arXiv subjects

Matthieu Barreau

Publications and source records attributed to Matthieu Barreau.

At least 19 recordsLinked to original sources

arXiv2026

Alternating Levenberg-Marquardt Training of Physics-Informed Neural Networks with Fourier-Enhanced Features

Physics-informed neural networks (PINNs) often fail to accurately resolve partial differential equations (PDEs) with high-frequency or multi-scale solutions, as well as strongly nonlinear problems. Two factors underlie this difficulty: spectral bias, the tendency of neural networks to underfit high-frequency features; and representation-coefficient coupling, the entanglement of representation learning and coefficient fitting within a single nonconvex optimization objective. In this work, we propose the Fourier-enhanced alternating Levenberg--Marquardt PINN (FALM-PINN), an optimization framework that decouples representation learning from coefficient fitting. The upper-level problem learns a Fourier-enhanced basis that enriches the latent space with high-frequency components, while the lower-level problem resolves the coupling by fitting the projection coefficients on this basis, solving a nonlinear least-squares problem with the Levenberg--Marquardt algorithm. The framework applies to general nonlinear and coupled PDE systems, and reduces to a single-step convex optimization problem for linear PDEs. We prove global convergence of the alternating training scheme in both cases. Numerical examples on multiple challenging high-frequency and nonlinear PDEs show that FALM-PINN achieves relative $L^2$ errors up to two orders of magnitude lower than state-of-the-art baselines.

cs.LG↗

arXiv2026

Learning Neural Maximal Lyapunov Functions on $\mathsf{SO}(n)$

Establishing stability guarantees for dynamical systems on Lie groups is a fundamental challenge, as classical Lyapunov methods developed for Euclidean spaces do not directly transfer to curved geometries. In this paper, we propose a framework for learning maximal Lyapunov functions for systems evolving on the special orthogonal group $\mathsf{SO}(n)$. Theoretically, we introduce a neural Lyapunov architecture based on the logarithmic map with proven approximation capabilities, and we formulate the learning problem via a Zubov-type characterization of the maximal region of attraction. A key technical contribution is the derivation of explicit, numerically tractable formulas for the derivative of the logarithmic map, enabling training through a two-phase algorithm that balances computational efficiency and accuracy. Empirically, we validate the approach on a low-dimensional nonlinear system.

math.OC↗

arXiv2026

Locally Stable Neural ODEs with Characterized Region of Attraction

We propose a class of neural ODEs that universally approximates locally exponentially stable dynamics and the region of attraction from trajectory data. The model dynamics are constrained by the gradient field of a jointly learned maximal Lyapunov function. Under this constraint, we show that exponentially stable dynamics can be approximated arbitrarily well within the region of attraction. Furthermore, the region of attraction of the constrained model is exactly characterized by the 1-sublevel set of the jointly learned Lyapunov function, and we derive conditions under which it approximates the true region of attraction arbitrarily well. We validate the approach experimentally on nonlinear systems with nonconvex regions of attraction.

math.OC↗

arXiv2026

Hybrid System Identification of Electric Freight Transition Dynamics via SINDy and Neural ODEs

The transition to electric heavy-duty freight is constrained by a strong interdependence between vehicle adoption and charging infrastructure deployment. While system dynamics models are well-suited to simulate these socio-technical feedback loops, their reliance on conditional logic and heuristic rules makes the underlying dynamics difficult to interpret mathematically. This limits the direct application of formal systems and control theory. This paper proposes a solution to that problem through a general hybrid system identification framework that extracts a continuous-time analytical surrogate from simulations of complex system dynamics. The approach combines the sparse identification of nonlinear dynamics algorithm with neural ordinary differential equations to form a grey-box model. The first component identifies the dominant interpretable dynamics under causal coupling constraints, while the neural residual captures unmodelled nonlinearities. The framework is trained using multiple shooting over a 40-year horizon. The resulting model reproduces the training trajectories with a normalized root-mean-square error below 4\%, while maintaining reliable predictive accuracy when evaluated on unseen initial conditions.

math.OC↗

arXiv2026

Model-free Anomaly Detection for Dynamical Systems with Gaussian Processes

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.

eess.SY↗

arXiv2026

Detectability of Subtle Anomalies in Dynamical Systems via Log-Likelihood Ratio

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.

eess.SY↗

arXiv2026

Region of Attraction Estimation for Linear Quadratic Regulator, Linear and Robust Model Predictive Control on a Two-Wheeled Inverted Pendulum

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.

eess.SY↗

arXiv2026

Curriculum-Learned Vanishing Stacked Residual PINNs for Hyperbolic PDE State Reconstruction

Modeling distributed dynamical systems governed by hyperbolic partial differential equations (PDEs) remains challenging due to discontinuities and shocks that hinder the convergence of traditional physics-informed neural networks (PINNs). The recently proposed vanishing stacked residual PINN (VSR-PINN) embeds a vanishing-viscosity mechanism within stacked residual refinements to enable a smooth transition from the parabolic to hyperbolic regime. This paper integrates three curriculum-learning methods as primal-dual (PD) optimization, causality progression, and adaptive sampling into the VSR-PINN. The PD strategy balances physics and data losses, the causality scheme unlocks deeper stacks by respecting temporal and gradient evolution, and adaptive sampling targets high residuals. Numerical experiments on traffic reconstruction confirm that enforcing causality systematically reduces the median point-wise MSE and its variability across runs, yielding improvements of nearly one order of magnitude over non-causal training in both the baseline and PD variants.

cs.NE↗

arXiv2025

Modeling and Physics-Enhanced Fault Detection in Wastewater Pump Stations

Monitoring wastewater pump stations is essential because they are critical infrastructure. However, monitoring is still often performed manually due to the lack of suitable algorithmic methods and data. This paper introduces a high-fidelity, physics-enhanced simulator of a three-pump wastewater station that captures transient hydro-mechanical dynamics at a one-second resolution. The simulator is fully parameter-driven, adaptable to other wastewater stations, and capable of generating datasets for data-driven analytics. It can also generate balanced faulty datasets when real failures are scarce or confidential. A comparison with high-frequency SCADA data from a municipal station shows strong agreement across key operational metrics. Furthermore, the paper proposes robust statistical and mathematical frameworks for fault detection and isolation, including a nested-model F-test to detect pump degradation or system faults, and a tangent residual approach to distinguish pump faults from system faults using operating-point kinematics. This framework enables what-if studies, facilitates early fault diagnosis based on flow rate and head, and provides actionable insights for condition-based maintenance in wastewater pumping infrastructure.

eess.SY↗

arXiv2025

Iterative Training of Physics-Informed Neural Networks with Fourier-enhanced Features

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.

cs.LG↗

arXiv2025

Physics-Informed Detection of Friction Anomalies in Satellite Reaction Wheels

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%.

eess.SY↗

arXiv2025

Source-Guided Flow Matching

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.

cs.LG↗

arXiv2025

Data-driven multi-agent modelling of calcium interactions in cell culture: PINN vs Regularized Least-squares

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.

q-bio.QM↗

arXiv2025

Closed-Loop Neural Operator-Based Observer of Traffic Density

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.

math.OC↗

arXiv2025

Online Traffic Density Estimation using Physics-Informed Neural Networks

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.

cs.LG↗

arXiv2025

Vanishing Stacked-Residual PINN for State Reconstruction of Hyperbolic Systems

In a more connected world, modeling multi-agent systems with hyperbolic partial differential equations (PDEs) offers a compact, physics-consistent description of collective dynamics. However, classical control tools need adaptation for these complex systems. Physics-informed neural networks (PINNs) provide a powerful framework to fix this issue by inferring solutions to PDEs by embedding governing equations into the neural network. A major limitation of original PINNs is their inability to capture steep gradients and discontinuities in hyperbolic PDEs. To tackle this problem, we propose a stacked residual PINN method enhanced with a vanishing viscosity mechanism. Initially, a basic PINN with a small viscosity coefficient provides a stable, low-fidelity solution. Residual correction blocks with learnable scaling parameters then iteratively refine this solution, progressively decreasing the viscosity coefficient to transition from parabolic to hyperbolic PDEs. Applying this method to traffic state reconstruction improved results by an order of magnitude in relative $\mathcal{L}^2$ error, demonstrating its potential to accurately estimate solutions where original PINNs struggle with instability and low fidelity.

eess.SY↗

arXiv2025

Optimal Sensor Placement in Power Transformers Using Physics-Informed Neural Networks

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.

cs.LG↗

arXiv2025

A Control Perspective on Training PINNs

We investigate the training of Physics-Informed Neural Networks (PINNs) from a control-theoretic perspective. Using gradient descent with resampling, we interpret the training dynamics as asymptotically equivalent to a stochastic control-affine system, where sampling effects act as process disturbances and measurement noise. Within this framework, we introduce two controllers for dynamically adapting the physics weight: an integral controller and a leaky integral controller. We theoretically analyze their asymptotic properties under the accuracy-robustness trade-off, and we evaluate them on a toy example. Numerical evidence suggests that the integral controller achieves accurate and robust convergence when the physical model is correct, whereas the leaky integrator provides improved performance in the presence of model mismatch. This work represents a first step toward convergence guarantees and principled training algorithms tailored to the distinct characteristics of PINN tasks.

cs.LG↗