Deep neural networks are an attractive alternative for simulating complex dynamical systems, as in comparison to traditional scientific computing methods, they offer reduced computational costs during inference and can be trained directly from observational data. Existing methods, however, cannot extrapolate accurately and are prone to error accumulation in long-time integration. Herein, we address this issue by combining neural operators with recurrent neural networks, learning the operator mapping, while offering a recurrent structure to capture temporal dependencies. The integrated framework is shown to stabilize the solution and reduce error accumulation for both interpolation and extrapolation of the Korteweg-de Vries equation.
Spatio-temporal dynamics of physical processes are generally modeled using partial differential equations (PDEs). Though the core dynamics follows some principles of physics, real-world physical processes are often driven by unknown external sources. In such cases, developing a purely analytical model becomes very difficult and data-driven modeling can be of assistance. In this paper, we present a hybrid framework combining physics-based numerical models with deep learning for source identification and forecasting of spatio-temporal dynamical systems with unobservable time-varying external sou…
Munich Data Science Institute（慕尼黑数据科学研究所）；School of Computation, Information and Technology, Technical University of Munich（技术大学慕尼黑计算、信息与技术学院）；Interdisciplinary Center for Scientific Computing (IWR)（跨学科科学计算中心）；Department of Mathematics and Computer Science, Heidelberg University（海德堡大学数学与计算机科学系）
Qom University of Technology（库姆理工大学）；London South Bank University（伦敦南岸大学）
Integration of renewable energy sources and emerging loads like electric vehicles to smart grids brings more uncertainty to the distribution system management. Demand Side Management (DSM) is one of the approaches to reduce the uncertainty. Some applications like Nonintrusive Load Monitoring (NILM) can support DSM, however they require accurate forecasting on high resolution data. This is challenging when it comes to single loads like one residential household due to its high volatility. In this paper, we review some of the existing Deep Learning-based methods and present our solution using Ti…
University of Toronto（多伦多大学）；MILA, Université de Montréal（蒙特利尔大学MILA）；CIFAR（加拿大高等研究院）；Carnegie Mellon University（卡内基梅隆大学）
We introduce a general and simple structural design called Multiplicative Integration (MI) to improve recurrent neural networks (RNNs). MI changes the way in which information from difference sources flows and is integrated in the computational building block of an RNN, while introducing almost no extra parameters. The new structure can be easily embedded into many popular RNN models, including LSTMs and GRUs. We empirically analyze its learning behaviour and conduct evaluations on several tasks using different RNN models. Our experimental results demonstrate that Multiplicative Integration ca…
Competence Center Machine Learning Rhine-Ruhr（莱茵-鲁尔机器学习能力中心）；Fraunhofer Center for Machine Learning（弗劳恩霍夫机器学习中心）；Fraunhofer IAIS（弗劳恩霍夫智能分析与信息系统研究所）；University of Bonn（波恩大学）；Berlin Center for Machine Learning（柏林机器学习中心）；TU Berlin（柏林工业大学）
Recent progress in recommender system research has shown the importance of including temporal representations to improve interpretability and performance. Here, we incorporate temporal representations in continuous time via recurrent point process for a dynamical model of reviews. Our goal is to characterize how changes in perception, user interest and seasonal effects affect review text.
Neural dynamical systems are dynamical systems that are described at least in part by neural networks. The class of continuous-time neural dynamical systems must, however, be numerically integrated for simulation and learning. Here, we present a compact neural circuit for two common numerical integrators: the explicit fixed-step Runge-Kutta method of any order and the semi-implicit/predictor-corrector Adams-Bashforth-Moulton method. Modeled as constant-sized recurrent networks embedding a continuous neural differential equation, they achieve fully neural temporal output. Using the polynomial c…
Laboratoire de Physique de l’Ecole Normale Supérieure（巴黎高等师范学院物理实验室）；CNRS（法国国家科学研究中心）；Sorbonne Université（索邦大学）
We study the learning dynamics and the representations emerging in Recurrent Neural Networks trained to integrate one or multiple temporal signals. Combining analytical and numerical investigations, we characterize the conditions under which a RNN with n neurons learns to integrate D(n) scalar signals of arbitrary duration. We show, both for linear and ReLU neurons, that its internal state lives close to a D-dimensional manifold, whose shape is related to the activation function. Each neuron therefore carries, to various degrees, information about the value of all integrals. We discuss the dee…
New York University（纽约大学）；Facebook AI Research（脸书人工智能研究院）；Tianjin University（天津大学）
We propose Symplectic Recurrent Neural Networks (SRNNs) as learning algorithms that capture the dynamics of physical systems from observed trajectories. An SRNN models the Hamiltonian function of the system by a neural network and furthermore leverages symplectic integration, multiple-step training and initial state optimization to address the challenging numerical issues associated with Hamiltonian systems. We show that SRNNs succeed reliably on complex and noisy Hamiltonian systems. We also show how to augment the SRNN integration scheme in order to handle stiff dynamical systems such as bou…
In this paper, we investigate the parameter identification problem in dynamical systems through a deep learning approach. Focusing mainly on second-order, linear time-invariant dynamical systems, the topic of damping factor identification is studied. By utilizing a six-layer deep neural network with different recurrent cells, namely GRUs, LSTMs or BiLSTMs; and by feeding input-output sequence pairs captured from a dynamical system simulator, we search for an effective deep recurrent architecture in order to resolve damping factor identification problem. Our study results show that, although pr…
Recurrent neural networks have gained widespread use in modeling sequence data across various domains. While many successful recurrent architectures employ a notion of gating, the exact mechanism that enables such remarkable performance is not well understood. We develop a theory for signal propagation in recurrent networks after random initialization using a combination of mean field theory and random matrix theory. To simplify our discussion, we introduce a new RNN cell with a simple gating mechanism that we call the minimalRNN and compare it with vanilla RNNs. Our theory allows us to define…
University of Minnesota（明尼苏达大学）；United States Geologic Survey（美国地质调查局）；University of Wisconsin-Madison（威斯康星大学麦迪逊分校）
In this paper, we introduce a novel framework for combining scientific knowledge within physics-based models and recurrent neural networks to advance scientific discovery in many dynamical systems. We will first describe the use of outputs from physics-based models in learning a hybrid-physics-data model. Then, we further incorporate physical knowledge in real-world dynamical systems as additional constraints for training recurrent neural networks. We will apply this approach on modeling lake temperature and quality where we take into account the physical constraints along both the depth dimen…
Complex nonlinear dynamics are ubiquitous in many fields. Moreover, we rarely have access to all of the relevant state variables governing the dynamics. Delay embedding allows us, in principle, to account for unobserved state variables. Here we provide an algebraic approach to delay embedding that permits explicit approximation of error. We also provide the asymptotic dependence of the first order approximation error on the system size. More importantly, this formulation of delay embedding can be directly implemented using a Recurrent Neural Network (RNN). This observation expands the interpre…
University of British Columbia（英属哥伦比亚大学）；Google Brain（谷歌大脑）
Recurrent neural networks have gained widespread use in modeling sequential data. Learning long-term dependencies using these models remains difficult though, due to exploding or vanishing gradients. In this paper, we draw connections between recurrent networks and ordinary differential equations. A special form of recurrent networks called the AntisymmetricRNN is proposed under this theoretical framework, which is able to capture long-term dependencies thanks to the stability property of its underlying differential equation. Existing approaches to improving RNN trainability often incur signif…
Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）；Max Planck ETH Center for Learning Systems（马克斯·普朗克-苏黎世联邦理工学院学习系统中心）
We introduce a method which enables a recurrent dynamics model to be temporally abstract. Our approach, which we call Adaptive Skip Intervals (ASI), is based on the observation that in many sequential prediction tasks, the exact time at which events occur is irrelevant to the underlying objective. Moreover, in many situations, there exist prediction intervals which result in particularly easy-to-predict transitions. We show that there are prediction tasks for which we gain both computational efficiency and prediction accuracy by allowing the model to make predictions at a sampling rate which i…
The capability of recurrent neural networks to approximate trajectories of a random dynamical system, with random inputs, on non-compact domains, and over an indefinite or infinite time horizon is considered. The main result states that certain random trajectories over an infinite time horizon may be approximated to any desired accuracy, uniformly in time, by a certain class of deep recurrent neural networks, with simple feedback structures. The formulation here contrasts with related literature on this topic, much of which is restricted to compact state spaces and finite time intervals. The m…
Gated recurrent units (GRUs) are specialized memory elements for building recurrent neural networks. Despite their incredible success on various tasks, including extracting dynamics underlying neural data, little is understood about the specific dynamics representable in a GRU network. As a result, it is both difficult to know a priori how successful a GRU network will perform on a given task, and also their capacity to mimic the underlying behavior of their biological counterparts. Using a continuous time analysis, we gain intuition on the inner workings of GRU networks. We restrict our prese…
Many real-world systems studied are governed by complex, nonlinear dynamics. By modeling these dynamics, we can gain insight into how these systems work, make predictions about how they will behave, and develop strategies for controlling them. While there are many methods for modeling nonlinear dynamical systems, existing techniques face a trade off between offering interpretable descriptions and making accurate predictions. Here, we develop a class of models that aims to achieve both simultaneously, smoothly interpolating between simple descriptions and more complex, yet also more accurate mo…
Technical University of Munich（慕尼黑工业大学）；Max-Planck Institute for Quantum Optics（马克斯·普朗克量子光学研究所）
We utilize machine learning models which are based on recurrent neural networks to optimize dynamical decoupling (DD) sequences. DD is a relatively simple technique for suppressing the errors in quantum memory for certain noise models. In numerical simulations, we show that with minimum use of prior knowledge and starting from random sequences, the models are able to improve over time and eventually output DD-sequences with performance better than that of the well known DD-families. Furthermore, our algorithm is easy to implement in experiments to find solutions tailored to the specific hardwa…
Arizona State University（亚利桑那州立大学）；Samsung Research America（三星美国研究院）
Recurrent neural networks (RNNs) achieve cutting-edge performance on a variety of problems. However, due to their high computational and memory demands, deploying RNNs on resource constrained mobile devices is a challenging task. To guarantee minimum accuracy loss with higher compression rate and driven by the mobile resource requirement, we introduce a novel model compression approach DirNet based on an optimized fast dictionary learning algorithm, which 1) dynamically mines the dictionary atoms of the projection dictionary matrix within layer to adjust the compression rate 2) adaptively chan…
Constraining Chaos: Enforcing dynamical invariants in the training of recurrent neural networks
University of California San Diego（加利福尼亚大学圣迭戈分校）；Cooperative Institute for Research in Environmental Sciences（环境科学研究合作研究所）；University of Colorado Boulder（科罗拉多大学博尔德分校）；National Oceanic and Atmospheric Administration（美国国家海洋和大气管理局）；Scripps Institution of Oceanography（斯克里普斯海洋研究所）
Recurrent neural networks (RNNs) are powerful models for processing time-series data, but it remains challenging to understand how they function. Improving this understanding is of substantial interest to both the machine learning and neuroscience communities. The framework of reverse engineering a trained RNN by linearizing around its fixed points has provided insight, but the approach has significant challenges. These include difficulty choosing which fixed point to expand around when studying RNN dynamics and error accumulation when reconstructing the nonlinear dynamics with the linearized …
The Pennsylvania State University（宾夕法尼亚州立大学）；Children’s Hospital of Philadelphia（费城儿童医院）
This work proposes a novel neural network architecture, called the Dynamically Controlled Recurrent Neural Network (DCRNN), specifically designed to model dynamical systems that are governed by ordinary differential equations (ODEs). The current state vectors of these types of dynamical systems only depend on their state-space models, along with the respective inputs and initial conditions. Long Short-Term Memory (LSTM) networks, which have proven to be very effective for memory-based tasks, may fail to model physical processes as they tend to memorize, rather than learn how to capture the inf…
Partially observed control problems are a challenging aspect of reinforcement learning. We extend two related, model-free algorithms for continuous control -- deterministic policy gradient and stochastic value gradient -- to solve partially observed domains using recurrent neural networks trained with backpropagation through time. We demonstrate that this approach, coupled with long-short term memory is able to solve a variety of physical control problems exhibiting an assortment of memory requirements. These include the short-term integration of information from noisy sensors and the identifi…
University of Central Florida（中佛罗里达大学）；Huawei Cloud（华为云）
Recent advancements in recurrent neural network (RNN) research have demonstrated the superiority of utilizing multiscale structures in learning temporal representations of time series. Currently, most of multiscale RNNs use fixed scales, which do not comply with the nature of dynamical temporal patterns among sequences. In this paper, we propose Adaptively Scaled Recurrent Neural Networks (ASRNN), a simple but efficient way to handle this problem. Instead of using predefined scales, ASRNNs are able to learn and adjust scales based on different temporal contexts, making them more flexible in mo…
The behavior of recurrent neural network for the data-driven simulation of noisy dynamical systems is studied by training a set of Long Short-Term Memory Networks (LSTM) on the Mackey-Glass time series with a wide range of noise level. It is found that, as the training noise becomes larger, LSTM learns to depend more on its autonomous dynamics than the noisy input data. As a result, LSTM trained on noisy data becomes less susceptible to the perturbation in the data, but has a longer relaxation timescale. On the other hand, when trained on noiseless data, LSTM becomes extremely sensitive to a s…
University of California, Berkeley（加利福尼亚大学伯克利分校）；University of Illinois, Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
Convolutional and recurrent neural networks have been widely employed to achieve state-of-the-art performance on classification tasks. However, it has also been noted that these networks can be manipulated adversarially with relative ease, by carefully crafted additive perturbations to the input. Though several experimentally established prior works exist on crafting and defending against attacks, it is also desirable to have theoretical guarantees on the existence of adversarial examples and robustness margins of the network to such examples. We provide both in this paper. We focus specifical…
Conventional hyperparameter optimization methods are computationally intensive and hard to generalize to scenarios that require dynamically adapting hyperparameters, such as life-long learning. Here, we propose an online hyperparameter optimization algorithm that is asymptotically exact and computationally tractable, both theoretically and practically. Our framework takes advantage of the analogy between hyperparameter optimization and parameter learning in recurrent neural networks (RNNs). It adapts a well-studied family of online learning algorithms for RNNs to tune hyperparameters and netwo…
University Paderborn（帕德博恩大学）；Heinz Nixdorf Institute（海因茨·尼克斯多夫研究所）；Paderborn University（帕德博恩大学）
While trade-offs between modeling effort and model accuracy remain a major concern with system identification, resorting to data-driven methods often leads to a complete disregard for physical plausibility. To address this issue, we propose a physics-guided hybrid approach for modeling non-autonomous systems under control. Starting from a traditional physics-based model, this is extended by a recurrent neural network and trained using a sophisticated multi-objective strategy yielding physically plausible models. While purely data-driven methods fail to produce satisfying results, experiments c…
Texas A&M University – Corpus Christi（得克萨斯农工大学科珀斯克里斯蒂分校）；Clarkson University（克拉克森大学）
Classical methods of solving spatiotemporal dynamical systems include statistical approaches such as autoregressive integrated moving average, which assume linear and stationary relationships between systems' previous outputs. Development and implementation of linear methods are relatively simple, but they often do not capture non-linear relationships in the data. Thus, artificial neural networks (ANNs) are receiving attention from researchers in analyzing and forecasting dynamical systems. Recurrent neural networks (RNN), derived from feed-forward ANNs, use internal memory to process variable…
UCLA（加州大学洛杉矶分校）；Google Research（谷歌研究院）；UT Southwestern Medical Center（德克萨斯大学西南医学中心）；UT Austin（德克萨斯大学奥斯汀分校）
The activity of the grid cell population in the medial entorhinal cortex (MEC) of the mammalian brain forms a vector representation of the self-position of the animal. Recurrent neural networks have been proposed to explain the properties of the grid cells by updating the neural activity vector based on the velocity input of the animal. In doing so, the grid cell system effectively performs path integration. In this paper, we investigate the algebraic, geometric, and topological properties of grid cells using recurrent network models. Algebraically, we study the Lie group and Lie algebra of th…
One of the most influential results in neural network theory is the universal approximation theorem [1, 2, 3] which states that continuous functions can be approximated to within arbitrary accuracy by single-hidden-layer feedforward neural networks. The purpose of this paper is to establish a result in this spirit for the approximation of general discrete-time linear dynamical systems - including time-varying systems - by recurrent neural networks (RNNs). For the subclass of linear time-invariant (LTI) systems, we devise a quantitative version of this statement. Specifically, measuring the com…
Surrogate testing techniques have been used widely to investigate the presence of dynamical nonlinearities, an essential ingredient of deterministic chaotic processes. Traditional surrogate testing subscribes to statistical hypothesis testing and investigates potential differences in discriminant statistics between the given empirical sample and its surrogate counterparts. The choice and estimation of the discriminant statistics can be challenging across short time series. Also, conclusion based on a single empirical sample is an inherent limitation. The present study proposes a recurrent neur…
Imperial College London（伦敦帝国理工学院）；Institut Polytechnique de Paris（巴黎理工学院）；EDF R&D（法国电力公司研发部）
Data assimilation techniques are widely used to predict complex dynamical systems with uncertainties, based on time-series observation data. Error covariance matrices modelling is an important element in data assimilation algorithms which can considerably impact the forecasting accuracy. The estimation of these covariances, which usually relies on empirical assumptions and physical constraints, is often imprecise and computationally expensive especially for systems of large dimension. In this work, we propose a data-driven approach based on long short term memory (LSTM) recurrent neural networ…
A common difficulty in applications of machine learning is the lack of any general principle for guiding the choices of key parameters of the underlying neural network. Focusing on a class of recurrent neural networks - reservoir computing systems that have recently been exploited for model-free prediction of nonlinear dynamical systems, we uncover a surprising phenomenon: the emergence of an interval in the spectral radius of the neural network in which the prediction error is minimized. In a three-dimensional representation of the error versus time and spectral radius, the interval correspon…
Massachusetts Institute of Technology（麻省理工学院）；Google Inc.（谷歌公司）；IBM Research（IBM研究院）
To understand the fundamental trade-offs between training stability, temporal dynamics and architectural complexity of recurrent neural networks~(RNNs), we directly analyze RNN architectures using numerical methods of ordinary differential equations~(ODEs). We define a general family of RNNs--the ODERNNs--by relating the composition rules of RNNs to integration methods of ODEs at discrete time steps. We show that the degree of RNN's functional nonlinearity $n$ and the range of its temporal memory $t$ can be mapped to the corresponding stage of Runge-Kutta recursion and the order of time-deriva…
Federal University of Ceará（塞阿拉联邦大学）；University of Sheffield（谢菲尔德大学）
We define Recurrent Gaussian Processes (RGP) models, a general family of Bayesian nonparametric models with recurrent GP priors which are able to learn dynamical patterns from sequential data. Similar to Recurrent Neural Networks (RNNs), RGPs can have different formulations for their internal states, distinct inference methods and be extended with deep structures. In such context, we propose a novel deep RGP model whose autoregressive states are latent, thereby performing representation and dynamical learning simultaneously. To fully exploit the Bayesian nature of the RGP model we develop the …
IBM T.J. Watson Research Center（IBM T.J.沃森研究中心）；Massachusetts Institute of Technology（麻省理工学院）
We propose a recurrent neural network for a "model-free" simulation of a dynamical system with unknown parameters without prior knowledge. The deep learning model aims to jointly learn the nonlinear time marching operator and the effects of the unknown parameters from a time series dataset. We assume that the time series data set consists of an ensemble of trajectories for a range of the parameters. The learning task is formulated as a statistical inference problem by considering the unknown parameters as random variables. A latent variable is introduced to model the effects of the unknown par…
University of Potsdam（波茨坦大学）；Vrije Universiteit Amsterdam（阿姆斯特丹自由大学）；Ambrosys GmbH（Ambrosys有限公司）
We investigate the predictive power of recurrent neural networks for oscillatory systems not only on the attractor, but in its vicinity as well. For this we consider systems perturbed by an external force. This allows us to not merely predict the time evolution of the system, but also study its dynamical properties, such as bifurcations, dynamical response curves, characteristic exponents etc. It is shown that they can be effectively estimated even in some regions of the state space where no input data were given. We consider several different oscillatory examples, including self-sustained, ex…
Loyola Marymount University（洛约拉马利蒙特大学）；California State University, Long Beach（加州州立大学长滩分校）
We introduce an exceptionally simple gated recurrent neural network (RNN) that achieves performance comparable to well-known gated architectures, such as LSTMs and GRUs, on the word-level language modeling task. We prove that our model has simple, predicable and non-chaotic dynamics. This stands in stark contrast to more standard gated architectures, whose underlying dynamical systems exhibit chaotic behavior.
KTH Royal Institute of Technology（瑞典皇家理工学院）；Uppsala University（乌普萨拉大学）
Stochastic nonlinear dynamical systems are ubiquitous in modern, real-world applications. Yet, estimating the unknown parameters of stochastic, nonlinear dynamical models remains a challenging problem. The majority of existing methods employ maximum likelihood or Bayesian estimation. However, these methods suffer from some limitations, most notably the substantial computational time for inference coupled with limited flexibility in application. In this work, we propose DeepBayes estimators that leverage the power of deep recurrent neural networks in learning an estimator. The method consists o…
Università di Firenze（佛罗伦萨大学）；INFN（意大利国家核物理研究所）；CSDC（复杂系统动力学研究中心）；University of Namur（那慕尔大学）；University of Lisbon（里斯本大学）；Instituto de Telecomunicacoes（电信研究所）
A novel strategy to automated classification is introduced which exploits a fully trained dynamical system to steer items belonging to different categories toward distinct asymptotic attractors. These latter are incorporated into the model by taking advantage of the spectral decomposition of the operator that rules the linear evolution across the processing network. Non-linear terms act for a transient and allow to disentangle the data supplied as initial condition to the discrete dynamical system, shaping the boundaries of different attractors. The network can be equipped with several memory …
Faculty of Physics and Astronomy, Heidelberg University（海德堡大学物理与天文学学院）；Institute of Neuroinformatics, University of Zurich and ETH Zurich（苏黎世大学与苏黎世联邦理工学院神经信息学研究所）
A major tenet in theoretical neuroscience is that cognitive and behavioral processes are ultimately implemented in terms of the neural system dynamics. Accordingly, a major aim for the analysis of neurophysiological measurements should lie in the identification of the computational dynamics underlying task processing. Here we advance a state space model (SSM) based on generative piecewise-linear recurrent neural networks (PLRNN) to assess dynamics from neuroimaging data. In contrast to many other nonlinear time series models which have been proposed for reconstructing latent dynamics, our mode…
Stability is a fundamental property of dynamical systems, yet to this date it has had little bearing on the practice of recurrent neural networks. In this work, we conduct a thorough investigation of stable recurrent models. Theoretically, we prove stable recurrent neural networks are well approximated by feed-forward networks for the purpose of both inference and training by gradient descent. Empirically, we demonstrate stable recurrent models often perform as well as their unstable counterparts on benchmark sequence tasks. Taken together, these findings shed light on the effective power of r…
University of Stuttgart（斯图加特大学）；ETAS GmbH（ETAS有限公司）
Modeling sequential data has become more and more important in practice. Some applications are autonomous driving, virtual sensors and weather forecasting. To model such systems, so called recurrent models are frequently used. In this paper we introduce several new Deep recurrent Gaussian process (DRGP) models based on the Sparse Spectrum Gaussian process (SSGP) and the improved version, called variational Sparse Spectrum Gaussian process (VSSGP). We follow the recurrent structure given by an existing DRGP based on a specific variational sparse Nyström approximation, the recurrent Gaussian pro…
Bosch Center for Artificial Intelligence（博世人工智能中心）
Deep neural networks, and in particular recurrent networks, are promising candidates to control autonomous agents that interact in real-time with the physical world. However, this requires a seamless integration of temporal features into the network's architecture. For the training of and inference with recurrent neural networks, they are usually rolled out over time, and different rollouts exist. Conventionally during inference, the layers of a network are computed in a sequential manner resulting in sparse temporal integration of information and long response times. In this study, we present…
