arXiv subjects

David Pardo

Publications and source records attributed to David Pardo.

At least 19 recordsLinked to original sources

arXiv2026

Parametric Neural r-Adaptivity for Isogeometric Analysis via Residual Minimization

We propose an r-adaptive neural algorithm for Isogeometric Analysis (IGA) based on residual minimization. The boundary-value problem is solved using a standard conforming Galerkin formulation, while a neural network relocates the interior knots. A strong-form residual in the sense of physics-informed neural networks (PINNs) controls a norm stronger than the energy (H^1) error. We therefore weight it by classical a posteriori theory: element residuals scaled by the local mesh size, interface flux jumps, and Neumann boundary residuals yield a computable estimator of the energy error, which we minimize with respect to the knots. For coercive problems on admissible mesh families, this estimator is reliable and locally efficient up to oscillation terms; beyond that regime, the same loss remains well-defined and extends differentiable r-adaptivity to indefinite and advection-dominated problems. In the parametric setting, the network maps each parameter to a knot-density function in a single evaluation; since it outputs a density rather than a fixed-dimensional vector of knot locations, one trained network produces an admissible mesh at any refinement level. Mesh gradients are obtained by reverse-mode automatic differentiation through the discrete solution equation. Numerical experiments in one and two dimensions illustrate that the method concentrates degrees of freedom near singularities, material interfaces, and boundary layers, improving accuracy for a fixed number of degrees of freedom.

math.NA↗

arXiv2026

Uncertainty-aware damage identification in short-span bridges via physics-informed variational autoencoder

Vibration-based damage identification in civil infrastructure is a challenging, ill-posed inverse problem due to measurement noise, sparse sensor arrays, and environmental variability. While deep learning is powerful for system identification, deterministic approaches lack reliable uncertainty quantification and can yield physically inconsistent results. This work proposes a robust probabilistic Scientific Machine Learning (SciML) framework: a physics-informed Gaussian copula variational autoencoder (PI-GCVAE) for structural health monitoring (SHM). First, we eliminate the need for data-driven surrogates by embedding a differentiable numerical eigenvalue solver directly into the VAE architecture. This ensures that latent space samples satisfy the governing equations of structural dynamics, reducing the trainable parameter space and improving generalization. Second, we replace the conventional independence assumption of latent variables with a Gaussian copula. This model captures complex, physics-dependent spatial cross-correlations between adjacent structural elements, defining feasible solutions while accounting for inherent system variability and measurement errors. Third, compared with alternatives such as Gaussian mixtures, our copula-based VAE provides an efficient distributional model for high-dimensional, strongly correlated latent spaces. We validate the approach using a synthetic dataset of a simply supported bridge subjected to various damage scenarios and corrupted with stochastic Gaussian noise. Synthetic data enables exhaustive validation against ground-truth stiffness values unavailable in practice. Results demonstrate that the PI-GCVAE accurately recovers the true posterior distribution, achieving 77.2% coverage. The proposed framework provides a reliable, scalable tool for early-stage damage diagnosis in operating bridges.

cs.LG↗

arXiv2026

A Green-Integral-Constrained Neural Solver with Stochastic Physics-Informed Regularization

Standard physics-informed neural networks (PINNs) struggle to simulate highly oscillatory Helmholtz solutions in heterogeneous media because pointwise minimization of second-order PDE residuals is computationally expensive, biased toward smooth solutions, and requires artificial absorbing boundary layers to restrict the solution. To overcome these challenges, we introduce a Green-Integral (GI) neural solver for the acoustic Helmholtz equation. It departs from the PDE-residual-based formulation by enforcing wave physics through an integral representation that imposes a nonlocal constraint. Oscillatory behavior and outgoing radiation are encoded directly through the integral kernel, eliminating second-order spatial derivatives and enforcing physical solutions without additional boundary layers. Theoretically, optimizing this GI loss via a neural network acts as a spectrally tuned preconditioned iteration, enabling convergence in heterogeneous media where the classical Born series diverges. By exploiting FFT-based convolution to accelerate the GI loss evaluation, our approach substantially reduces GPU memory usage and training time. However, this efficiency relies on a fixed regular grid, which can limit local resolution. To improve local accuracy in strong scattering regions, we also propose a hybrid GI+PDE loss, enforcing a lightweight Helmholtz residual at a small number of nonuniformly sampled collocation points. We evaluate our method on seismic benchmark models characterized by structural contrasts and subwavelength heterogeneity at frequencies up to 20Hz. GI-based training consistently outperforms PDE-based PINNs, reducing computational cost by over a factor of ten. In models with localized scattering, the hybrid loss yields the most accurate reconstructions, providing a stable, efficient, and physically grounded alternative.

cs.LG↗

arXiv2026

Robust Deep FOSLS for Transmission Problems

This work presents a robust, energy-based deep learning framework for solving transmission problems in heterogeneous media, including cases with discontinuous material scenarios. We introduce a weighted First-Order System Least-Squares (FOSLS) formulation involving an energy-norm Poincar\'e constant and prove its equivalence to a natural energy norm of the underlying equations, with constants independent of material parameters. As a result, the optimization landscape remains aligned with a meaningful error approximation even under high material contrast, where standard neural network losses often deteriorate. We further prove that the FOSLS formulation, together with its integral-loss representation, exhibits a passive variance reduction property, whereby the gradient variance progressively decreases as the loss diminishes, in contrast to methods such as VPINNs and Deep Ritz. From a numerical standpoint, we adopt a reduced-order perspective by constructing a low-dimensional space described by a neural network. The optimal coefficients are computed via a least-squares solver, and the space is subsequently improved through gradient-based updates. By selecting the activation function ReQU, the method mitigates the spurious overshoots typically observed in smooth networks when approximating discontinuities. Numerical experiments in 1D and 2D interface settings corroborate these findings.

math.NA↗

arXiv2026

RUNNs: Ritz-Uzawa Neural Networks for Solving Variational Problems

Solving Partial Differential Equations (PDEs) using neural networks presents different challenges, including integration errors and spectral bias, often leading to poor approximations. In addition, standard neural network-based methods, such as Physics-Informed Neural Networks (PINNs), often lack stability when dealing with PDEs characterized by low-regularity solutions. To address these limitations, we introduce the Ritz--Uzawa Neural Networks (RUNNs) framework, an iterative methodology to solve strong, weak, and ultra-weak variational formulations. Rewriting the PDE as a sequence of Ritz-type minimization problems within a Uzawa loop provides an iterative framework that, in specific cases, reduces both bias and variance during training. We demonstrate that the strong formulation offers a passive variance reduction mechanism, whereas variance remains persistent in weak and ultra-weak regimes. Furthermore, we address the spectral bias of standard architectures through a data-driven frequency tuning strategy. By initializing a Sinusoidal Fourier Feature Mapping based on the Normalized Cumulative Power Spectral Density (NCPSD) of previous residuals or their proxies, the network dynamically adapts its bandwidth to capture high-frequency components and severe singularities. Numerical experiments demonstrate the robustness of RUNNs, accurately resolving highly oscillatory solutions and successfully recovering a discontinuous $L^2$ solution from a distributional $H^{-2}$ source -- a scenario where standard energy-based methods fail.

math.NA↗

arXiv2026

A Least-Squares-Based Regularity-Conforming Neural Networks (LS-ReCoNNs) for Solving Parametric Transmission Problems

This article focuses on solving parametric transmission problems in one and two spatial dimensions. These problems belong to a class of partial differential equations that arise in the modeling of physical systems with heterogeneous materials. They often exhibit discontinuities across interfaces and singularities at points where interfaces intersect. To address these problems, we propose a new deep learning approach named {\it{Least-Squares-Based Regularity-Conforming Neural Network (LS-ReCoNN)}}. This approach proposes a loss function that is shown to be a consistent upper bound for the energy-norm error. The method represents the solution as the sum of a principal component and a singular component. The principal component is decomposed into smooth and gradient-jump parts, which capture both the regular solution behavior and reduced regularity across interfaces in one- and two-dimensional problems. The singular component is introduced to model junction singularities and it is approximated using basis functions computed from a one-dimensional finite element eigenvalue problem. For the principal component, a separated representation is employed, consisting of parameter-dependent coefficients and space-dependent functions. A deep neural network approximates the space-dependent functions, while the parameter-dependent coefficients are determined by a least-squares solver, where the optimal coefficients for each parameter instance are obtained online by solving a low-dimensional least-squares problem. Numerical experiments in one and two dimensions demonstrate that LS-ReCoNN effectively captures singularities while maintaining solution accuracy across a wide range of parameter values.

math.NA↗

arXiv2025

A Matter of Height: The Impact of a Robotic Object on Human Compliance

Robots come in various forms and have different characteristics that may shape the interaction with them. In human-human interactions, height is a characteristic that shapes human dynamics, with taller people typically perceived as more persuasive. In this work, we aspired to evaluate if the same impact replicates in a human-robot interaction and specifically with a highly non-humanoid robotic object. The robot was designed with modules that could be easily added or removed, allowing us to change its height without altering other design features. To test the impact of the robot's height, we evaluated participants' compliance with its request to volunteer to perform a tedious task. In the experiment, participants performed a cognitive task on a computer, which was framed as the main experiment. When done, they were informed that the experiment was completed. While waiting to receive their credits, the robotic object, designed as a mobile robotic service table, entered the room, carrying a tablet that invited participants to complete a 300-question questionnaire voluntarily. We compared participants' compliance in two conditions: A Short robot composed of two modules and 95cm in height and a Tall robot consisting of three modules and 132cm in height. Our findings revealed higher compliance with the Short robot's request, demonstrating an opposite pattern to human dynamics. We conclude that while height has a substantial social impact on human-robot interactions, it follows a unique pattern of influence. Our findings suggest that designers cannot simply adopt and implement elements from human social dynamics to robots without testing them first.

cs.RO↗

arXiv2025

Bayesian Physics Informed Neural Networks for Reliable Transformer Prognostics

Scientific Machine Learning (SciML) integrates physics and data into the learning process, offering improved generalization compared with purely data-driven models. Despite its potential, applications of SciML in prognostics remain limited, partly due to the complexity of incorporating partial differential equations (PDEs) for ageing physics and the scarcity of robust uncertainty quantification methods. This work introduces a Bayesian Physics-Informed Neural Network (B-PINN) framework for probabilistic prognostics estimation. By embedding Bayesian Neural Networks into the PINN architecture, the proposed approach produces principled, uncertainty-aware predictions. The method is applied to a transformer ageing case study, where insulation degradation is primarily driven by thermal stress. The heat diffusion PDE is used as the physical residual, and different prior distributions are investigated to examine their impact on predictive posterior distributions and their ability to encode a priori physical knowledge. The framework is validated against a finite element model developed and tested with real measurements from a solar power plant. Results, benchmarked against a dropout-PINN baseline, show that the proposed B-PINN delivers more reliable prognostic predictions by accurately quantifying predictive uncertainty. This capability is crucial for supporting robust and informed maintenance decision-making in critical power assets.

cs.LG↗

arXiv2025

Efficient Numerical Integration for Finite Element Trunk Spaces in 2D and 3D using Machine Learning: A new Optimisation Paradigm to Construct Application-Specific Quadrature Rules

Finite element methods usually construct basis functions and quadrature rules for multidimensional domains via tensor products of one-dimensional counterparts. While straightforward, this approach results in integration spaces larger than necessary, especially as the polynomial degree $p$ or the spatial dimension increases, leading to considerable computational overhead. This work starts from the hypothesis that reducing the dimensionality of the polynomial space can lead to quadrature rules with fewer points and lower computational cost, while preserving the exactness of numerical integration. We use trunk spaces that exclude high-degree monomials that do not improve the approximation quality of the discrete space. These reduced spaces retain sufficient expressive power and allow us to construct smaller (more economical) integration domains. Given a maximum degree $p$, we define trial and test spaces $U$ and $V$ as 2D or 3D trunk spaces and form the integration space $\mathcal{S} = U \otimes V$. We then construct exact quadrature rules by solving a non-convex optimisation problem over the number of points $q$, their coordinates, and weights. We use a shallow neural network with linear activations to parametrise the rule, and a random restart strategy to mitigate convergence to poor local minima. When necessary, we dynamically increase $q$ to achieve exact integration. Our construction reaches machine-precision accuracy (errors below 1e-22) using significantly fewer points than standard tensor-product Gaussian quadrature: up to 30\% reduction in 2D for $p \leq 10$, and 50\% in 3D for $p \leq 6$. These results show that combining the mathematical understanding of polynomial structure with numerical optimisation can lead to a practical and extensible methodology for improving the adaptiveness, efficiency, and scalability of quadrature rules for high-order finite element simulations.

math.NA↗

arXiv2025

An $r$-adaptive finite element method using neural networks for parametric self-adjoint elliptic problem

This work proposes an $r$-adaptive finite element method (FEM) using neural networks (NNs). The method employs the Ritz energy functional as the loss function, currently limiting its applicability to symmetric and coercive problems, such as those arising from self-adjoint elliptic problems. The objective of the NN optimization is to determine the mesh node locations. For simplicity in two-dimensional problems, these locations are assumed to form a tensor product structure. The method is designed to solve parametric partial differential equations (PDEs). For each PDE parameter instance, the optimal $r$-adapted mesh generated by the NN is then solved with a standard FEM. The construction of FEM matrices and load vectors is implemented such that their derivatives with respect to mesh node locations, required for NN training, can be efficiently computed using automatic differentiation. However, the linear equation solver does not need to be differentiable, enabling the use of efficient, readily available `out-of-the-box' solvers. Consequently, the proposed approach retains the robustness and reliability guarantees of the FEM for each parameter instance, while the NN optimization adaptively adjusts the mesh node locations. The method's performance is demonstrated on parametric Poisson problems using one- and two-dimensional tensor product meshes.

math.NA↗

arXiv2025

Least-Squares-Embedded Optimization for Accelerated Convergence of PINNs in Acoustic Wavefield Simulations

Physics-Informed Neural Networks (PINNs) have shown promise in solving partial differential equations (PDEs), including the frequency-domain Helmholtz equation. However, standard training of PINNs using gradient descent (GD) suffers from slow convergence and instability, particularly for high-frequency wavefields. For scattered acoustic wavefield simulation based on Helmholtz equation, we derive a hybrid optimization framework that accelerates training convergence by embedding a least-squares (LS) solver directly into the GD loss function. This formulation enables optimal updates for the linear output layer. Our method is applicable with or without perfectly matched layers (PML), and we provide practical tensor-based implementations for both scenarios. Numerical experiments on benchmark velocity models demonstrate that our approach achieves faster convergence, higher accuracy, and improved stability compared to conventional PINN training. In particular, our results show that the LS-enhanced method converges rapidly even in cases where standard GD-based training fails. The LS solver operates on a small normal matrix, ensuring minimal computational overhead and making the method scalable for large-scale wavefield simulations.

cs.LG↗

arXiv2025

Stochastic Quadrature Rules for Solving PDEs using Neural Networks

We examine the challenges associated with numerical integration when applying Neural Networks to solve Partial Differential Equations (PDEs). We specifically investigate the Deep Ritz Method (DRM), chosen for its practical applicability and known sensitivity to integration inaccuracies. Our research demonstrates that both standard deterministic integration techniques and biased stochastic quadrature methods can lead to incorrect solutions. In contrast, employing high-order, unbiased stochastic quadrature rules defined on integration meshes in low dimensions is shown to significantly enhance convergence rates at a comparable computational expense with respect to low-order methods like Monte Carlo. Additionally, we introduce novel stochastic quadrature approaches designed for triangular and tetrahedral mesh elements, offering increased adaptability for handling complex geometric domains. We highlight that the variance inherent in the stochastic gradient acts as a bottleneck for convergence. Furthermore, we observe that for gradient-based optimisation, the crucial factor is the accurate integration of the gradient, rather than just minimizing the quadrature error of the loss function itself.

math.NA↗

arXiv2025

Gabor-Enhanced Physics-Informed Neural Networks for Fast Simulations of Acoustic Wavefields

Physics-Informed Neural Networks (PINNs) have gained increasing attention for solving partial differential equations, including the Helmholtz equation, due to their flexibility and mesh-free formulation. However, their low-frequency bias limits their accuracy and convergence speed for high-frequency wavefield simulations. To alleviate these problems, we propose a simplified PINN framework that incorporates Gabor functions, designed to capture the oscillatory and localized nature of wavefields more effectively. Unlike previous attempts that rely on auxiliary networks to learn Gabor parameters, we redefine the network's task to map input coordinates to a custom Gabor coordinate system, simplifying the training process without increasing the number of trainable parameters compared to a simple PINN. We validate the proposed method across multiple velocity models, including the complex Marmousi and Overthrust models, and demonstrate its superior accuracy, faster convergence, and better robustness features compared to both traditional PINNs and earlier Gabor-based PINNs. Additionally, we propose an efficient integration of a Perfectly Matched Layer (PML) to enhance wavefield behavior near the boundaries. These results suggest that our approach offers an efficient and accurate alternative for scattered wavefield modeling and lays the groundwork for future improvements in PINN-based seismic applications.

physics.geo-ph↗

arXiv2024

A Least-Squares-Based Neural Network (LS-Net) for Solving Linear Parametric PDEs

Developing efficient methods for solving parametric partial differential equations is crucial for addressing inverse problems. This work introduces a Least-Squares-based Neural Network (LS-Net) method for solving linear parametric PDEs. It utilizes a separated representation form for the parametric PDE solution via a deep neural network and a least-squares solver. In this approach, the output of the deep neural network consists of a vector-valued function, interpreted as basis functions for the parametric solution space, and the least-squares solver determines the optimal solution within the constructed solution space for each given parameter. The LS-Net method requires a quadratic loss function for the least-squares solver to find optimal solutions given the set of basis functions. In this study, we consider loss functions derived from the Deep Fourier Residual and Physics-Informed Neural Networks approaches. We also provide theoretical results similar to the Universal Approximation Theorem, stating that there exists a sufficiently large neural network that can theoretically approximate solutions of parametric PDEs with the desired accuracy. We illustrate the LS-net method by solving one- and two-dimensional problems. Numerical results clearly demonstrate the method's ability to approximate parametric solutions.

math.NA↗

arXiv2024

Optimizing Variational Physics-Informed Neural Networks Using Least Squares

Variational Physics-Informed Neural Networks often suffer from poor convergence when using stochastic gradient-descent-based optimizers. By introducing a Least Squares solver for the weights of the last layer of the neural network, we improve the convergence of the loss during training in most practical scenarios. This work analyzes the computational cost of the resulting hybrid Least-Squares/Gradient-Descent optimizer and explains how to implement it efficiently. In particular, we show that a traditional implementation based on backward-mode automatic differentiation leads to a prohibitively expensive algorithm. To remedy this, we propose using either forward-mode automatic differentiation or an ultraweak-type scheme that avoids the differentiation of trial functions in the discrete weak formulation. The proposed alternatives are up to one hundred times faster than the traditional one, recovering a computational cost-per-iteration similar to that of a conventional gradient-descent-based optimizer alone. To support our analysis, we derive computational estimates and conduct numerical experiments in one- and two-dimensional problems.

math.NA↗

arXiv2024

Regularity-Conforming Neural Networks (ReCoNNs) for solving Partial Differential Equations

Whilst the Universal Approximation Theorem guarantees the existence of approximations to Sobolev functions -- the natural function spaces for PDEs -- by Neural Networks (NNs) of sufficient size, low-regularity solutions may lead to poor approximations in practice. For example, classical fully-connected feed-forward NNs fail to approximate continuous functions whose gradient is discontinuous when employing strong formulations like in Physics Informed Neural Networks (PINNs). In this article, we propose the use of regularity-conforming neural networks, where a priori information on the regularity of solutions to PDEs can be employed to construct proper architectures. We illustrate the potential of such architectures via a two-dimensional (2D) transmission problem, where the solution may admit discontinuities in the gradient across interfaces, as well as power-like singularities at certain points. In particular, we formulate the weak transmission problem in a PINNs-like strong formulation with interface and continuity conditions. Such architectures are partially explainable; discontinuities are explicitly described, allowing the introduction of novel terms into the loss function. We demonstrate via several model problems in one and two dimensions the advantages of using regularity-conforming architectures in contrast to classical architectures. The ideas presented in this article easily extend to problems in higher dimensions.

math.NA↗

arXiv2024

Reducing Spatial Discretization Error on Coarse CFD Simulations Using an OpenFOAM-Embedded Deep Learning Framework

We propose a method for reducing the spatial discretization error of coarse computational fluid dynamics (CFD) problems by enhancing the quality of low-resolution simulations using deep learning. We feed the model with fine-grid data after projecting it to the coarse-grid discretization. We substitute the default differencing scheme for the convection term by a feed-forward neural network that interpolates velocities from cell centers to face values to produce velocities that approximate the down-sampled fine-grid data well. The deep learning framework incorporates the open-source CFD code OpenFOAM, resulting in an end-to-end differentiable model. We automatically differentiate the CFD physics using a discrete adjoint code version. We present a fast communication method between TensorFlow (Python) and OpenFOAM (c++) that accelerates the training process. We applied the model to the flow past a square cylinder problem, reducing the error from 120% to 25% in the velocity for simulations inside the training distribution compared to the traditional solver using an x8 coarser mesh. For simulations outside the training distribution, the error reduction in the velocities was about 50%. The training is affordable in terms of time and data samples since the architecture exploits the local features of the physics.

cs.LG↗

arXiv2024

Residual-based Attention Physics-informed Neural Networks for Spatio-Temporal Ageing Assessment of Transformers Operated in Renewable Power Plants

Transformers are crucial for reliable and efficient power system operations, particularly in supporting the integration of renewable energy. Effective monitoring of transformer health is critical to maintain grid stability and performance. Thermal insulation ageing is a key transformer failure mode, which is generally tracked by monitoring the hotspot temperature (HST). However, HST measurement is complex, costly, and often estimated from indirect measurements. Existing HST models focus on space-agnostic thermal models, providing worst-case HST estimates. This article introduces a spatio-temporal model for transformer winding temperature and ageing estimation, which leverages physics-based partial differential equations (PDEs) with data-driven Neural Networks (NN) in a Physics Informed Neural Networks (PINNs) configuration to improve prediction accuracy and acquire spatio-temporal resolution. The computational accuracy of the PINN model is improved through the implementation of the Residual-Based Attention (PINN-RBA) scheme that accelerates the PINN model convergence. The PINN-RBA model is benchmarked against self-adaptive attention schemes and classical vanilla PINN configurations. For the first time, PINN based oil temperature predictions are used to estimate spatio-temporal transformer winding temperature values, validated through PDE numerical solution and fiber optic sensor measurements. Furthermore, the spatio-temporal transformer ageing model is inferred, which supports transformer health management decision-making. Results are validated with a distribution transformer operating on a floating photovoltaic power plant.

cs.LG↗