arXiv subjects

Pietro Zanotta

Publications and source records attributed to Pietro Zanotta.

4 recordsLinked to original sources

arXiv2026

CINOC: Cardinality-Invariant Neural Operator Policies for Scalable PDE Control

Controlling partial differential equations (PDEs) with learning-based policies remains fundamentally limited by fixed-dimensional representations: policies trained for a specific sensor, actuator, or agent configuration typically fail when the configuration changes. This limitation is particularly severe in multi-agent PDE control, where policies do not scale across population sizes without retraining. We address this challenge by introducing Cardinality Invariant Neural Operator Control (CINOC), reformulating PDE control as an operator learning problem that maps state fields to continuous control functions and trains them end-to-end through differentiable PDE solvers, yielding policies that naturally adapt to varying sensor and actuator configurations. Remarkably, CINOC policies trained on small swarms exhibit cardinality invariance, allowing for zero-shot transfer to significantly larger populations as well as robustness to partial agent failure. This scalability arises from agents sharing a common policy and coordinating through their physical environment, which produces an emergent self-normalization effect. To explain this phenomenon, we provide a theorem grounded in mean-field theory demonstrating that policy gradients computed from finite-agent systems converge to those of a continuous control limit. Empirically, we validate CINOC on tracking, stabilization, and density transport across linear, nonlinear, chaotic, and turbulent PDEs.

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arXiv2026

SCORE: Statistical Certification of Regions of Attraction via Extreme Value Theory

Certifying the Region of Attraction (ROA) for high-dimensional nonlinear dynamical systems remains a severe computational bottleneck. Traditional deterministic verification methods, such as Sum-of-Squares (SOS) programming and Satisfiability Modulo Theories (SMT), provide hard guarantees but suffer from the curse of dimensionality, typically failing to scale beyond 20 dimensions. To overcome these limitations, we propose SCORE, a statistical certification framework that shifts from seeking deterministic guarantees to bounding the worst-case safety violation with high statistical confidence. By integrating Projected Stochastic Gradient Langevin Dynamics (PSGLD) with Extreme Value Theory (EVT), we frame ROA certification as a constrained extreme-value estimation problem on the sublevel set boundary. We theoretically demonstrate that modeling the optimization process as a stochastic diffusion on a compact manifold places the local maxima of the Lyapunov derivative into the Weibull maximum domain of attraction. Since the Weibull domain features a finite right endpoint, we can compute a rigorous statistical upper bound on the global maximum of the Lyapunov derivative. Numerical experiments validate that our EVT-based approach achieves certification tightness competitive to exact SOS programming on a 2D Van der Pol benchmark. Furthermore, we demonstrate unprecedented scalability by successfully certifying a dense, unstructured 500-dimensional ODE system up to a confidence level of 99.99\%, effectively bypassing the severe combinatorial constraints that limit existing formal verification pipelines.

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arXiv2025

Quantum Orthogonal Separable Physics-Informed Neural Networks

This paper introduces Quantum Orthogonal Separable Physics-Informed Neural Networks (QO-SPINNs), a novel architecture for solving Partial Differential Equations, integrating quantum computing principles to address the computational bottlenecks of classical methods. We leverage a quantum algorithm for accelerating matrix multiplication within each layer, achieving a $\mathcal O(d\log d/ε^2)$ complexity, a significant improvement over the classical $\mathcal O(d^2)$ complexity, where $d$ is the dimension of the matrix, $ε$ the accuracy level. This is accomplished by using a Hamming weight-preserving quantum circuit and a unary basis for data encoding, with a comprehensive theoretical analysis of the overall architecture provided. We demonstrate the practical utility of our model by applying it to solve both forward and inverse PDE problems. Furthermore, we exploit the inherent orthogonality of our quantum circuits (which guarantees a spectral norm of 1) to develop a novel uncertainty quantification method. Our approach adapts the Spectral Normalized Gaussian Process for SPINNs, eliminating the need for the computationally expensive spectral normalization step. By using a Quantum Orthogonal SPINN architecture based on stacking, we provide a robust and efficient framework for uncertainty quantification (UQ) which, to our knowledge, is the first UQ method specifically designed for Separable PINNs. Numerical results based on classical simulation of the quantum circuits, are presented to validate the theoretical claims and demonstrate the efficacy of the proposed method.

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arXiv2025

On the Potential of Quantum Computing in Classical Program Analysis

Classical program analysis techniques, such as abstract interpretation and symbolic execution, are essential for ensuring software correctness, optimizing performance, and enabling compiler optimizations. However, these techniques face computational limitations when analyzing programs with large or exponential state spaces, limiting their effectiveness in ensuring system reliability. Quantum computing, with its parallelism and ability to process superposed states, offers a promising solution to these challenges. In this work, we present QEX, a design that uses quantum computing to analyze classical programs. By synthesizing quantum circuits that encode program states in superposition and trace data dependency between program variables through entanglement, QEX enables the simultaneous exploration of program behaviors, significantly improving scalability and precision. This advancement has broad applications, from debugging and security verification to optimizing compilers for next-generation hardware. As a proof-of-concept, we evaluated QEX on 22 benchmark programs, demonstrating its effectiveness in analyzing program states. To support more language features and make QEX realized sooner in Fault-Tolerant Quantum Computing (FTQC), we propose QEX-H which hybridizes QEX with classical analysis techniques. To our knowledge, this work is the first proposal to use quantum computing for classical program analysis.

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