arXiv subjects

Umberto Biccari

Publications and source records attributed to Umberto Biccari.

At least 19 recordsLinked to original sources

arXiv2026

Bi-HYCO: Bi-Objective Cooperative Learning for PDE Parameter Identification under Fragmented Observations

Physical and synthetic models may describe complementary aspects of the same PDE-governed system while receiving different, possibly fragmented, observations. We propose Bi-Objective HYCO (Bi-HYCO), a cooperative framework that retains both representations and their local observational objectives while coupling their predicted states at unlabeled interaction points. These points contain no measurements and do not augment the data; they provide a communication mechanism in the common state space. The two criteria form a vector-valued objective, and weighted scalarizations provide computational realizations. For the deterministic shared-observation algorithm with fixed interaction points, we prove sufficient decrease and finite length of the whole alternating sequence, which converges to a mixed critical point under the stated Kurdyka-Lojasiewicz-type assumptions. Elliptic transmission and two-dimensional Navier-Stokes experiments assess parameter and state reconstruction, noise and scalarization effects, and PINN/XPINN references. Ablations show that removing state interaction while retaining aggregation deteriorates parameter recovery in the tested configurations, particularly for Navier-Stokes.

cs.LG↗

arXiv2026

Boundary observation and control for fractional heat and wave equations

We establish boundary null controllability of the heat equation driven by the integral fractional Laplacian $(-Δ)^s$ on a bounded smooth domain for every $T>0$ and every fractional exponent $s\in(1/2,1)$. The control acts through the singular boundary trace naturally associated with the fractional Dirichlet problem. The main ingredient is a frequency-dependent boundary observability inequality for the associated fractional wave equation, obtained by combining multiplier arguments with the fractional Pohozaev identity. In contrast with the classical wave equation, the observability time deteriorates with the spectral cutoff, reflecting the slow propagation of high-frequency fractional waves. We transfer this estimate to the parabolic problem by transmutation and combine the resulting low-frequency controllability estimates with the high-frequency dissipation of the fractional heat semigroup through a Lebeau-Robbiano iteration. The balance between these two mechanisms yields null controllability precisely in the range $s>1/2$.

math.AP↗

arXiv2026

Interior interpretability with attention rollout: contraction and propagation profiles in Transformers

Feature-attribution methods assign scores relating input variables to a model's output, but do not by themselves characterize how explicitly defined interaction operators compose across its intermediate layers. We introduce \emph{interior interpretability}, a propagation-based perspective on internal model organization, and instantiate it for tabular Transformers using attention rollout. We interpret rollout as a row-stochastic operator encoding attention-mediated propagation between feature tokens. By applying classical Doeblin--Dobrushin contraction theory, we show that a rollout operator with a small Dobrushin coefficient is quantitatively close to a rank-one stochastic matrix whose common row is determined by its normalized column sums. This result gives a structural interpretation to the corresponding rollout propagation profile. In Transformers trained for metabolomic age prediction, the measured rollout contraction strengthens with depth. Trained and randomly initialized models also exhibit different propagation profiles, although the present experiments do not establish the predictive relevance of individual rollout-ranked variables. Exploratory comparisons with PCA and GradientExplainer approximations to SHAP reveal localized agreement among highly ranked variables but weak agreement across complete rankings. Attention rollout is therefore used here as a diagnostic of attention-mediated propagation, not as a causal explanation or faithful attribution of the complete Transformer.

cs.LG↗

arXiv2026

Robust stabilization of hyperbolic PDE-ODE systems via Neural Operator-approximated gain kernels

This paper investigates the mean square exponential stabilization problem for a class of coupled PDE-ODE systems with Markov jump parameters. The considered system consists of multiple coupled hyperbolic PDEs and a finite-dimensional ODE, where all system parameters evolve according to a homogeneous continuous-time Markov process. The control design is based on a backstepping approach. To address the computational complexity of solving kernel equations, a DeepONet framework is proposed to learn the mapping from system parameters to the backstepping kernels. By employing Lyapunov-based analysis, we further prove that the controller obtained from the neural operator ensures stability of the closed-loop stochastic system. Numerical simulations demonstrate that the proposed approach achieves more than two orders of magnitude speedup compared to traditional numerical solvers, while maintaining high accuracy and ensuring robust closed-loop stability under stochastic switching.

math.OC↗

arXiv2026

Learning the Riccati solution operator for time-varying LQR via Deep Operator Networks

We propose a computational framework for replacing the repeated numerical solution of differential Riccati equations in finite-horizon Linear Quadratic Regulator (LQR) problems by a learned operator surrogate. Instead of solving a nonlinear matrix-valued differential equation for each new system instance, we construct offline an approximation of the associated solution operator mapping time-dependent system parameters to the Riccati trajectory. The resulting model enables fast online evaluation of approximate optimal feedbacks across a wide class of systems, thereby shifting the computational burden from repeated numerical integration to a one-time learning stage. From a theoretical perspective, we establish control-theoretic guarantees for this operator-based approximation. In particular, we derive bounds quantifying how operator approximation errors propagate to feedback performance, trajectory accuracy, and cost suboptimality, and we prove that exponential stability of the closed-loop system is preserved under sufficiently accurate operator approximation. These results provide a framework to assess the reliability of data-driven approximations in optimal control. On the computational side, we design tailored DeepONet architectures for matrix-valued, time-dependent problems and introduce a progressive learning strategy to address scalability with respect to the system dimension. Numerical experiments on both time-invariant and time-varying LQR problems demonstrate that the proposed approach achieves high accuracy and strong generalization across a wide range of system configurations, while delivering substantial computational speedups compared to classical solvers. The method offers an effective and scalable alternative for parametric and real-time optimal control applications.

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arXiv2026

Computational performance of the MMOC in the inverse design of the Doswell frontogenesis equation

Inverse design of transport equations can be addressed by using a gradient-adjoint methodology. In this methodology numerical schemes used for the adjoint resolution determine the direction of descent in its iterative algorithm, and consequently the CPU time consumed by the inverse design. As the CPU time constitutes a known bottleneck, it is important to employ light and quick schemes to the adjoint problem. In this regard, we proposed to use the Modified Method of Characteristics (MMOC). Despite not preserving identity conservation, the MMOC is computationally competitive. In this work we investigated the advantage of using the MMOC in comparison with the Lax-Friedrichs and Lax-Wendro? schemes for the inverse design problem. By testing the Doswell frontogenesis equation, we observed that the MMOC can provide more efficient and accurate computation under some simulation conditions.

math.NA↗

arXiv2026

Blow-up results for a logarithmic pseudo-parabolic $p(.)$-Laplacian type equation

In this paper, we consider an initial-boundary value problem for the following mixed pseudo-parabolic $p(.)$-Laplacian type equation with logarithmic nonlinearity: $$ u_t-Δu_t-\mbox{div}\left(\left\vert \nabla u\right\vert^{p(.)-2}\nabla u\right) =|u|^{q(.)-2}u\ln(|u|), \quad (x,t)\inΩ\times(0,+\infty),$$ where $Ω\subset\mathbb{R}^n$ is a bounded and regular domain, and the variable exponents $p(.)$ and $q(.)$ satisfy suitable regularity assumptions. By adapting the first-order differential inequality method, we establish a blow-up criterion for the solutions and obtain an upper bound for the blow-up time. In a second moment, we show that blow-up may be prevented under appropriate smallness conditions on the initial datum, in which case we also establish decay estimates in the $H_0^1(Ω)$-norm as $t\to+\infty$. This decay result is illustrated by a two-dimensional numerical example.

math.AP↗

arXiv2026

Fair feature attribution for multi-output prediction: a Shapley-based perspective

In this article, we provide an axiomatic characterization of feature attribution for multi-output predictors within the Shapley framework. While SHAP explanations are routinely computed independently for each output coordinate, the theoretical necessity of this practice has remained unclear. By extending the classical Shapley axioms to vector-valued cooperative games, we establish a rigidity theorem showing that any attribution rule satisfying efficiency, symmetry, dummy player, and additivity must necessarily decompose component-wise across outputs. Consequently, any joint-output attribution rule must relax at least one of the classical Shapley axioms. This result identifies a previously unformalized structural constraint in Shapley-based interpretability, clarifying the precise scope of fairness-consistent explanations in multi-output learning. Numerical experiments on a biomedical benchmark illustrate that multi-output models can yield computational savings in training and deployment, while producing SHAP explanations that remain fully consistent with the component-wise structure imposed by the Shapley axioms.

cs.LG↗

arXiv2026

Operator learning for prescribed-time stabilization of reaction-diffusion systems

This paper addresses boundary prescribed-time stabilization of a one-dimensional heat equation with spatially and temporally varying coefficients. In contrast to asymptotic or exponential stabilization, prescribed-time stabilization ensures convergence to equilibrium within a user-defined time that is independent of the initial condition, a property that is particularly attractive in applications with stringent transient performance requirements. The backstepping design for this problem requires solving, at each time instant, a two-dimensional time-dependent kernel Partial Differential Equation (PDE) whose solution continuously varies with the plant coefficients. The repeated numerical solution of this parabolic kernel PDE results in a prohibitive computational burden, thereby limiting real-time applicability. To overcome this limitation, we propose a neural-operator-based approximation of the mapping from the time-varying system coefficient to the corresponding backstepping kernel. The operator is trained offline using representative solutions of the kernel PDE and subsequently deployed online to generate the required time-varying kernels in real time. We establish, via Lyapunov analysis, that the resulting neural-operator-based controller preserves prescribed-time stability provided that the operator approximation error satisfies an explicit bound. Furthermore, we investigate a direct approximation of the full feedback law mapping the plant parameter functions and state measurements to the boundary control input. For this setting, we prove semiglobal practical prescribed-time stability of the closed-loop system. Numerical experiments demonstrate that the proposed approach reduces the computational cost of kernel generation by several orders of magnitude, thereby enabling real-time prescribed-time stabilization for heat equations with spatially and temporally varying coefficients.

math.OC↗

arXiv2025

Spiking Neural Networks: a theoretical framework for Universal Approximation and training

Spiking Neural Networks (SNNs) are widely regarded as a biologically-inspired and energy-efficient alternative to classical artificial neural networks. Yet, their theoretical foundations remain only partially understood. In this work, we develop a rigorous mathematical analysis of a representative SNN architecture based on Leaky Integrate-and-Fire (LIF) neurons with threshold-reset dynamics. Our contributions are twofold. First, we establish a universal approximation theorem showing that SNNs can approximate continuous functions on compact domains to arbitrary accuracy. The proof relies on a constructive encoding of target values via spike timing and a careful interplay between idealized $δ$-driven dynamics and smooth Gaussian-regularized models. Second, we analyze the quantitative behavior of spike times across layers, proving well-posedness of the hybrid dynamics and deriving conditions under which spike counts remain stable, decrease, or in exceptional cases increase due to resonance phenomena or overlapping inputs. Together, these results provide a principled foundation for understanding both the expressive power and the dynamical constraints of SNNs, offering theoretical guarantees for their use in classification and signal processing tasks.

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arXiv2025

A Multi-Objective Optimization framework for Decentralized Learning with coordination constraints

This article introduces a generalized framework for Decentralized Learning formulated as a Multi-Objective Optimization problem, in which both distributed agents and a central coordinator contribute independent, potentially conflicting objectives over a shared model parameter space. Unlike traditional approaches that merge local losses under a common goal, our formulation explicitly incorporates coordinator-side criteria, enabling more flexible and structured training dynamics. To navigate the resulting trade-offs, we explore scalarization strategies, particularly weighted sums, to construct tractable surrogate problems. These yield solutions that are provably Pareto optimal under standard convexity and smoothness assumptions, while embedding global preferences directly into local updates. We propose a decentralized optimization algorithm with convergence guarantees, and demonstrate its empirical performance through simulations, highlighting the impact of the coordinator's influence on local agent behavior. The proposed approach offers a principled and customizable strategy for balancing personalization, fairness, and coordination in decentralized learning systems.

math.OC↗

arXiv2023

Gaussian Beam ansatz for finite difference wave equations

This work is concerned with the construction of Gaussian Beam (GB) solutions for the numerical approximation of wave equations, semi-discretized in space by finite difference schemes. GB are high-frequency solutions whose propagation can be described, both at the continuous and at the semi-discrete levels, by microlocal tools along the bi-characteristics of the corresponding Hamiltonian. Their dynamics differ in the continuous and the semi-discrete setting, because of the high-frequency gap between the Hamiltonians. In particular, numerical high-frequency solutions can exhibit spurious pathological behaviors, such as lack of propagation in space, contrary to the classical space-time propagation properties of continuous waves. This gap between the behavior of continuous and numerical waves introduces also significant analytical difficulties, since classical GB constructions cannot be immediately extrapolated to the finite difference setting, and need to be properly tailored to accurately detect the propagation properties in discrete media. Our main objective in this paper is to present a general and rigorous construction of the GB ansatz for finite difference wave equations, and corroborate this construction through accurate numerical simulations.

math.AP↗

arXiv2023

A two-stage numerical approach for the sparse initial source identification of a diffusion-advection equation

We consider the problem of identifying a sparse initial source condition to achieve a given state distribution of a diffusion-advection partial differential equation after a given final time. The initial condition is assumed to be a finite combination of Dirac measures. The locations and intensities of this initial condition are required to be identified. This problem is known to be exponentially ill-posed because of the strong diffusive and smoothing effects. We propose a two-stage numerical approach to treat this problem. At the first stage, to obtain a sparse initial condition with the desire of achieving the given state subject to a certain tolerance, we propose an optimal control problem involving sparsity-promoting and ill-posedness-avoiding terms in the cost functional, and introduce a generalized primal-dual algorithm for this optimal control problem. At the second stage, the initial condition obtained from the optimal control problem is further enhanced by identifying its locations and intensities in its representation of the combination of Dirac measures. This two-stage numerical approach is shown to be easily implementable and its efficiency in short time horizons is promisingly validated by the results of numerical experiments. Some discussions on long time horizons are also included.

math.OC↗

arXiv2023

Stochastic optimization methods for the simultaneous control of parameter-dependent systems

We address the application of stochastic optimization methods for the simultaneous control of parameter-dependent systems. In particular, we focus on the classical Stochastic Gradient Descent (SGD) approach of Robbins and Monro, and on the recently developed Continuous Stochastic Gradient (CSG) algorithm. We consider the problem of computing simultaneous controls through the minimization of a cost functional defined as the superposition of individual costs for each realization of the system. We compare the performances of these stochastic approaches, in terms of their computational complexity, with those of the more classical Gradient Descent (GD) and Conjugate Gradient (CG) algorithms, and we discuss the advantages and disadvantages of each methodology. In agreement with well-established results in the machine learning context, we show how the SGD and CSG algorithms can significantly reduce the computational burden when treating control problems depending on a large amount of parameters. This is corroborated by numerical experiments.

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arXiv2022

Null Controllability for a Degenerate Structured Population Model

In this paper, we consider the infinite dimensional linear control system describing population models structured by age, size, and spatial position. The diffusion coefficient is degenerate at a point of the domain or both extreme points. Moreover, the control is localized in the space variable as well as with respect to the age and size. For each control support, we give an estimate of the time needed to control the system to zero. We establish the null controllability of the model by using a technique that avoids the explicit use of parabolic Carleman estimates. Indeed, our argument relies on a method that combines final-state observability estimates with the use of the characteristic method.

math.OC↗

arXiv2022

Controllability and Positivity Constraints in Population Dynamics with age, size Structuring and Diffusion

In this article, we consider the infinite dimensional linear control system describing the Population Models Structured by Age, Size, and Spatial Position. The control is localized in the space variable as well as with respect to the age and size. For each control support, we give an estimate of the time needed to control the system to zero. We prove the null controllability of the model, using a technique avoids the explicit use of parabolic Carleman estimates. Indeed, this method combines final-state observability estimates with the use of characteristics and with $L^{\infty}$ estimates of the associated semigroup.

math.OC↗

arXiv2021

Multilevel Selective Harmonic Modulation by Duality

We address the Selective Harmonic Modulation (SHM) problem in power electronic engineering, consisting in designing a multilevel staircase control signal with some prescribed frequencies to improve the performances of a converter. In this work, SHM is addressed through an optimal control methodology based on duality, in which the admissible controls are piece-wise constant functions, taking values only in a given finite set. To fulfill this constraint, the cornerstone of our approach is the introduction of a special penalization in the cost functional, in the form of a piece-wise affine approximation of a parabola. In this manner, we build optimal multilevel controls having the desired staircase structure.

math.OC↗

arXiv2021

Multilevel Selective Harmonic Modulation via Optimal Control

We consider the Selective Harmonic Modulation (SHM) problem, consisting in the design of a staircase control signal with some prescribed frequency components. In this work, we propose a novel methodology to address SHM as an optimal control problem in which the admissible controls are piecewise constant functions, taking values only in a given finite set. In order to fulfill this constraint, we introduce a cost functional with piecewise affine penalization for the control, which, by means of Pontryagin's maximum principle, makes the optimal control have the desired staircase form. Moreover, the addition of the penalization term for the control provides uniqueness and continuity of the solution with respect to the target frequencies. Another advantage of our approach is that the number of switching angles and the waveform need not be determined a priori. Indeed, the solution to the optimal control problem is the entire control signal, and therefore, it determines the waveform and the location of the switches. We also provide several numerical examples in which the SHM problem is solved by means of our approach.

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