Generative adversarial networks (GANs) have shown promising results when applied on partial differential equations and financial time series generation. We investigate if GANs can also be used to approximate one-dimensional Ito stochastic differential equations (SDEs). We propose a scheme that approximates the path-wise conditional distribution of SDEs for large time steps. Standard GANs are only able to approximate processes in distribution, yielding a weak approximation to the SDE. A conditional GAN architecture is proposed that enables strong approximation. We inform the discriminator of this GAN with the map between the prior input to the generator and the corresponding output samples, i.e. we introduce a `supervised GAN'. We compare the input-output map obtained with the standard GAN and supervised GAN and show experimentally that the standard GAN may fail to provide a path-wise approximation. The GAN is trained on a dataset obtained with exact simulation. The architecture was tested on geometric Brownian motion (GBM) and the Cox-Ingersoll-Ross (CIR) process. The supervised GAN outperformed the Euler and Milstein schemes in strong error on a discretisation with large time steps. It also outperformed the standard conditional GAN when approximating the conditional distribution. We also demonstrate how standard GANs may give rise to non-parsimonious input-output maps that are sensitive to perturbations, which motivates the need for constraints and regularisation on GAN generators.
Mathematical Institute, University of Oxford（牛津大学数学研究所）；The Alan Turing Institute（艾伦·图灵研究所）；Stanford（斯坦福大学）
Stochastic differential equations (SDEs) are a staple of mathematical modelling of temporal dynamics. However, a fundamental limitation has been that such models have typically been relatively inflexible, which recent work introducing Neural SDEs has sought to solve. Here, we show that the current classical approach to fitting SDEs may be approached as a special case of (Wasserstein) GANs, and in doing so the neural and classical regimes may be brought together. The input noise is Brownian motion, the output samples are time-evolving paths produced by a numerical solver, and by parameterising …
University of Oxford（牛津大学）；The Alan Turing Insitute（艾伦图灵研究所）；Stanford（斯坦福大学）
Neural SDEs combine many of the best qualities of both RNNs and SDEs: memory efficient training, high-capacity function approximation, and strong priors on model space. This makes them a natural choice for modelling many types of temporal dynamics. Training a Neural SDE (either as a VAE or as a GAN) requires backpropagating through an SDE solve. This may be done by solving a backwards-in-time SDE whose solution is the desired parameter gradients. However, this has previously suffered from severe speed and accuracy issues, due to high computational cost and numerical truncation errors. Here, we…
Maxwell Institute for Mathematical Sciences（麦克斯韦数学科学研究所）
We consider the filtering and prediction problem for a diffusion process. The signal and observation are modeled by stochastic differential equations (SDEs) driven by correlated Wiener processes. In classical estimation theory, measure-valued stochastic partial differential equations (SPDEs) are derived for the filtering and prediction measures. These equations can be hard to solve numerically. We provide an approximation algorithm using conditional generative adversarial networks (GANs) in combination with signatures, an object from rough path theory. The signature of a sufficiently smooth pa…
Universitat Autònoma de Barcelona（巴塞罗那自治大学）；Universitat de Barcelona（巴塞罗那大学）
(Conditional) Generative Adversarial Networks (GANs) have found great success in recent years, due to their ability to approximate (conditional) distributions over extremely high dimensional spaces. However, they are highly unstable and computationally expensive to train, especially in the time series setting. Recently, it has been proposed the use of a key object in rough path theory, called the signature of a path, which is able to convert the min-max formulation given by the (conditional) GAN framework into a classical minimization problem. However, this method is extremely expensive in ter…
Télécom ParisTech（巴黎高等电信学院）；Institute for Infocomm Research（信息通信研究院）；CentraleSupélec（中央理工-高等电力学院）；Nanyang Technological University（南洋理工大学）
GANS are powerful generative models that are able to model the manifold of natural images. We leverage this property to perform manifold regularization by approximating the Laplacian norm using a Monte Carlo approximation that is easily computed with the GAN. When incorporated into the feature-matching GAN of Improved GAN, we achieve state-of-the-art results for GAN-based semi-supervised learning on the CIFAR-10 dataset, with a method that is significantly easier to implement than competing methods.
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
Inferring the most likely configuration for a subset of variables of a joint distribution given the remaining ones - which we refer to as co-generation - is an important challenge that is computationally demanding for all but the simplest settings. This task has received a considerable amount of attention, particularly for classical ways of modeling distributions like structured prediction. In contrast, almost nothing is known about this task when considering recently proposed techniques for modeling high-dimensional distributions, particularly generative adversarial nets (GANs). Therefore, in…
Department of Bioengineering, University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校生物工程系）
The Ideal Observer (IO) performance has been advocated when optimizing medical imaging systems for signal detection tasks. However, analytical computation of the IO test statistic is generally intractable. To approximate the IO test statistic, sampling-based methods that employ Markov-Chain Monte Carlo (MCMC) techniques have been developed. However, current applications of MCMC techniques have been limited to several object models such as a lumpy object model and a binary texture model, and it remains unclear how MCMC methods can be implemented with other more sophisticated object models. Deep…
Institute for Infocomm Research, A*STAR（新加坡科技研究局资讯通信研究院）
Generative Adversarial Networks are powerful generative models that are able to model the manifold of natural images. We leverage this property to perform manifold regularization by approximating a variant of the Laplacian norm using a Monte Carlo approximation that is easily computed with the GAN. When incorporated into the semi-supervised feature-matching GAN we achieve state-of-the-art results for GAN-based semi-supervised learning on CIFAR-10 and SVHN benchmarks, with a method that is significantly easier to implement than competing methods. We also find that manifold regularization improv…
University of Essex（埃塞克斯大学）；University of Macau（澳门大学）；Boston College（波士顿学院）
The generative adversarial networks (GANs) have recently been applied to estimating the distribution of independent and identically distributed data, and have attracted a lot of research attention. In this paper, we use the blocking technique to demonstrate the effectiveness of GANs for estimating the distribution of stationary time series. Theoretically, we derive a non-asymptotic error bound for the Deep Neural Network (DNN)-based GANs estimator for the stationary distribution of the time series. Based on our theoretical analysis, we propose an algorithm for estimating the change point in ti…
Learning Kinetic Monte Carlo stochastic dynamics with Deep Generative Adversarial Networks
Department of Physics, University of Genova（基因瓦大学物理系）；Department of Materials Science, University of Milano-Bicocca（米兰-比科卡大学材料科学系）；Institut Lumière Matière, UMR5306 Université Lyon 1—CNRS（光与物质研究所，里昂1大学-CNRS）
Uber AI Labs（Uber人工智能实验室）；Cornell University（康奈尔大学）
Generative adversarial networks (GANs) can implicitly learn rich distributions over images, audio, and data which are hard to model with an explicit likelihood. We present a practical Bayesian formulation for unsupervised and semi-supervised learning with GANs. Within this framework, we use stochastic gradient Hamiltonian Monte Carlo to marginalize the weights of the generator and discriminator networks. The resulting approach is straightforward and obtains good performance without any standard interventions such as feature matching, or mini-batch discrimination. By exploring an expressive pos…
Deep unfolding of MCMC kernels: scalable, modular & explainable GANs for high-dimensional posterior sampling
Maxwell Institute for Mathematical Sciences（麦克斯韦数学科学研究所）；School of Mathematical and Computer Sciences, Heriot-Watt University（赫瑞-瓦特大学数学与计算机科学学院）；School of Mathematics, University of Edinburgh（爱丁堡大学数学学院）
Training-free score-based diffusion for parameter-dependent stochastic dynamical systems
Fusion Energy Division, Oak Ridge National Laboratory（奥克兰国家实验室融合能源部门）；Department of Mechanical and Aerospace Engineering, University of Tennessee（田纳西大学机械与航空航天工程系）
The precise simulation of particle transport through detectors remains a key element for the successful interpretation of high energy physics results. However, Monte Carlo based simulation is extremely demanding in terms of computing resources. This challenge motivates investigations of faster, alternative approaches for replacing the standard Monte Carlo approach. We apply Generative Adversarial Networks (GANs), a deep learning technique, to replace the calorimeter detector simulations and speeding up the simulation time by orders of magnitude. We follow a previous approach which used three-d…
We introduce the Metropolis-Hastings generative adversarial network (MH-GAN), which combines aspects of Markov chain Monte Carlo and GANs. The MH-GAN draws samples from the distribution implicitly defined by a GAN's discriminator-generator pair, as opposed to standard GANs which draw samples from the distribution defined only by the generator. It uses the discriminator from GAN training to build a wrapper around the generator for improved sampling. With a perfect discriminator, this wrapped generator samples from the true distribution on the data exactly even when the generator is imperfect. W…
Semi-supervision in Machine Learning can be used in searches for new physics where the signal plus background regions are not labelled. This strongly reduces model dependency in the search for signals Beyond the Standard Model. This approach displays the drawback in that over-fitting can give rise to fake signals. Tossing toy Monte Carlo (MC) events can be used to estimate the corresponding trials factor through a frequentist inference. However, MC events that are based on full detector simulations are resource intensive. Generative Adversarial Networks (GANs) can be used to mimic MC generator…
Molecular Generative Adversarial Network with Multi-Property Optimization
Cloud Computing Technology and Application Department, Shandong Institute of Commerce and Technology（山东商业科技学院云计算技术与应用系）；Graduate School of Advanced Science and Engineering, Hiroshima University（广岛大学研究生院）；D3 Center, The University of Osaka（大阪大学D3中心）；Graduate School of Informatics, Nagoya University（名古屋大学研究生院）
Université Gustave Eiffel（古斯塔夫·埃菲尔大学）；EDF Lab（法国电力实验室）；FiME (Laboratoire de Finance des Marchés de l’Énergie)（能源市场金融实验室）
We introduce three new generative models for time series that are based on Euler discretization of Stochastic Differential Equations (SDEs) and Wasserstein metrics. Two of these methods rely on the adaptation of generative adversarial networks (GANs) to time series. The third algorithm, called Conditional Euler Generator (CEGEN), minimizes a dedicated distance between the transition probability distributions over all time steps. In the context of Ito processes, we provide theoretical guarantees that minimizing this criterion implies accurate estimations of the drift and volatility parameters. …
CERN（欧洲核子研究中心）；EPFL（洛桑联邦理工学院）；Cambridge Quantum Computing Ltd（剑桥量子计算公司）；University of Cambridge（剑桥大学）；University of Oviedo（奥维耶多大学）；University of Strathclyde（斯特拉斯克莱德大学）；University College London（伦敦大学学院）
Generative models, and Generative Adversarial Networks (GAN) in particular, are being studied as possible alternatives to Monte Carlo simulations. It has been proposed that, in certain circumstances, simulation using GANs can be sped-up by using quantum GANs (qGANs). We present a new design of qGAN, the dual-Parameterized Quantum Circuit(PQC) GAN, which consists of a classical discriminator and two quantum generators which take the form of PQCs. The first PQC learns a probability distribution over N-pixel images, while the second generates normalized pixel intensities of an individual image fo…
Due to the intractable partition function, training energy-based models (EBMs) by maximum likelihood requires Markov chain Monte Carlo (MCMC) sampling to approximate the gradient of the Kullback-Leibler divergence between data and model distributions. However, it is non-trivial to sample from an EBM because of the difficulty of mixing between modes. In this paper, we propose to learn a variational auto-encoder (VAE) to initialize the finite-step MCMC, such as Langevin dynamics that is derived from the energy function, for efficient amortized sampling of the EBM. With these amortized MCMC sampl…
Delft University of Technology（代尔夫特理工大学）；Rabobank（荷兰合作银行）
We propose an accurate data-driven numerical scheme to solve Stochastic Differential Equations (SDEs), by taking large time steps. The SDE discretization is built up by means of a polynomial chaos expansion method, on the basis of accurately determined stochastic collocation (SC) points. By employing an artificial neural network to learn these SC points, we can perform Monte Carlo simulations with large time steps. Error analysis confirms that this data-driven scheme results in accurate SDE solutions in the sense of strong convergence, provided the learning methodology is robust and accurate. …
Monte Carlo sampling for Bayesian posterior inference is a common approach used in machine learning. The Markov Chain Monte Carlo procedures that are used are often discrete-time analogues of associated stochastic differential equations (SDEs). These SDEs are guaranteed to leave invariant the required posterior distribution. An area of current research addresses the computational benefits of stochastic gradient methods in this setting. Existing techniques rely on estimating the variance or covariance of the subsampling error, and typically assume constant variance. In this article, we propose …
Bosch Center for Artificial Intelligence（博世人工智能中心）；University of Southern Denmark（南丹麦大学）；Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）
Neural Stochastic Differential Equations (NSDEs) model the drift and diffusion functions of a stochastic process as neural networks. While NSDEs are known to make accurate predictions, their uncertainty quantification properties have been remained unexplored so far. We report the empirical finding that obtaining well-calibrated uncertainty estimations from NSDEs is computationally prohibitive. As a remedy, we develop a computationally affordable deterministic scheme which accurately approximates the transition kernel, when dynamics is governed by a NSDE. Our method introduces a bidimensional m…
Georgia Institute of Technology（佐治亚理工学院）；Shenzhen Institute of Artificial Intelligence and Robotics for Society（深圳人工智能与机器人研究院）；The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））
This article considers the popular MCMC method of unadjusted Langevin Monte Carlo (LMC) and provides a non-asymptotic analysis of its sampling error in 2-Wasserstein distance. The proof is based on a refinement of mean-square analysis in Li et al. (2019), and this refined framework automates the analysis of a large class of sampling algorithms based on discretizations of contractive SDEs. Using this framework, we establish an $\tilde{O}(\sqrt{d}/ε)$ mixing time bound for LMC, without warm start, under the common log-smooth and log-strongly-convex conditions, plus a growth condition on the 3rd-…
