Matthieu Barreau

John Cao, Muhammad Umar B. Niazi, Matthieu Barreau et al.

This paper presents a novel observer-based approach to detect and isolate faulty sensors in nonlinear systems. The proposed sensor fault detection and isolation (s-FDI) method applies to a general class of nonlinear systems. Our focus is on s-FDI for two types of faults: complete failure and sensor degradation. The key aspect of this approach lies in the utilization of a neural network-based Kazantzis-Kravaris/Luenberger (KKL) observer. The neural network is trained to learn the dynamics of the observer, enabling accurate output predictions of the system. Sensor faults are detected by comparing the actual output measurements with the predicted values. If the difference surpasses a theoretical threshold, a sensor fault is detected. To identify and isolate which sensor is faulty, we compare the numerical difference of each sensor meassurement with an empirically derived threshold. We derive both theoretical and empirical thresholds for detection and isolation, respectively. Notably, the proposed approach is robust to measurement noise and system uncertainties. Its effectiveness is demonstrated through numerical simulations of sensor faults in a network of Kuramoto oscillators.

Matthieu Barreau, Haoming Shen

Physics-Informed Neural Networks (PINNs) have emerged as powerful tools for integrating physics-based models with data by minimizing both data and physics losses. However, this multi-objective optimization problem is notoriously challenging, with some benchmark problems leading to unfeasible solutions. To address these issues, various strategies have been proposed, including adaptive weight adjustments in the loss function. In this work, we introduce clear definitions of accuracy and robustness in the context of PINNs and propose a novel training algorithm based on the Primal-Dual (PD) optimization framework. Our approach enhances the robustness of PINNs while maintaining comparable performance to existing weight-balancing methods. Numerical experiments demonstrate that the PD method consistently achieves reliable solutions across all investigated cases, even in the low-data regime, and can be easily implemented, facilitating its practical adoption. The code is available at https://github.com/haoming-SHEN/Accuracy-and-Robustness-of-Weight-Balancing-Methods-for-Training-PINNs.git.

Matthieu Barreau, Nicola Bastianello

In this paper, we address the problem of discovering maximal Lyapunov functions, as a means of determining the region of attraction of a dynamical system. To this end, we design a novel neural network architecture, which we prove to be a universal approximator of (maximal) Lyapunov functions. The architecture combines a local quadratic approximation with the output of a neural network, which models global higher-order terms in the Taylor expansion. We formulate the problem of training the Lyapunov function as an unsupervised optimization problem with dynamical constraints, which can be solved leveraging techniques from physics-informed learning. We propose and analyze a tailored training algorithm, based on the primal-dual algorithm, that can efficiently solve the problem. Additionally, we show how the learning problem formulation can be adapted to integrate data, when available. We apply the proposed approach to different classes of systems, showing that it matches or outperforms state-of-the-art alternatives in the accuracy of the approximated regions of attraction.

Alejandro Penacho Riveiros, Nicola Bastianello, Matthieu Barreau

In this paper we address the problem of detecting differences or anomalies in a dynamical system, based on historical data of nominal operations. This problem encompasses quality control, where newly manufactured systems are tested against desired nominal operations, and the detection of changes in the dynamics due to degradation or repairs. We propose a model free approach based on Gaussian processes (GPs). The idea is to train offline a GP based on nominal data, which is then deployed online to detect whether measurements of the system state are compatible with nominal operations or if they deviate. Detecting this deviation is made more challenging by the presence of process and measurement noise, which might obfuscate deviations in the dynamics. The detection then is based on a threshold that ensures a specific false positive rate. We showcase the promising performance of the proposed method with two systems, and highlight several interesting future research questions.

Alejandro Penacho Riveiros, Nicola Bastianello, Karl H. Johansson et al.

As the number of satellites in orbit has increased exponentially in recent years, ensuring their correct functionality has started to require automated methods to decrease human workload. In this work, we present an algorithm that analyzes the on-board data related to friction from the Reaction Wheel Assemblies (RWA) of a satellite and determines their operating status, distinguishing between nominal status and several possible anomalies that require preventive measures to be taken. The algorithm first uses a model based on hybrid systems theory to extract the information relevant to the problem. The extraction process combines techniques in changepoint detection, dynamic programming, and maximum likelihood in a structured way. A classifier then uses the extracted information to determine the status of the RWA. This last classifier has been previously trained with a labelled dataset produced by a high-fidelity simulator, comprised for the most part of nominal data. The final algorithm combines model-based and data-based approaches to obtain satisfactory results with an accuracy around 95%.

Alejandro Penacho Riveiros, Matthieu Barreau, Nicola Bastianello

Industrial control applications require detecting system anomalies as accurately and quickly as possible to enable prompt maintenance. In this context, it is common to consider several possible plant models, each linked to a different anomaly. The log-likelihood ratio method can then be used to identify the most accurate model and thereby classify which anomaly, if any, has occurred. Although the method has been applied to a wide variety of systems, there is no formal analysis of what makes anomalies more or less prone to detection. In this paper, we investigate a real-time anomaly detector based on the log-likelihood ratio and provide a theoretical characterization of its error rate when it is applied to linear Gaussian systems. We showcase the performance of this algorithm and the characterization obtained, and demonstrate how the latter can be leveraged for observer design.

M. Umar B. Niazi, John Cao, Matthieu Barreau et al.

This paper proposes a novel learning approach for designing Kazantzis-Kravaris/Luenberger (KKL) observers for autonomous nonlinear systems. The design of a KKL observer involves finding an injective map that transforms the system state into a higher-dimensional observer state, whose dynamics is linear and stable. The observer's state is then mapped back to the original system coordinates via the inverse map to obtain the state estimate. However, finding this transformation and its inverse is quite challenging. We propose to sequentially approximate these maps by neural networks that are trained using physics-informed learning. We generate synthetic data for training by numerically solving the system and observer dynamics. Theoretical guarantees for the robustness of state estimation against approximation error and system uncertainties are provided. Additionally, a systematic method for optimizing observer performance through parameter selection is presented. The effectiveness of the proposed approach is demonstrated through numerical simulations on benchmark examples and its application to sensor fault detection and isolation in a network of Kuramoto oscillators using learned KKL observers.

Lorenzo Fici, Dalim Wahby, Alvaro Detailleur et al.

Nonlinear underactuated systems such as two-wheeled inverted pendulums (TWIPs) exhibit a limited region of attraction (RoA), which defines the set of initial conditions from which the closed-loop system converges to the equilibrium. The RoA of nonlinear and constrained systems is generally nonconvex and analytically intractable, requiring numerical or approximate estimation methods. This work investigates the estimation of the RoA for a TWIP stabilized under three model-based control strategies: saturated linear quadratic regulator (LQR), linear model predictive control (MPC), and constraint tightening MPC (CTMPC). We first derive a Lyapunov-based invariant set that provides a certified inner approximation of the RoA. Since this analytical bound is highly conservative, a Monte Carlo-based estimation procedure is then employed to obtain a more representative approximation of the RoA, capturing how the controllers behave beyond the analytically guaranteed region. The proposed methodology combines analytical guarantees with data-driven estimation, providing both a formally certified inner bound and an empirical characterization of the RoA, offering a practical way to evaluate controller performance without relying solely on conservative analytical bounds or purely empirical simulation.

Zifan Wang, Alice Harting, Matthieu Barreau et al.

Guidance of generative models is typically achieved by modifying the probability flow vector field through the addition of a guidance field. In this paper, we instead propose the Source-Guided Flow Matching (SGFM) framework, which modifies the source distribution directly while keeping the pre-trained vector field intact. This reduces the guidance problem to a well-defined problem of sampling from the source distribution. We theoretically show that SGFM recovers the desired target distribution exactly. Furthermore, we provide bounds on the Wasserstein error for the generated distribution when using an approximate sampler of the source distribution and an approximate vector field. The key benefit of our approach is that it allows the user to flexibly choose the sampling method depending on their specific problem. To illustrate this, we systematically compare different sampling methods and discuss conditions for asymptotically exact guidance. Moreover, our framework integrates well with optimal flow matching models since the straight transport map generated by the vector field is preserved. Experimental results on synthetic 2D benchmarks, physics-informed generative tasks, and imaging inverse problems demonstrate the effectiveness and flexibility of the proposed framework.

Dennis Wilkman, Kateryna Morozovska, Karl Henrik Johansson et al.

Recent works on the application of Physics-Informed Neural Networks to traffic density estimation have shown to be promising for future developments due to their robustness to model errors and noisy data. In this paper, we introduce a methodology for online approximation of the traffic density using measurements from probe vehicles in two settings: one using the Greenshield model and the other considering a high-fidelity traffic simulation. The proposed method continuously estimates the real-time traffic density in space and performs model identification with each new set of measurements. The density estimate is updated in almost real-time using gradient descent and adaptive weights. In the case of full model knowledge, the resulting algorithm has similar performance to the classical open-loop one. However, in the case of model mismatch, the iterative solution behaves as a closed-loop observer and outperforms the baseline method. Similarly, in the high-fidelity setting, the proposed algorithm correctly reproduces the traffic characteristics.

Yulun Wu, Miguel Aguiar, Karl H. Johansson et al.

Spectral bias, the tendency of neural networks to learn low-frequency features first, is a well-known issue with many training algorithms for physics-informed neural networks (PINNs). To overcome this issue, we propose IFeF-PINN, an algorithm for iterative training of PINNs with Fourier-enhanced features. The key idea is to enrich the latent space using high-frequency components through Random Fourier Features. This creates a two-stage training problem: (i) estimate a basis in the feature space, and (ii) perform regression to determine the coefficients of the enhanced basis functions. For an underlying linear model, it is shown that the latter problem is convex, and we prove that the iterative training scheme converges. Furthermore, we empirically establish that Random Fourier Features enhance the expressive capacity of the network, enabling accurate approximation of high-frequency PDEs. Through extensive numerical evaluation on classical benchmark problems, the superior performance of our method over state-of-the-art algorithms is shown, and the improved approximation across the frequency domain is illustrated.

Aurora Poggi, Giuseppe Alessio D'Inverno, Hjalmar Brismar et al.

Data-driven discovery of dynamics in biological systems allows for better observation and characterization of processes, such as calcium signaling in cell culture. Recent advancements in techniques allow the exploration of previously unattainable insights of dynamical systems, such as the Sparse Identification of Non-Linear Dynamics (SINDy), overcoming the limitations of more classic methodologies. The latter requires some prior knowledge of an effective library of candidate terms, which is not realistic for a real case study. Using inspiration from fields like traffic density estimation and control theory, we propose a methodology for characterization and performance analysis of calcium delivery in a family of cells. In this work, we compare the performance of the Constrained Regularized Least-Squares Method (CRLSM) and Physics-Informed Neural Networks (PINN) for system identification and parameter discovery for governing ordinary differential equations (ODEs). The CRLSM achieves a fairly good parameter estimate and a good data fit when using the learned parameters in the Consensus problem. On the other hand, despite the initial hypothesis, PINNs fail to match the CRLSM performance and, under the current configuration, do not provide fair parameter estimation. However, we have only studied a limited number of PINN architectures, and it is expected that additional hyperparameter tuning, as well as uncertainty quantification, could significantly improve the performance in future works.

Alice Harting, Karl Henrik Johansson, Matthieu Barreau

We consider the problem of traffic density estimation with sparse measurements from stationary roadside sensors. Our approach uses Fourier neural operators to learn macroscopic traffic flow dynamics from high-fidelity data. During inference, the operator functions as an open-loop predictor of traffic evolution. To close the loop, we couple the open-loop operator with a correction operator that combines the predicted density with sparse measurements from the sensors. Simulations with the SUMO software indicate that, compared to open-loop observers, the proposed closed-loop observer exhibits classical closed-loop properties such as robustness to noise and ultimate boundedness of the error. This shows the advantages of combining learned physics with real-time corrections, and opens avenues for accurate, efficient, and interpretable data-driven observers.

Sirui Li, Federica Bragone, Matthieu Barreau et al.

Our work aims at simulating and predicting the temperature conditions inside a power transformer using Physics-Informed Neural Networks (PINNs). The predictions obtained are then used to determine the optimal placement for temperature sensors inside the transformer under the constraint of a limited number of sensors, enabling efficient performance monitoring. The method consists of combining PINNs with Mixed Integer Optimization Programming to obtain the optimal temperature reconstruction inside the transformer. First, we extend our PINN model for the thermal modeling of power transformers to solve the heat diffusion equation from 1D to 2D space. Finally, we construct an optimal sensor placement model inside the transformer that can be applied to problems in 1D and 2D.

Francis Tembo, Federica Bragone, Tor Laneryd et al.

Power transformers are subjected to electrical currents and temperature fluctuations that, if not properly controlled, can lead to major deterioration of their insulation system. Therefore, monitoring the temperature of a power transformer is fundamental to ensure a long-term operational life. Models presented in the IEC 60076-7 and IEEE standards, for example, monitor the temperature by calculating the top-oil and the hot-spot temperatures. However, these models are not very accurate and rely on the power transformers' properties. This paper focuses on finding an alternative method to predict the top-oil temperatures given previous measurements. Given the large quantities of data available, machine learning methods for time series forecasting are analyzed and compared to the real measurements and the corresponding prediction of the IEC standard. The methods tested are Artificial Neural Networks (ANNs), Time-series Dense Encoder (TiDE), and Temporal Convolutional Networks (TCN) using different combinations of historical measurements. Each of these methods outperformed the IEC 60076-7 model and they are extended to estimate the temperature rise over ambient. To enhance prediction reliability, we explore the application of quantile regression to construct prediction intervals for the expected top-oil temperature ranges. The best-performing model successfully estimates conditional quantiles that provide sufficient coverage.

Sirui Li, Federica Bragone, Matthieu Barreau et al.

Physics-Informed Neural Networks (PINNs) are a powerful deep learning method capable of providing solutions and parameter estimations of physical systems. Given the complexity of their neural network structure, the convergence speed is still limited compared to numerical methods, mainly when used in applications that model realistic systems. The network initialization follows a random distribution of the initial weights, as in the case of traditional neural networks, which could lead to severe model convergence bottlenecks. To overcome this problem, we follow current studies that deal with optimal initial weights in traditional neural networks. In this paper, we use a convex optimization model to improve the initialization of the weights in PINNs and accelerate convergence. We investigate two optimization models as a first training step, defined as pre-training, one involving only the boundaries and one including physics. The optimization is focused on the first layer of the neural network part of the PINN model, while the other weights are randomly initialized. We test the methods using a practical application of the heat diffusion equation to model the temperature distribution of power transformers. The PINN model with boundary pre-training is the fastest converging method at the current stage.