Model order reduction of deep structured state-space models: A system-theoretic approach

With a specific emphasis on control design objectives, achieving accurate system modeling with limited complexity is crucial in parametric system identification. The recently introduced deep structured state-space models (SSM), which feature linear dynamical blocks as key constituent components, offer high predictive performance. However, the learned representations often suffer from excessively large model orders, which render them unsuitable for control design purposes. The current paper addresses this challenge by means of system-theoretic model order reduction techniques that target the linear dynamical blocks of SSMs. We introduce two regularization terms which can be incorporated into the training loss for improved model order reduction. In particular, we consider modal $\ell_1$ and Hankel nuclear norm regularization to promote sparsity, allowing one to retain only the relevant states without sacrificing accuracy. The presented regularizers lead to advantages in terms of parsimonious representations and faster inference resulting from the reduced order models. The effectiveness of the proposed methodology is demonstrated using real-world ground vibration data from an aircraft.

Synthetic data generation for system identification: leveraging knowledge transfer from similar systems

This paper addresses the challenge of overfitting in the learning of dynamical systems by introducing a novel approach for the generation of synthetic data, aimed at enhancing model generalization and robustness in scenarios characterized by data scarcity. Central to the proposed methodology is the concept of knowledge transfer from systems within the same class. Specifically, synthetic data is generated through a pre-trained meta-model that describes a broad class of systems to which the system of interest is assumed to belong. Training data serves a dual purpose: firstly, as input to the pre-trained meta model to discern the system's dynamics, enabling the prediction of its behavior and thereby generating synthetic output sequences for new input sequences; secondly, in conjunction with synthetic data, to define the loss function used for model estimation. A validation dataset is used to tune a scalar hyper-parameter balancing the relative importance of training and synthetic data in the definition of the loss function. The same validation set can be also used for other purposes, such as early stopping during the training, fundamental to avoid overfitting in case of small-size training datasets. The efficacy of the approach is shown through a numerical example that highlights the advantages of integrating synthetic data into the system identification process.

On the adaptation of in-context learners for system identification

In-context system identification aims at constructing meta-models to describe classes of systems, differently from traditional approaches that model single systems. This paradigm facilitates the leveraging of knowledge acquired from observing the behaviour of different, yet related dynamics. This paper discusses the role of meta-model adaptation. Through numerical examples, we demonstrate how meta-model adaptation can enhance predictive performance in three realistic scenarios: tailoring the meta-model to describe a specific system rather than a class; extending the meta-model to capture the behaviour of systems beyond the initial training class; and recalibrating the model for new prediction tasks. Results highlight the effectiveness of meta-model adaptation to achieve a more robust and versatile meta-learning framework for system identification.

In-context learning for model-free system identification

In traditional system identification, we estimate a model of an unknown dynamical system based on given input/output sequences and available physical knowledge. Yet, is it also possible to understand the intricacies of dynamical systems not solely from their input/output patterns, but by observing the behavior of other systems within the same class? This central question drives the study presented in this paper. In response to this query, we introduce a novel paradigm for system identification, addressing two primary tasks: one-step-ahead prediction and multi-step simulation. Unlike conventional methods, we do not directly estimate a model for the specific system. Instead, we pretrain a meta model that represents a class of dynamical systems. This meta model is trained from a potentially infinite stream of synthetic data, generated by systems randomly extracted from a certain distribution. At its core, the meta model serves as an implicit representation of the main characteristics of a class of dynamical systems. When provided with a brief context from a new system - specifically, a short input/output sequence - the meta model implicitly discerns its dynamics, enabling predictions of its behavior. The proposed approach harnesses the power of Transformer architectures, renowned for their in-context learning capabilities in Natural Language Processing tasks. For one-step prediction, a GPT-like decoder-only architecture is utilized, whereas the simulation problem employs an encoder-decoder structure. Initial experimental results affirmatively answer our foundational question, opening doors to fresh research avenues in system identification.

Neural State-Space Models: Empirical Evaluation of Uncertainty Quantification

Effective quantification of uncertainty is an essential and still missing step towards a greater adoption of deep-learning approaches in different applications, including mission-critical ones. In particular, investigations on the predictive uncertainty of deep-learning models describing non-linear dynamical systems are very limited to date. This paper is aimed at filling this gap and presents preliminary results on uncertainty quantification for system identification with neural state-space models. We frame the learning problem in a Bayesian probabilistic setting and obtain posterior distributions for the neural network's weights and outputs through approximate inference techniques. Based on the posterior, we construct credible intervals on the outputs and define a surprise index which can effectively diagnose usage of the model in a potentially dangerous out-of-distribution regime, where predictions cannot be trusted.

Learning neural state-space models: do we need a state estimator?

In recent years, several algorithms for system identification with neural state-space models have been introduced. Most of the proposed approaches are aimed at reducing the computational complexity of the learning problem, by splitting the optimization over short sub-sequences extracted from a longer training dataset. Different sequences are then processed simultaneously within a minibatch, taking advantage of modern parallel hardware for deep learning. An issue arising in these methods is the need to assign an initial state for each of the sub-sequences, which is required to run simulations and thus to evaluate the fitting loss. In this paper, we provide insights for calibration of neural state-space training algorithms based on extensive experimentation and analyses performed on two recognized system identification benchmarks. Particular focus is given to the choice and the role of the initial state estimation. We demonstrate that advanced initial state estimation techniques are really required to achieve high performance on certain classes of dynamical systems, while for asymptotically stable ones basic procedures such as zero or random initialization already yield competitive performance.

On the adaptation of recurrent neural networks for system identification

This paper presents a transfer learning approach which enables fast and efficient adaptation of Recurrent Neural Network (RNN) models of dynamical systems. A nominal RNN model is first identified using available measurements. The system dynamics are then assumed to change, leading to an unacceptable degradation of the nominal model performance on the perturbed system. To cope with the mismatch, the model is augmented with an additive correction term trained on fresh data from the new dynamic regime. The correction term is learned through a Jacobian Feature Regression (JFR) method defined in terms of the features spanned by the model's Jacobian with respect to its nominal parameters. A non-parametric view of the approach is also proposed, which extends recent work on Gaussian Process (GP) with Neural Tangent Kernel (NTK-GP) to the RNN case (RNTK-GP). This can be more efficient for very large networks or when only few data points are available. Implementation aspects for fast and efficient computation of the correction term, as well as the initial state estimation for the RNN model are described. Numerical examples show the effectiveness of the proposed methodology in presence of significant system variations.

Deep learning with transfer functions: new applications in system identification

This paper presents a linear dynamical operator described in terms of a rational transfer function, endowed with a well-defined and efficient back-propagation behavior for automatic derivatives computation. The operator enables end-to-end training of structured networks containing linear transfer functions and other differentiable units {by} exploiting standard deep learning software. Two relevant applications of the operator in system identification are presented. The first one consists in the integration of {prediction error methods} in deep learning. The dynamical operator is included as {the} last layer of a neural network in order to obtain the optimal one-step-ahead prediction error. The second one considers identification of general block-oriented models from quantized data. These block-oriented models are constructed by combining linear dynamical operators with static nonlinearities described as standard feed-forward neural networks. A custom loss function corresponding to the log-likelihood of quantized output observations is defined. For gradient-based optimization, the derivatives of the log-likelihood are computed by applying the back-propagation algorithm through the whole network. Two system identification benchmarks are used to show the effectiveness of the proposed methodologies.

dynoNet: a neural network architecture for learning dynamical systems

This paper introduces a network architecture, called dynoNet, utilizing linear dynamical operators as elementary building blocks. Owing to the dynamical nature of these blocks, dynoNet networks are tailored for sequence modeling and system identification purposes. The back-propagation behavior of the linear dynamical operator with respect to both its parameters and its input sequence is defined. This enables end-to-end training of structured networks containing linear dynamical operators and other differentiable units, exploiting existing deep learning software. Examples show the effectiveness of the proposed approach on well-known system identification benchmarks. Examples show the effectiveness of the proposed approach against well-known system identification benchmarks.

Continuous-time system identification with neural networks: model structures and fitting criteria

This paper presents tailor-made neural model structures and two custom fitting criteria for learning dynamical systems. The proposed framework is based on a representation of the system behavior in terms of continuous-time state-space models. The sequence of hidden states is optimized along with the neural network parameters in order to minimize the difference between measured and estimated outputs, and at the same time to guarantee that the optimized state sequence is consistent with the estimated system dynamics. The effectiveness of the approach is demonstrated through three case studies, including two public system identification benchmarks based on experimental data.