arXiv subjects

Nicola Demo

Publications and source records attributed to Nicola Demo.

At least 19 recordsLinked to original sources

arXiv2026

Trainable Spline Representations for Physics-Informed Learning

This work introduces Physics-Informed Splines (PI-Splines), a structured spline-based architecture for physics-informed learning. Instead of representing the solution of a differential equation with a neural network, PI-Splines directly parametrize the unknown field through a tensor-product B-spline expansion with trainable control coefficients. This formulation preserves the residual-based training paradigm of Physics-Informed Neural Networks while providing compact support, explicit smoothness control, analytical derivatives, and a direct geometric interpretation of the trainable parameters. When compatible with the spline representation, boundary conditions can be imposed strongly by fixing suitable boundary control coefficients. The proposed method is evaluated on several benchmark problems of increasing difficulty and compared with standard physics-informed frameworks under matched governing equations, collocation sets, loss terms, and optimization procedures, so as to isolate the effect of the approximation architecture. Numerical experiments show that PI-Splines provide a competitive and stable alternative to neural physics-informed architectures, particularly in settings where structured representations, locality, and parameter efficiency are desirable.

cs.LG↗

arXiv2026

Constraint-driven Optimization and Parametrization of Industrial NURBS Geometries via Neural Deformation Field

This work presents a differentiable framework for the parametrization and shape optimization of industrial CAD geometries represented by multi-patch NURBS surfaces. The method enables the deformation of complex CAD models through a physics-informed geometric parametrization, allowing direct morphing driven by physical constraints without the need to prescribe a predefined deformation strategy. A neural displacement field, implemented as a multi-layer perceptron acting on the NURBS control points, provides a compact parametrization of the admissible design space while preserving patch connectivity. Global geometric quantities relevant to hydrostatic design, including displaced volume, wetted surface area and buoyancy centroid, are formulated as differentiable integral operators evaluated on the parametric domain. These quantities are computed through Gauss-Legendre quadrature combined with analytical B-spline derivatives for surface metric evaluation, allowing gradient propagation to the deformation parameters while limiting the computational overhead of automatic differentiation. The proposed framework operates directly on CAD representations without intermediate mesh generation. Numerical experiments on a modified KVLCC2 hull demonstrate the capability of the method to satisfy competing hydrostatic constraints while producing smooth CAD-compatible geometries and showing stable convergence across multiple random initializations.

math.NA↗

arXiv2026

Machine Learning-based quadratic closures for non-intrusive Reduced Order Models

In the present work, we introduce a data-driven approach to enhance the accuracy of non-intrusive Reduced Order Models (ROMs). In particular, we focus on ROMs built using Proper Orthogonal Decomposition (POD) in an under-resolved and marginally-resolved regime, i.e. when the number of modes employed is not enough to capture the system dynamics. We propose a method to re-introduce the contribution of neglected modes through a quadratic correction term, given by the action of a quadratic operator on the POD coefficients. Differently from the state-of-the-art methodologies, where the operator is learned via least-squares optimisation, we propose to parametrise the operator by a Multi-Input Operators Network (MIONet). This way, we are able to build models with higher generalisation capabilities, where the operator itself is continuous in space -- thus agnostic of the domain discretisation -- and parameter-dependent. We test our model on two standard benchmarks in fluid dynamics and show that the correction term improves the accuracy of standard POD-based ROMs.

math.NA↗

arXiv2025

BARNN: A Bayesian Autoregressive and Recurrent Neural Network

Autoregressive and recurrent networks have achieved remarkable progress across various fields, from weather forecasting to molecular generation and Large Language Models. Despite their strong predictive capabilities, these models lack a rigorous framework for addressing uncertainty, which is key in scientific applications such as PDE solving, molecular generation and Machine Learning Force Fields. To address this shortcoming we present BARNN: a variational Bayesian Autoregressive and Recurrent Neural Network. BARNNs aim to provide a principled way to turn any autoregressive or recurrent model into its Bayesian version. BARNN is based on the variational dropout method, allowing to apply it to large recurrent neural networks as well. We also introduce a temporal version of the "Variational Mixtures of Posteriors" prior (tVAMP-prior) to make Bayesian inference efficient and well-calibrated. Extensive experiments on PDE modelling and molecular generation demonstrate that BARNN not only achieves comparable or superior accuracy compared to existing methods, but also excels in uncertainty quantification and modelling long-range dependencies.

cs.LG↗

arXiv2025

Non-intrusive model reduction of advection-dominated hyperbolic problems using neural network shift augmented manifold transformation

Advection-dominated problems are predominantly noticed in nature, engineering systems, and various industrial processes. Traditional linear compression methods, such as proper orthogonal decomposition (POD) and reduced basis (RB) methods are ill-suited for these problems, due to slow Kolmogorov $n$-width decay. This results in inefficient and inaccurate reduced order models (ROMs). There are few non-linear approaches to accelerate the Kolmogorov $n$-width decay. In this work, we use a neural network shift augmented transformation technique that employs automatic shift detection. This approach leverages a deep-learning framework to derive a parameter-dependent mapping between the original manifold $\mathcal{M}$ and the transformed manifold $\tilde{\mathcal{M}}$. We apply a linear compression method to obtain a low-dimensional linear approximation subspace of the transformed manifold $\tilde{\mathcal{M}}$. Furthermore, we construct non-intrusive reduced order models on the resulting transformed linear approximation subspace and employ automatic shift detection for predictions in the online stage. We propose a complete framework, the neural network shift-augmented proper orthogonal decomposition-based reduced order model (NNsPOD-ROM) algorithm, comprising both offline and online stages for model reduction of advection-dominated problems. We test our proposed methodology on numerous experiments to evaluate its performance on the 1D linear advection equation, a higher order method benchmark case - the 2D isentropic convective vortex, and 2D two-phase flow.

math.NA↗

arXiv2025

Kinetic data-driven approach to turbulence subgrid modeling

Numerical simulations of turbulent flows are well known to pose extreme computational challenges due to the huge number of dynamical degrees of freedom required to correctly describe the complex multi-scale statistical correlations of the velocity. On the other hand, kinetic mesoscale approaches based on the Boltzmann equation, have the potential to describe a broad range of flows, stretching well beyond the special case of gases close to equilibrium, which results in the ordinary Navier-Stokes dynamics. Here we demonstrate that, by properly tuning, a kinetic approach can statistically reproduce the quantitative dynamics of the larger scales in turbulence, thereby providing an alternative, computationally efficient and physically rooted approach towards subgrid scale (SGS) modeling in turbulence. More specifically we show that by leveraging on data from fully resolved Direct Numerical Simulation (DNS) we can learn a collision operator for the discretized Boltzmann equation solver (the lattice Boltzmann method), which effectively implies a turbulence subgrid closure model. The mesoscopic nature of our formulation makes the learning problem fully local in both space and time, leading to reduced computational costs and enhanced generalization capabilities. We show that the model offers superior performance compared to traditional methods, such as the Smagorinsky model, being less dissipative and, therefore, being able to more closely capture the intermittency of higher-order velocity correlations. This foundational work lays the basis for extending the proposed framework to different turbulent flow settings and -- most importantly -- to develop new classes of hybrid data-driven kinetic-based models capable of faithfully capturing the complex macroscopic dynamics of diverse physical systems such as emulsions, non-Newtonian fluid and multiphase systems.

physics.flu-dyn↗

arXiv2025

Generative Adversarial Reduced Order Modelling

In this work, we present GAROM, a new approach for reduced order modelling (ROM) based on generative adversarial networks (GANs). GANs have the potential to learn data distribution and generate more realistic data. While widely applied in many areas of deep learning, little research is done on their application for ROM, i.e. approximating a high-fidelity model with a simpler one. In this work, we combine the GAN and ROM framework, by introducing a data-driven generative adversarial model able to learn solutions to parametric differential equations. The latter is achieved by modelling the discriminator network as an autoencoder, extracting relevant features of the input, and applying a conditioning mechanism to the generator and discriminator networks specifying the differential equation parameters. We show how to apply our methodology for inference, provide experimental evidence of the model generalisation, and perform a convergence study of the method.

cs.LG↗

arXiv2024

Data-driven Discovery of Delay Differential Equations with Discrete Delays

The Sparse Identification of Nonlinear Dynamics (SINDy) framework is a robust method for identifying governing equations, successfully applied to ordinary, partial, and stochastic differential equations. In this work we extend SINDy to identify delay differential equations by using an augmented library that includes delayed samples and Bayesian optimization. To identify a possibly unknown delay we minimize the reconstruction error over a set of candidates. The resulting methodology improves the overall performance by remarkably reducing the number of calls to SINDy with respect to a brute force approach. We also address a multivariate setting to identify multiple unknown delays and (non-multiplicative) parameters. Several numerical tests on delay differential equations with different long-term behavior, number of variables, delays, and parameters support the use of Bayesian optimization highlighting both the efficacy of the proposed methodology and its computational advantages. As a consequence, the class of discoverable models is significantly expanded.

math.NA↗

arXiv2024

PyDMD: A Python package for robust dynamic mode decomposition

The dynamic mode decomposition (DMD) is a simple and powerful data-driven modeling technique that is capable of revealing coherent spatiotemporal patterns from data. The method's linear algebra-based formulation additionally allows for a variety of optimizations and extensions that make the algorithm practical and viable for real-world data analysis. As a result, DMD has grown to become a leading method for dynamical system analysis across multiple scientific disciplines. PyDMD is a Python package that implements DMD and several of its major variants. In this work, we expand the PyDMD package to include a number of cutting-edge DMD methods and tools specifically designed to handle dynamics that are noisy, multiscale, parameterized, prohibitively high-dimensional, or even strongly nonlinear. We provide a complete overview of the features available in PyDMD as of version 1.0, along with a brief overview of the theory behind the DMD algorithm, information for developers, tips regarding practical DMD usage, and introductory coding examples. All code is available at https://github.com/PyDMD/PyDMD .

stat.CO↗

arXiv2024

A Shape Optimization Pipeline for Marine Propellers by means of Reduced Order Modeling Techniques

In this paper, we propose a shape optimization pipeline for propeller blades, applied to naval applications. The geometrical features of a blade are exploited to parametrize it, allowing to obtain deformed blades by perturbating their parameters. The optimization is performed using a genetic algorithm that exploits the computational speed-up of reduced order models to maximize the efficiency of a given propeller. A standard offline-online procedure is exploited to construct the reduced-order model. In an expensive offline phase, the full order model, which reproduces an open water test, is set up in the open-source software OpenFOAM and the same full order setting is used to run the CFD simulations for all the deformed propellers. The collected high-fidelity snapshots and the deformed parameters are used in the online stage to build the non-intrusive reduced-order model. This paper provides a proof of concept of the pipeline proposed, where the optimized propeller improves the efficiency of the original propeller.

math.OC↗

arXiv2024

A Dynamic Mode Decomposition Extension for the Forecasting of Parametric Dynamical Systems

Dynamic mode decomposition (DMD) has recently become a popular tool for the non-intrusive analysis of dynamical systems. Exploiting Proper Orthogonal Decomposition (POD) as a dimensionality reduction technique, DMD is able to approximate a dynamical system as a sum of spatial basis evolving linearly in time, thus enabling a better understanding of the physical phenomena and forecasting of future time instants. In this work we propose an extension of DMD to parameterized dynamical systems, focusing on the future forecasting of the output of interest in a parametric context. Initially all the snapshots -- for different parameters and different time instants -- are projected to a reduced space; then DMD, or one of its variants, is employed to approximate reduced snapshots for future time instants. Exploiting the low dimension of the reduced space the predicted reduced snapshots are then combined using regression techniques, thus enabling the possibility to approximate any untested parametric configuration in future. This paper depicts in detail the algorithmic core of this method; we also present and discuss three test cases for our algorithm: a simple dynamical system with a linear parameter dependency, a heat problem with nonlinear parameter dependency and a fluid dynamics problem with nonlinear parameter dependency.

math.NA↗

arXiv2023

A DeepONet multi-fidelity approach for residual learning in reduced order modeling

In the present work, we introduce a novel approach to enhance the precision of reduced order models by exploiting a multi-fidelity perspective and DeepONets. Reduced models provide a real-time numerical approximation by simplifying the original model. The error introduced by the such operation is usually neglected and sacrificed in order to reach a fast computation. We propose to couple the model reduction to a machine learning residual learning, such that the above-mentioned error can be learned by a neural network and inferred for new predictions. We emphasize that the framework maximizes the exploitation of high-fidelity information, using it for building the reduced order model and for learning the residual. In this work, we explore the integration of proper orthogonal decomposition (POD), and gappy POD for sensors data, with the recent DeepONet architecture. Numerical investigations for a parametric benchmark function and a nonlinear parametric Navier-Stokes problem are presented.

math.NA↗

arXiv2023

An extended physics informed neural network for preliminary analysis of parametric optimal control problems

In this work we propose an extension of physics informed supervised learning strategies to parametric partial differential equations. Indeed, even if the latter are indisputably useful in many applications, they can be computationally expensive most of all in a real-time and many-query setting. Thus, our main goal is to provide a physics informed learning paradigm to simulate parametrized phenomena in a small amount of time. The physics information will be exploited in many ways, in the loss function (standard physics informed neural networks), as an augmented input (extra feature employment) and as a guideline to build an effective structure for the neural network (physics informed architecture). These three aspects, combined together, will lead to a faster training phase and to a more accurate parametric prediction. The methodology has been tested for several equations and also in an optimal control framework.

cs.LG↗

arXiv2023

A Continuous Convolutional Trainable Filter for Modelling Unstructured Data

Convolutional Neural Network (CNN) is one of the most important architectures in deep learning. The fundamental building block of a CNN is a trainable filter, represented as a discrete grid, used to perform convolution on discrete input data. In this work, we propose a continuous version of a trainable convolutional filter able to work also with unstructured data. This new framework allows exploring CNNs beyond discrete domains, enlarging the usage of this important learning technique for many more complex problems. Our experiments show that the continuous filter can achieve a level of accuracy comparable to the state-of-the-art discrete filter, and that it can be used in current deep learning architectures as a building block to solve problems with unstructured domains as well.

cs.LG↗

arXiv2023

A Graph-based Framework for Complex System Simulating and Diagnosis with Automatic Reconfiguration

Fault detection has a long tradition: the necessity to provide the most accurate diagnosis possible for a process plant criticality is somehow intrinsic in its functioning. Continuous monitoring is a possible way for early detection. However, it is somehow fundamental to be able to actually simulate failures. Reproducing the issues remotely allows to quantify in advance their consequences, causing literally no real damage. Within this context, signed directed graphs have played an essential role within the years, managing to model with a relatively simple theory diverse elements of an industrial network, as well as the logic relations between them.\\ In this work we present a quantitative approach, employing directed graphs to the simulation and automatic reconfiguration of a fault in a network. To model the typical operation of industrial plants, we propose several additions with respect to the standard graphs: 1. a quantitative measure to control the overall residual capacity, 2. nodes of different categories - and then different behaviors - and 3. a fault propagation procedure based on the predecessors and the redundancy of the system. The obtained graph is able to mimic the behaviour of the real target plant when one or more faults occur. Additionally, we also implement a generative approach capable to activate a particular category of nodes in order to contain the issue propagation, equipping the network with the capability of reconfigure itself and resulting then in a mathematical tool useful not only for simulating and monitoring, but also to design and optimize complex plants. The final asset of the system is provided in output with its complete diagnostics, and a detailed description of the steps that have been carried out to obtain the final realization.

eess.SY↗

arXiv2022

Towards a machine learning pipeline in reduced order modelling for inverse problems: neural networks for boundary parametrization, dimensionality reduction and solution manifold approximation

In this work, we propose a model order reduction framework to deal with inverse problems in a non-intrusive setting. Inverse problems, especially in a partial differential equation context, require a huge computational load due to the iterative optimization process. To accelerate such a procedure, we apply a numerical pipeline that involves artificial neural networks to parametrize the boundary conditions of the problem in hand, compress the dimensionality of the (full-order) snapshots, and approximate the parametric solution manifold. It derives a general framework capable to provide an ad-hoc parametrization of the inlet boundary and quickly converges to the optimal solution thanks to model order reduction. We present in this contribution the results obtained by applying such methods to two different CFD test cases.

math.NA↗

arXiv2022

A Proper Orthogonal Decomposition approach for parameters reduction of Single Shot Detector networks

As a major breakthrough in artificial intelligence and deep learning, Convolutional Neural Networks have achieved an impressive success in solving many problems in several fields including computer vision and image processing. Real-time performance, robustness of algorithms and fast training processes remain open problems in these contexts. In addition object recognition and detection are challenging tasks for resource-constrained embedded systems, commonly used in the industrial sector. To overcome these issues, we propose a dimensionality reduction framework based on Proper Orthogonal Decomposition, a classical model order reduction technique, in order to gain a reduction in the number of hyperparameters of the net. We have applied such framework to SSD300 architecture using PASCAL VOC dataset, demonstrating a reduction of the network dimension and a remarkable speedup in the fine-tuning of the network in a transfer learning context.

cs.CV↗

arXiv2021

A Dimensionality Reduction Approach for Convolutional Neural Networks

The focus of this paper is the application of classical model order reduction techniques, such as Active Subspaces and Proper Orthogonal Decomposition, to Deep Neural Networks. We propose a generic methodology to reduce the number of layers of a pre-trained network by combining the aforementioned techniques for dimensionality reduction with input-output mappings, such as Polynomial Chaos Expansion and Feedforward Neural Networks. The necessity of compressing the architecture of an existing Convolutional Neural Network is motivated by its application in embedded systems with specific storage constraints. Our experiment shows that the reduced nets obtained can achieve a level of accuracy similar to the original Convolutional Neural Network under examination, while saving in memory allocation.

cs.LG↗