Search arXiv⌕ Search

arXiv subjects

Matteo Cozzi

Publications and source records attributed to Matteo Cozzi.

At least 19 recordsLinked to original sources
arXiv2026

Certified Access: Test-Time Compute on Conservative Physics-Structured AI Memory

Test-time compute mechanisms, such as retries, confidence-based escalation and non-local routing, are conventionally deployed as black-box additions to trained models. They typically add forward passes or halting heads, but give no structural account of what the extra compute buys or what it costs against the model's stability guarantees. We therefore propose a theoretical and empirical framework for test-time compute on a Latent Dynamics Model used as the memory. When an encoder's latent state $(q,p)$ is advanced by a damped symplectic (velocity-Verlet) step of a learned Hamiltonian, every mechanism that expands access has an exact phase-volume Jacobian, a bounded energy change and an exact latch-transport law. We split reachability failures into escape failures and reach failures. A Lorentz squeeze resolves escape failures by injecting bounded, re-absorbable energy. A gated canonical translation (a one-step channel across $C_T$) resolves reach failures at $\detJ=1$ exactly. We verify this certificate stack on a designed analytic testbed. Reaching just beyond $C_T$ with the squeeze requires energy that grows quadratically in the excess distance. The gated channel's energy cost equals the potential difference between its endpoints, which is zero on the matched channel measured here, and so does not grow with distance in this testbed. An untrained state-replacing map annihilates phase volume and erases stored charge, while the gated channel transports it. On a learned physics-structured memory, we show calibrated compute rationing and evaluate the scaling limits of non-local routing against a physics-free baseline. The main contribution is the formalization and verification of the certificate stack, not benchmark performance.

cs.AI↗
arXiv2026

FactoryNet: A Large-Scale Dataset toward Industrial Time-Series Foundation Models

Industrial machines are actuated systems, yet most open time-series corpora record sensor outcomes alone; the few that do log commanded signals expose them in vendor-specific forms that differ from machine to machine, so commanded intent and measured response cannot be compared across embodiments. We introduce FactoryNet, a pretraining corpus for industrial time-series: 113M datapoints across 57k end-to-end task executions (15k real, 42k synthetic) on seven embodiments and four tasks, with 27 annotated anomaly types alongside healthy baselines and counterfactual pairs. Every signal in every source is mapped into a single control-theoretic schema, Setpoint, Effort, Feedback, Context (S-E-F-C), which expresses any actuated system in a common representational frame and makes commanded-versus-realized dynamics readable as explicit prediction residuals. Using the corpus, we find that a task-phase model trained on simulation, with no real phase labels, recovers roughly two thirds of the real-data ceiling on both robots, and that transfer holds within a robot family but rarely across manufacturers. Fault signatures behave differently: an uncalibrated simulator gives no detectable fault signal on either robot, and aligning the feature distributions does not repair it. What separation there is comes from Effort: motor current distinguishes faults where measured tool motion is close to chance, and calibrating the simulator to reproduce that one channel recovers a payload fault on the real robot. We read this as simulation capturing the stable structure of a task while fault behaviour stays diverse and fault-type specific. We release FactoryNet as a growing, multi-embodiment substrate for industrial foundation models.

cs.LG↗
arXiv2026

KITE: A Knowledge-Injected Temporal Encoder for Time-Series Event Forecasting

Forecasting rare, high-cost events in multivariate time-series systems is hard because their precursors are seldom observed, so a purely data-driven model has little to learn from. The engineers who run these systems, however, hold knowledge that precedes such events, information absent from the scarce labels. We call this a semantic prior and inject it into a self-supervised event forecaster through two switchable mechanisms, a covariate mechanism and an attention-bias mechanism, each the identity at its null, so one architecture serves every prior type with no per-dataset architecture search. A prior helps when it supplies information the observed window does not already contain: in a short context an informative prior beats no prior in-domain (pooled h-AUROC over the no-prior baseline: a periodicity clock +14%, p<0.001; a conservation residual +47% at a 0.25 label fraction), and a working prior transfers out-of-domain (+53%), substituting for feature transfer, while priors the window already recovers stay inert and the attention-bias mechanism shows no measurable gain. KITE is a switchable, attributable route to contextual conditioning, and a substrate for scaling toward cross-domain, pretrained time-series foundation world models that carry their governing structure explicitly.

cs.LG↗
arXiv2026

A fractional De Giorgi isoperimetric type inequality

We establish an isoperimetric type inequality for the level sets of functions in fractional Sobolev spaces. This answers a question posed by the first author in a previous paper. To obtain it, we work out a slight modification of some estimates for nonlocal interaction functionals established by Savin and Valdinoci. We also show how said isoperimetric inequality leads to the Hölder continuity of functions in (weak) fractional De Giorgi classes.

math.AP↗
arXiv2025

Optimal boundary regularity for mixed local and nonlocal equations

We provide sharp boundary regularity estimates for solutions to elliptic equations driven by an integro-differential operator obtained as the sum of a Laplacian with a nonlocal operator generalizing a fractional Laplacian. Our approach makes use of weighted Hölder spaces as well as regularity estimates for the Laplacian in this context and a fixed-point argument. We show the optimality of the obtained estimates by means of a counterexample that we have striven to keep as explicit as possible.

math.AP↗
arXiv2025

A quantitative study of radial symmetry for solutions to semilinear equations in $\mathbb{R}^n$

A celebrated result by Gidas, Ni & Nirenberg asserts that positive classical solutions, decaying at infinity, to semilinear equations $Δu +f(u)=0$ in $\mathbb{R}^n$ must be radial and radially decreasing. In this paper, we consider both energy solutions in $\mathcal{D}^{1,2}(\mathbb{R}^n)$ and non-energy local weak solutions to small perturbations of these equations, and study its quantitative stability counterpart. To the best of our knowledge, the present work provides the first quantitative stability result for non-energy solutions to semilinear equations involving the Laplacian, even for the critical nonlinearity.

math.AP↗
arXiv2024

A quantitative version of the Gidas-Ni-Nirenberg Theorem

A celebrated result by Gidas, Ni & Nirenberg asserts that classical positive solutions to semilinear equations $- Δu = f(u)$ in a ball vanishing at the boundary must be radial and radially decreasing. In this paper we consider small perturbations of this equation and study the quantitative stability counterpart of this result.

math.AP↗
arXiv2023

Global gradient regularity and a Hopf lemma for quasilinear operators of mixed local-nonlocal type

We address some regularity issues for mixed local-nonlocal quasilinear operators modeled upon the sum of a $p$-Laplacian and of a fractional $(s, q)$-Laplacian. Under suitable assumptions on the right-hand sides and the outer data, we show that weak solutions of the Dirichlet problem are $C^{1, θ}$-regular up to the boundary. In addition, we establish a Hopf type lemma for positive supersolutions. Both results hold assuming the boundary of the reference domain to be merely of class $C^{1, α}$, while for the regularity result we also require that $p > s q$.

math.AP↗
arXiv2023

Non-uniqueness for the nonlocal Liouville equation in $\mathbb{R}$ and applications

We construct multiple solutions to the nonlocal Liouville equation \begin{equation} \label{eqk} \tag{L} (-Δ)^{\frac{1}{2}} u = K(x) e^u \quad \mbox{ in } \mathbb{R}. \end{equation} More precisely, for $K$ of the form $K(x) = 1+\varepsilon κ(x)$ with $\varepsilon \in (0,1)$ small and $κ\in C^{1,α}(\mathbb{R}) \cap L^{\infty}(\mathbb{R})$ for some $α> 0$, we prove existence of multiple solutions to \eqref{eqk} bifurcating from the bubbles. These solutions provide examples of flat metrics in the half-plane with prescribed geodesic curvature $K(x)$ on its boundary. Furthermore, they imply the existence of multiple ground state soliton solutions for the Calogero-Moser derivative NLS.

math.AP↗
arXiv2022

Blowing-up solutions for a nonlocal Liouville type equation

We consider the nonlocal Liouville type equation $$ (-Δ)^{\frac{1}{2}} u = \varepsilon κ(x) e^u, \quad u > 0, \quad \mbox{in } I, \qquad u = 0, \quad \mbox{in } \mathbb{R} \setminus I, $$ where $I$ is a union of $d \geq 2$ disjoint bounded intervals, $κ$ is a smooth bounded function with positive infimum and $\varepsilon > 0$ is a small parameter. For any integer $1 \leq m \leq d$, we construct a family of solutions $(u_\varepsilon)_{\varepsilon}$ which blow up at $m$ interior distinct points of $I$ and for which $\varepsilon \int_I κe^{u_\varepsilon} \, \rightarrow 2 m π$, as $\varepsilon \to 0$. Moreover, we show that, when $d = 2$ and $m$ is suitably large, no such construction is possible.

math.AP↗
arXiv2020

On nonlocal minimal graphs

We develop a functional analytic approach for the study of nonlocal minimal graphs. Through this, we establish existence and uniqueness results, a priori estimates, comparison principles, rearrangement inequalities, and the equivalence of several notions of minimizers and solutions.

math.AP↗
arXiv2020

An elliptic boundary value problem with fractional nonlinearity

We investigate existence and uniqueness of solutions to second-order elliptic boundary value problems containing a power nonlinearity applied to a fractional Laplacian. We detect the critical power separating the existence from the non-existence regimes. For the existence results, we make use of a particular class of weighted Sobolev spaces to compensate boundary singularities which are naturally built in the problem.

math.AP↗
arXiv2020

A fractional Michael-Simon Sobolev inequality on convex hypersurfaces

The classical Michael-Simon and Allard inequality is a Sobolev inequality for functions defined on a submanifold of Euclidean space. It is governed by a universal constant independent of the manifold, but displays on the right-hand side an additional $L^p$ term weighted by the mean curvature of the underlying manifold. We prove here a fractional version of this inequality on hypersurfaces of Euclidean space that are boundaries of convex sets. It involves the Gagliardo semi-norm of the function, as well as its $L^p$ norm weighted by the fractional mean curvature of the hypersurface. As an application, we establish a new upper bound for the maximal time of existence in the smooth fractional mean curvature flow of a convex set. The bound depends on the perimeter of the initial set instead of on its diameter.

math.AP↗
arXiv2019

Long-time asymptotics for evolutionary crystal dislocation models

We consider a family of evolution equations that generalize the Peierls-Nabarro model for crystal dislocations. They can be seen as semilinear parabolic reaction-diffusion equations in which the diffusion is regulated by a fractional Laplace operator of order $2 s \in (0, 2)$ acting in one space dimension and the reaction is determined by a $1$-periodic multi-well potential. We construct solutions of these equations that represent the typical propagation of $N \ge 2$ equally oriented dislocations of size $1$. For large times, the dislocations occur around points that evolve according to a repulsive dynamical system. When $s \in (1/2, 1)$, these solutions are shown to be asymptotically stable with respect to odd perturbations.

math.AP↗
arXiv2018

Bernstein-Moser-type results for nonlocal minimal graphs

We prove a flatness result for entire nonlocal minimal graphs having some partial derivatives bounded from either above or below. This result generalizes fractional versions of classical theorems due to Bernstein and Moser. Our arguments rely on a general splitting result for blow-downs of nonlocal minimal graphs. Employing similar ideas, we establish that entire nonlocal minimal graphs bounded on one side by a cone are affine. Moreover, we show that entire graphs having constant nonlocal mean curvature are minimal, thus extending a celebrated result of Chern on classical CMC graphs.

math.AP↗
arXiv2018

On the growth of nonlocal catenoids

As well known, classical catenoids in ${\mathbf{R}}^3$ possess logarithmic growth at infinity. In this note we prove that the case of nonlocal minimal surfaces is significantly different, and indeed all nonlocal catenoids must grow at least linearly. More generally, we prove that stationary sets for the nonlocal perimeter functional which grow sublinearly at infinity are necessarily half-spaces.

math.AP↗
arXiv2018

Regularity results and Harnack inequalities for minimizers and solutions of nonlocal problems: a unified approach via fractional De Giorgi classes

We study energy functionals obtained by adding a possibly discontinuous potential to an interaction term modeled upon a Gagliardo-type fractional seminorm. We prove that minimizers of such non-differentiable functionals are locally bounded, Hölder continuous, and that they satisfy a suitable Harnack inequality. Hence, we provide an extension of celebrated results of M. Giaquinta and E. Giusti to the nonlocal setting. To do this, we introduce a particular class of fractional Sobolev functions, reminiscent of that considered by E. De Giorgi in his seminal paper of 1957. The flexibility of these classes allows us to also establish regularity of solutions to rather general nonlinear integral equations.

math.AP↗