We consider data-driven Bayesian state estimation from compressed measurements (BSCM) of a model-free process. The dimension of the temporal measurement vector is lower than that of the temporal state vector to be estimated, leading to an under-determined inverse problem. The underlying dynamical model of the state’s evolution is unknown for a ‘model-free process.’ Hence, it is difficult to use traditional model-driven methods, for example, Kalman and particle filters. Instead, we consider data-driven methods. We experimentally show that two existing unsupervised learning-based data-driven methods fail to address the BSCM problem in a model-free process. The methods are-data-driven nonlinear state estimation (DANSE) and deep Markov model (DMM). While DANSE provides good predictive/ forecasting performance to model the temporal measurement data as a time series, its unsupervised learning lacks suitable regularization for tackling the BSCM task. We then propose a semi-supervised learning approach and develop a semi-supervised learning-based DANSE method, referred to as SemiDANSE. In SemiDANSE, we use a large amount of unlabelled data along with a limited amount of labelled data, i.e., pairwise measurement-and-state data, which provides the desired regularization. Using benchmark chaotic dynamical systems, we empirically show that the data-driven SemiDANSE provides competitive state estimation performance for BSCM using a handful of different measurement systems, against a hybrid method called KalmanNet and two model-driven methods (extended Kalman filter and unscented Kalman filter) that know the dynamical models exactly.

The Kalman filter (KF) and its variants are among the most celebrated algorithms in signal processing. These methods are used for state estimation of dynamic systems by relying on mathematical representations in the form of simple state-space (SS) models, which may be crude and inaccurate descriptions of the underlying dynamics. Emerging data-centric artificial intelligence (AI) techniques tackle these tasks using deep neural networks (DNNs), which are model agnostic. Recent developments illustrate the possibility of fusing DNNs with classic Kalman-type filtering, obtaining systems that learn to track in partially known dynamics. This article provides a tutorial-style overview of design approaches for incorporating AI in aiding KF-type algorithms. We review both generic and dedicated DNN architectures suitable for state estimation and provide a systematic presentation of techniques for fusing AI tools with KFs and for leveraging partial SS modeling and data, categorizing design approaches into task oriented and SS model oriented. The usefulness of each approach in preserving the individual strengths of model-based KFs and data-driven DNNs is investigated in a qualitative and quantitative study (whose code is publicly available), illustrating the gains of hybrid model-based/data-driven designs. We also discuss existing challenges and future research directions that arise from fusing AI and Kalman-type algorithms.

The tasks of state estimation and parameter estimation are closely related and often treated jointly by researchers in the areas of signal processing, control, and system identification. In recent literature, there is a growing trend of applying machine learning and deep learning techniques to tackle the above problems. The thesis follows a similar line of thought and is primarily concerned with the development of deep learning-based approaches for addressing state estimation and parameter estimation, under modeling constraints. 

The first main contribution of the thesis concerns deep learning approaches for addressing Bayesian state estimation in a model-free scenario. We introduce DANSE -- a Data-driven Nonlinear State Estimation method where the underlying dynamical model is unknown. The core of the DANSE framework consists of directly modeling the prior distribution of the state using a recurrent neural network. We assume knowledge of the measurement system, which is considered noisy, linear, and additive. This assumption offers attractive advantages in Bayesian state estimation. Firstly, it enables DANSE to provide a closed-form posterior distribution of the state. Secondly, the training of DANSE can be carried out in an unsupervised manner as long as the noisy measurements contain sufficient information about the unknown state (either when we have a full-rank measurement system or an over-determined measurement system). In the case of under-determined measurements, the state estimation problem becomes challenging. To meet this challenge, we modify the existing unsupervised learning framework of DANSE by using a limited amount of labelled training data. This modification results in a semi-supervised learning framework, and the resulting method is named semi-supervised DANSE (SemiDANSE). Finally, ideas from particle filters and importance sampling are borrowed for handling noisy and nonlinear measurements. This results in a particle-based DANSE (pDANSE). pDANSE approximates the unknown posterior distribution using the first two statistical moments, while requiring no special knowledge of the underlying dynamic model. We employ nonlinear dynamical systems, such as chaotic attractors, in our experimental study. Such attractors are a popular choice for benchmarking state estimation performances. We compare the empirical performance of DANSE and its extensions, SemiDANSE and pDANSE, against those of classical, model-driven approaches such as the extended Kalman filter (EKF), the unscented Kalman filter (UKF), the particle filter (PF), data-driven, and recent hybrid approaches like KalmanNet.

The second main contribution of the thesis concerns deep learning-based parameter estimation. Challenges associated with classical parameter estimation techniques include computationally intensive inference and dependency on parameter initialization. For addressing these challenges, we propose DeepBayes -- a method for constructing an estimator using recurrent neural networks. For training the recurrent neural networks, a simulable model is used to generate a large quantity of synthetic, labelled data for offline training using parameters sampled from a non-informative uniform prior. The result is that once trained, DeepBayes can provide rapid inference without any special consideration regarding initialisation. Furthermore, we empirically demonstrate that DeepBayes can provide comparable parameter estimation performance compared to the mean-squared-error optimal Bayes estimator and relevant maximum-likelihood based estimators on popular nonlinear dynamical system benchmarks. 

Tillståndsestimering och parameterestimering är nära besläktade uppgifter som ofta behandlas gemensamt av forskare inom signalbehandling, reglerteknik och systemidentifiering. I den aktuella litteraturen blir det allt vanligare att använda metoder från maskininlärning och djupinlärning för att angripa dessa problem. Denna avhandling följer samma inriktning och fokuserar huvudsakligen på att utveckla djupinlärningsbaserade metoder för tillståndsestimering och parameterestimering under modelleringsbegränsningar.

Avhandlingens första huvudbidrag gäller djupinlärningsbaserade angreppssätt för Bayesiansk tillståndsestimering i en modellfri situation.  Vi introducerar DANSE (Data-driven Nonlinear State Estimation) där den underliggande dynamiska modellen är okänd. Kärnan i DANSE-metoden är att direkt modellera en a-priori sannolikhetsfördelning för tillståndsvariablerna med hjälp av återkommande neurala nätverk. Vi antar kännedom om mätmodellen, vilken antas vara linjär, additiv och brusig. Detta antagande ger flera fördelar vid Bayesiansk tillståndsestimering. För det första möjliggör det att DANSE kan ge en sluten form för tillståndsvariablernas a-posteriori sannolikhetsfördelning. För det andra kan DANSE tränas oövervakat, så länge de brusiga mätningarna innehåller tillräcklig information om det okända tillståndet (till exempel när mätmodellen har full rang eller är överbestämd). Vid underbestämda mätningar blir tillståndsestimeringsproblemet mer utmanande. För att hantera detta utvidgar vi den befintliga oövervakade inlärningsmetoden genom att använda en begränsad mängd annoterad träningsdata. Denna förändring resulterar i en semiövervakad metod, benämnd SemiDANSE. Slutligen lånar vi idéer från partikelfilter och viktningsmetoder för att hantera olinjära och brusiga mätningar. Detta leder till en partikelbaserad variant av DANSE, pDANSE. pDANSE approximerar den okända a-posteriori sannolikhetsfördelningen genom att beskriva den med hjälp av väntevärde och varians, utan krav på djupare kännedom om den underliggande dynamiska modellen. I våra experiment använder vi olinjära dynamiska system, såsom kaotiska attraktorer, vilka ofta används som referenssystem för att utvärdera tillståndsestimeringsprestanda. Vi jämför de empiriska resultaten för DANSE samt dess varianter SemiDANSE och pDANSE med klassiska, modellbaserade metoder, till exempel Extended Kalman Filter (EKF), Unscented Kalman Filter (UKF) och Particle Filter (PF), samt med rena datadrivna metoder och nya hybrida angreppssätt såsom KalmanNet. 

Avhandlingens andra huvudbidrag rör djupinlärningsbaserad parameterestimering. Klassiska parameterestimeringsmetoder präglas ofta av beräkningsintensiv inferens och av ett starkt beroende av parameterinitialisering. För att hantera dessa utmaningar introducerar vi DeepBayes, en metod för att konstruera en skattare med hjälp av återkommande neurala nätverk. För att träna dessa nätverk används en simulerbar modell för att generera stora mängder annoterad syntetisk  data för offline träning, där parametrarna dragits från en icke-informativ likformig a-priori-fördelning. Resultatet är att DeepBayes, när modellen väl är offline-tränad, kan erbjuda snabb inferens utan särskilda krav på initialisering. Vi visar dessutom empiriskt att DeepBayes kan uppnå jämförbar prestanda i parameterestimering med både den Bayesianska skattaren, som är optimal med avseende på medelkvadratfelet, och relevanta maximum likelihood-baserade skattare på välkända referenssystem för olinjära dynamiska system.

We introduce a model-free hidden sequence (MHS) estimation problem where the task is to estimate a long sequence of 'model-free' process hidden under additive Gaussian noise. To estimate the posterior of the hidden sequence from the noisy observation sequence, we have three main challenges: (a) the process is 'model-free', that means there is no underlying state-space-model (SSM), limiting the use of traditional SSM-informed model-based methods like Kalman filter (KF) and particle filter (PF); (b) only an unlabelled dataset comprised of noisy observation sequences is available as training data, and hence no supervised learning possible; and,

We consider the problem of causal filtering of a model-free process from (noisy) nonlinear measurements. The 'model-free process' means that we do not have a state-space model (SSM) of the process dynamics, limiting the use of traditional model-driven filters, such as unscented Kalman filter (UKF) and particle filter (PF). To address the problem we propose a particle-based data-driven nonlinear state estimation (pDANSE) method. In pDANSE, a recurrent neural network (RNN) provides the statistical parameters of a Gaussian prior of the underlying state, and particles are then drawn from the prior to compute the posterior moments. pDANSE is typically trained in a semi-supervised fashion. For our experiments we study the use of halfwave rectification as a nonlinear transformation of measurements. We first show that an unsupervised learning-based method under-performs, and subsequently the semi-supervised learning-based pDANSE performs satisfactorily. Using Lorenz-63 system as benchmark, pDANSE is found to be competitive against a model-driven PF that knows the exact SSM.

We address the tasks of Bayesian state estimation and forecasting for a model-free process in an unsupervised learning setup. For a model-free process, we do not have any a-priori knowledge of the process dynamics. In the article, we propose DANSE - a Data-driven Nonlinear State Estimation method. DANSE provides a closed-form posterior of the state of the model-free process, given linear measurements of the state. In addition, it provides a closed-form posterior for forecasting. A data-driven recurrent neural network (RNN) is used in DANSE to provide the parameters of a prior of the state. The prior depends on the past measurements as input, and then we find the closed-form posterior of the state using the current measurement as input. The data-driven RNN captures the underlying non-linear dynamics of the model-free process. The training of DANSE, mainly learning the parameters of the RNN, is executed using an unsupervised learning approach. In unsupervised learning, we have access to a training dataset comprising only a set of (noisy) measurement data trajectories, but we do not have any access to the state trajectories. Therefore, DANSE does not have access to state information in the training data and can not use supervised learning. Using simulated linear and non-linear process models (Lorenz attractor and Chen attractor), we evaluate the unsupervised learning-based DANSE. We show that the proposed DANSE, without knowledge of the process model and without supervised learning, provides a competitive performance against model-driven methods, such as the Kalman filter (KF), extended KF (EKF), unscented KF (UKF), a data-driven deep Markov model (DMM) and a recently proposed hybrid method called KalmanNet. In addition, we show that DANSE works for high-dimensional state estimation.

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. The method consists of first training a recurrent neural network to minimize the mean-squared estimation error over a set of synthetically generated data using models drawn from the model set of interest. The a priori trained estimator can then be used directly for inference by evaluating the network with the estimation data. The deep recurrent neural network architectures can be trained offline and ensure significant time savings during inference. We experiment with two popular recurrent neural networks — long short term memory network (LSTM) and gated recurrent unit (GRU). We demonstrate the applicability of our proposed method on different example models and perform detailed comparisons with state-of-the-art approaches. We also provide a study on a real-world nonlinear benchmark problem. The experimental evaluations show that the proposed approach is asymptotically as good as the Bayes estimator.

We consider reconstruction of an ambient signal in a compressed sensing (CS) setup where the ambient signal has a neural network based generative model. The generative model has a sparse-latent input and we refer to the generated ambient signal as generative sparse-latent signal (GSL). The proposed sparsity inducing reconstruction algorithm is inherently non-convex, and we show that a gradient based search provides a good reconstruction performance. We evaluate our proposed algorithm using simulated data.

We propose DANSE - a data-driven non-linear state estimation method. DANSE provides a closed-form posterior of the state of a model-free process, given linear measurements of the state in a Bayesian setup, like the celebrated Kalman filter (KF). Non-linear dynamics of the state are captured by data-driven recurrent neural networks (RNNs). The training of DANSE combines maximum-likelihood and gradient-descent in an unsupervised framework, i.e. only measurement data and no process data are required. Using simulated linear and non-linear process models, we demonstrate that DANSE - without knowledge of the process model - provides competitive performance against model-based approaches such as KF, unscented KF (UKF), extended KF (EKF), and a hybrid approach such as KalmanNet.

The Sun's activity, which is associated with the solar magnetic cycle, creates a dynamic environment in space known as space weather. Severe space weather can disrupt space-based and Earth-based technologies. Slow decadal-scale variations on solar-cycle timescales are important for radiative forcing of the Earth's atmosphere and impact satellite lifetimes and atmospheric dynamics. Predicting the solar magnetic cycle is therefore of critical importance for humanity. In this context, a novel development is the application of machine-learning algorithms for solar-cycle forecasting. Diverse approaches have been developed for this purpose; however, with no consensus across different techniques and physics-based approaches. Here, we first explore the performance of four different machine-learning algorithms - all of them belonging to a class called Recurrent Neural Networks (RNNs) - in predicting simulated sunspot cycles based on a widely studied, stochastically forced, nonlinear time-delay solar dynamo model. We conclude that the algorithm Echo State Network (ESN) performs the best, but predictability is limited to only one future sunspot cycle, in agreement with recent physical insights. Subsequently, we train the ESN algorithm and a modified version of it (MESN) with solar-cycle observations to forecast Cycles 22 - 25. We obtain accurate hindcasts for Solar Cycles 22 - 24. For Solar Cycle 25 the ESN algorithm forecasts a peak amplitude of 131 +/- 14 sunspots around July 2024 and indicates a cycle length of approximately 10 years. The MESN forecasts a peak of 137 +/- 2 sunspots around April 2024, with the same cycle length. Qualitatively, both forecasts indicate that Cycle 25 will be slightly stronger than Cycle 24 but weaker than Cycle 23. Our novel approach bridges physical model-based forecasts with machine-learning-based approaches, achieving consistency across these diverse techniques.