We present a fast randomized algorithm and discuss different examples: a random band matrix, discretizations of the local operator $-Δ+ V$ and the nonlocal operator $(-Δ)^{3/4} + V$.

math.NA↗

arXiv2018

Game-Theoretic Design of Optimal Two-Sided Rating Protocols for Service Exchange Dilemma in Crowdsourcing

Despite the increasing popularity and successful examples of crowdsourcing, it is stripped of aureole when collective efforts are derailed or severely hindered by elaborate sabotage. A service exchange dilemma arises when non-cooperation among self-interested users, and zero social welfare is obtained at myopic equilibrium. Traditional rating protocols are not effective to overcome the inefficiency of the socially undesirable equilibrium due to specific features of crowdsourcing: a large number of anonymous users having asymmetric service requirements, different service capabilities, and dynamically joining/leaving a crowdsourcing platform with imperfect monitoring. In this paper, we develop the first game-theoretic design of the two-sided rating protocol to stimulate cooperation among self-interested users, which consists of a recommended strategy and a rating update rule. The recommended strategy recommends a desirable behavior from three predefined plans according to intrinsic parameters, while the rating update rule involves the update of ratings of both users, and uses differential punishments that punish users with different ratings differently. By quantifying necessary and sufficient conditions for a sustainable social norm, we formulate the problem of designing an optimal two-sided rating protocol that maximizes the social welfare among all sustainable protocols, provide design guidelines for optimal two-sided rating protocols and a low-complexity algorithm to select optimal design parameters in an alternate manner. Finally, evaluation results show the validity and effectiveness of our protocol designed for service exchange dilemma in crowdsourcing.

cs.GT↗

arXiv2018

On discrete Wigner transforms

In this work, we derive a discrete analog of the Wigner transform over the space $(\mathbb{C}^p)^{\otimes N}$ for any prime $p$ and any positive integer $N$. We show that the Wigner transform over this space can be constructed as the inverse Fourier transform of the standard Pauli matrices for $p=2$ or more generally of the Heisenberg-Weyl group elements for $p > 2$. We connect our work to a previous construction by Wootters of a discrete Wigner transform by showing that for all $p$, Wootters' construction corresponds to taking the inverse symplectic Fourier transform instead of the inverse Fourier transform. Finally, we discuss some implications of these results for the numerical simulation of many-body quantum spin systems.

math-ph↗

arXiv2018

Solving for high dimensional committor functions using artificial neural networks

In this note we propose a method based on artificial neural network to study the transition between states governed by stochastic processes. In particular, we aim for numerical schemes for the committor function, the central object of transition path theory, which satisfies a high-dimensional Fokker-Planck equation. By working with the variational formulation of such partial differential equation and parameterizing the committor function in terms of a neural network, approximations can be obtained via optimizing the neural network weights using stochastic algorithms. The numerical examples show that moderate accuracy can be achieved for high-dimensional problems.

cs.LG↗

arXiv2018

Riesz Energy on the Torus: Regularity of Minimizers

We study sets of $N$ points on the $d-$dimensional torus $\mathbb{T}^d$ minimizing interaction functionals of the type \[ \sum_{i, j =1 \atop i \neq j}^{N}{ f(x_i - x_j)}. \] The main result states that for a class of functions $f$ that behave like Riesz energies $f(x) \sim \|x\|^{-s}$ for $0< s < d$, the minimizing configuration of points has optimal regularity w.r.t. a Fourier-analytic regularity measure that arises in the study of irregularities of distribution. A particular consequence is that they are optimal quadrature points in the space of trigonometric polynomials up to a certain degree. The proof extends to other settings and also covers less singular functions such as $f(x) = \exp\bigl(- N^{\frac{2}{d}} \|x\|^2 \bigr)$.

math-ph↗

arXiv2018

Fundamental limitations for measurements in quantum many-body systems

Dynamical measurement schemes are an important tool for the investigation of quantum many-body systems, especially in the age of quantum simulation. Here, we address the question whether generic measurements can be implemented efficiently if we have access to a certain set of experimentally realizable measurements and can extend it through time evolution. For the latter, two scenarios are considered (a) evolution according to unitary circuits and (b) evolution due to Hamiltonians that we can control in a time-dependent fashion. We find that the time needed to realize a certain measurement to a predefined accuracy scales in general exponentially with the system size -- posing a fundamental limitation. The argument is based, on the construction of $\varepsilon$-packings for manifolds of observables with identical spectra and a comparison of their cardinalities to those of $\varepsilon$-coverings for quantum circuits and unitary time-evolution operators. The former is related to the study of Grassmann manifolds.

quant-ph↗

arXiv2018

Accelerated sampling by infinite swapping of path integral molecular dynamics with surface hopping

To accelerate the thermal equilibrium sampling of multi-level quantum systems, the infinite swapping limit of a recently proposed multi-level ring polymer representation is investigated. In the infinite swapping limit, the ring polymer evolves according to an averaged Hamiltonian with respect to all possible surface index configurations of the ring polymer, thus connects the surface hopping approach to the mean-field path integral molecular dynamics. A multiscale integrator for the infinite swapping limit is also proposed to enable efficient sampling based on the limiting dynamics. Numerical results demonstrate the huge improvement of sampling efficiency of the infinite swapping compared with the direct simulation of path integral molecular dynamics with surface hopping.

physics.chem-ph↗

arXiv2018

A Quantum Kinetic Monte Carlo Method for Quantum Many-body Spin Dynamics

We propose a general framework of quantum kinetic Monte Carlo algorithm, based on a stochastic representation of a series expansion of the quantum evolution. Two approaches have been developed in the context of quantum many-body spin dynamics, using different decomposition of the Hamiltonian. The effectiveness of the methods is tested for many-body spin systems up to 40 spins.

physics.comp-ph↗