arXiv subjects

Piermario Vitullo

Publications and source records attributed to Piermario Vitullo.

5 recordsLinked to original sources

arXiv2026

Multiscale Mixed-Dimensional Simulation via Domain Decomposition and Non-Intrusive Neural Model Order Reduction

Many computational models arising in science and engineering exhibit a multiscale structure that makes the assembly or direct solution of the global problem computationally prohibitive. Domain Decomposition (DD) methods overcome this limitation by replacing the global problem with a sequence of coupled local problems, whose iterative solution reconstructs the global response. This work introduces a method in the family of Domain Decomposition Reduced Order Models (DD-ROMs), based on the observation that DD naturally localizes not only the solution operator but also its geometric and parametric dependence. The central idea is that DD transforms a globally intractable solution map into a family of locally representable operators learnable from affordable local data after identification with a common reference configuration, a concept that we formalize through the notion of local representability. Non-intrusive neural surrogates are then trained to approximate the fine-scale local operations and embedded into the iterative solver. The training algorithm is based on a cascaded strategy designed to match the distributions encountered by the deployed surrogate iteration. We interpret the resulting DD method as a perturbed fixed-point iteration and establish that the global error remains bounded by the surrogate approximation error. The framework is instantiated for mixed-dimensional elliptic problems coupling three-dimensional bulk domains with embedded one-dimensional inclusions, using a two-level non-overlapping Robin-Robin method. Numerical experiments show that the resulting DD-ROM is stable, achieves accurate approximation on unseen microscale geometries and features good scalability properties with respect to the number of subdomains, scaling to large size global problems while avoiding fine-scale operator assembly and local high-fidelity solvers in the online stage.

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arXiv2025

Physics-Informed Learning of Microvascular Flow Models using Graph Neural Networks

The simulation of microcirculatory blood flow in realistic vascular architectures poses significant challenges due to the multiscale nature of the problem and the topological complexity of capillary networks. In this work, we propose a novel deep learning-based reduced-order modeling strategy, leveraging Graph Neural Networks (GNNs) trained on synthetic microvascular graphs to approximate hemodynamic quantities on anatomically realistic domains. Our method combines algorithms for synthetic vascular generation with a physics-informed training procedure that integrates graph topological information and local flow dynamics. To ensure the physical reliability of the learned surrogates, we incorporate a physics-informed loss functional derived from the governing equations, allowing enforcement of mass conservation and rheological constraints. The resulting GNN architecture demonstrates robust generalization capabilities across diverse network configurations. The GNN formulation is validated on benchmark problems with linear and nonlinear rheology, showing accurate pressure and velocity field reconstruction with substantial computational gains over full-order solvers. The methodology showcases significant generalization capabilities with respect to vascular complexity, as highlighted by tests on data from the mouse cerebral cortex. This work establishes a new class of graph-based surrogate models for microvascular flow, grounded in physical laws and equipped with inductive biases that mirror mass conservation and rheological models, opening new directions for real-time inference in vascular modeling and biomedical applications.

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arXiv2025

Mathematical modeling and sensitivity analysis of hypoxia-activated drugs

Hypoxia-activated prodrugs offer a promising strategy for targeting oxygen-deficient regions in solid tumors, which are often resistant to conventional therapies. However, modeling their behavior is challenging because of the complex interplay between oxygen availability, drug activation, and cell survival. In this work, we develop a multiscale and mixed-dimensional model that couples spatially resolved drug and oxygen transport with pharmacokinetics and pharmacodynamics to simulate the cellular response. The model integrates blood flow, oxygen diffusion and consumption, drug delivery, and metabolism. To reduce computational cost, we mitigate the global nonlinearity through a one-way coupling of the multiscale and mixed/dimensional models with a reduced 0D model for the drug metabolism. The global sensitivity analysis is then used to identify key parameters influencing drug activation and therapeutic outcome. This approach enables efficient simulation and supports the design of optimized hypoxia-targeted therapies.

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arXiv2024

Deep learning enhanced cost-aware multi-fidelity uncertainty quantification of a computational model for radiotherapy

Forward uncertainty quantification (UQ) for partial differential equations is a many-query task that requires a significant number of model evaluations. The objective of this work is to mitigate the computational cost of UQ for a 3D-1D multiscale computational model of microcirculation. To this purpose, we present a deep learning enhanced multi-fidelity Monte Carlo (DL-MFMC) method that integrates the information of a multiscale full-order model (FOM) with that coming from a deep learning enhanced non-intrusive projection-based reduced order model (ROM). The latter is constructed by leveraging on proper orthogonal decomposition (POD) and mesh-informed neural networks (previously developed by the authors and co-workers), integrating diverse architectures that approximate POD coefficients while introducing fine-scale corrections for the microstructures. The DL-MFMC approach provides a robust estimator of specific quantities of interest and their associated uncertainties, with optimal management of computational resources. In particular, the computational budget is efficiently divided between training and sampling, ensuring a reliable estimation process suitably exploiting the ROM speed-up. Here, we apply the DL-MFMC technique to accelerate the estimation of biophysical quantities regarding oxygen transfer and radiotherapy outcomes. Compared to classical Monte Carlo methods, the proposed approach shows remarkable speed-ups and a substantial reduction of the overall computational cost.

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arXiv2023

Nonlinear model order reduction for problems with microstructure using mesh informed neural networks

Many applications in computational physics involve approximating problems with microstructure, characterized by multiple spatial scales in their data. However, these numerical solutions are often computationally expensive due to the need to capture fine details at small scales. As a result, simulating such phenomena becomes unaffordable for many-query applications, such as parametrized systems with multiple scale-dependent features. Traditional projection-based reduced order models (ROMs) fail to resolve these issues, even for second-order elliptic PDEs commonly found in engineering applications. To address this, we propose an alternative nonintrusive strategy to build a ROM, that combines classical proper orthogonal decomposition (POD) with a suitable neural network (NN) model to account for the small scales. Specifically, we employ sparse mesh-informed neural networks (MINNs), which handle both spatial dependencies in the solutions and model parameters simultaneously. We evaluate the performance of this strategy on benchmark problems and then apply it to approximate a real-life problem involving the impact of microcirculation in transport phenomena through the tissue microenvironment.

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