arXiv subjects

Carlos Uriarte

Publications and source records attributed to Carlos Uriarte.

13 recordsLinked to original sources

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 Poincaré 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↗

arXiv2025

Phase field modelling of the growth and detachment of bubbles in a hydrogen electrolyzer

We develop and implement numerically a phase field model for the growth and detachment of a gas bubble resting on an electrode and being filled with hydrogen produced by water electrolysis. The bubble is surrounded by a viscous liquid, has a prescribed static contact angle and is also subject to gravitational forces. We compute, as a function of the static contact angle, the time at which the bubble detaches from the substrate and what volume it has at that time. We also investigate de dependence of the detachment time on other parameters such as the applied voltage and the hydrogen ion concentration at the fluid bulk.

cond-mat.soft↗

arXiv2025

Consensus as cooling: a granular gas model for continuous opinions on structured networks

A continuous-opinion model accounting for the social compromise propensity is theoretically and numerically analysed. An agent's opinion is represented by a real number that can be changed through social interactions with her neighbours. The proposed dynamics depends on two fundamental parameters, $α\in[-1,1]$ and $β\ge 0$. If an interaction takes place between two agents, their relative opinions decreases an amount given by $α$. The probability of two neighbours to interact is proportional to the $β$-power of their relative opinions. We unveil the behaviour of the system for all physical relevant values of the parameters and several representative interaction networks. When $α\in(-1,1)$ and $β\ge 0$, the system always reaches consensus, with all agents having the mean initial opinion, provided the interaction network is connected. The approach to consensus can be characterized by means of the mean opinion and the temperature (or opinion dispersion) of each agent. Three scenarios have been identified. When the agents are well mixed, as with all-to-all interactions, a pre-consensus regime is seen, with all agents having zero mean opinion and the same temperature, following the Haff's law of granular gases. A similar regime is observed with Erdös-Rényi and Barabási-Albert networks: mean opinions are zero but agents with different degrees have different temperatures, though still following the Haff's law. Finally, the case of a square 2D lattice has been carefully analyzed, by starting from the derivation of closed set of hydrodynamic-like equations using the Chapman-Enskog method ...

physics.soc-ph↗

arXiv2025

Phase field modelling of the detachment of bubbles from a solid substrate

We develop and implement numerically a phase field model for the evolution and detachment of a gas bubble resting on a solid substrate and surrounded by a viscous liquid. The bubble has a static contact angle $θ$ and will be subject to gravitational forces. We compute, as a function of the static contact angle, the cricital Bond number over which bubbles detach from the substrate. Then, we perform similar studies for bubble resting on inclined substrates and bubbles under the action of an external flow. We provide approximate formulas for the critical Bond number under all these circumstances. Our method is also able to resolve the pinchoff of the bubble and the possible appearence of satellites.

physics.flu-dyn↗

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

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↗

arXiv2025

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

Solving Partial Differential Equations Using Artificial Neural Networks

Partial differential equations have a wide range of applications in modeling multiple physical, biological, or social phenomena. Therefore, we need to approximate the solutions of these equations in computationally feasible terms. Nowadays, among the most popular numerical methods for solving partial differential equations in engineering, we encounter the finite difference and finite element methods. An alternative numerical method that has recently gained popularity for numerically solving partial differential equations is the use of artificial neural networks. Artificial neural networks, or neural networks for short, are mathematical structures with universal approximation properties. In addition, thanks to the extraordinary computational development of the last decade, neural networks have become accessible and powerful numerical methods for engineers and researchers. For example, imaging and language processing are applications of neural networks today that show sublime performance inconceivable years ago. This dissertation contributes to the numerical solution of partial differential equations using neural networks with the following two-fold objective: investigate the behavior of neural networks as approximators of solutions of partial differential equations and propose neural-network-based methods for frameworks that are hardly addressable via traditional numerical methods. As novel neural-network-based proposals, we first present a method inspired by the finite element method when applying mesh refinements to solve parametric problems. Secondly, we propose a general residual minimization scheme based on a generalized version of the Ritz method. Finally, we develop a memory-based strategy to overcome a usual numerical integration limitation when using neural networks to solve partial differential equations.

math.NA↗

arXiv2024

Collocation-based Robust Variational Physics-Informed Neural Networks (CRVPINN)

Physics-Informed Neural Networks (PINNs) have been successfully applied to solve Partial Differential Equations (PDEs). Their loss function is founded on a strong residual minimization scheme. Variational Physics-Informed Neural Networks (VPINNs) are their natural extension to weak variational settings. In this context, the recent work of Robust Variational Physics-Informed Neural Networks (RVPINNs) highlights the importance of conveniently translating the norms of the underlying continuum-level spaces to the discrete level. Otherwise, VPINNs might become unrobust, implying that residual minimization might be highly uncorrelated with a desired minimization of the error in the energy norm. However, applying this robustness to VPINNs typically entails dealing with the inverse of a Gram matrix, usually producing slow convergence speeds during training. In this work, we accelerate the implementation of RVPINN, establishing a LU factorization of sparse Gram matrix in a kind of point-collocation scheme with the same spirit as original PINNs. We call out method the Collocation-based Robust Variational Physics Informed Neural Networks (CRVPINN). We test our efficient CRVPINN algorithm on Laplace, advection-diffusion, and Stokes problems in two spatial dimensions.

cs.LG↗

arXiv2023

A Deep Double Ritz Method (D$^2$RM) for solving Partial Differential Equations using Neural Networks

Residual minimization is a widely used technique for solving Partial Differential Equations in variational form. It minimizes the dual norm of the residual, which naturally yields a saddle-point (min-max) problem over the so-called trial and test spaces. In the context of neural networks, we can address this min-max approach by employing one network to seek the trial minimum, while another network seeks the test maximizers. However, the resulting method is numerically unstable as we approach the trial solution. To overcome this, we reformulate the residual minimization as an equivalent minimization of a Ritz functional fed by optimal test functions computed from another Ritz functional minimization. We call the resulting scheme the Deep Double Ritz Method (D$^2$RM), which combines two neural networks for approximating trial functions and optimal test functions along a nested double Ritz minimization strategy. Numerical results on different diffusion and convection problems support the robustness of our method, up to the approximation properties of the networks and the training capacity of the optimizers.

math.NA↗

arXiv2022

Progress on levitating a sphere in cryogenic fluids

We present the working prototype of a levitation system designed for investigation of flows in cryogenic helium fluids. The current device allows the levitation of a superconducting sphere and has several provisions made for allowing precise control over its motion. We report on progress in the detection and control systems of the prototype and demonstrate how uniform circular motion can be implemented.

cond-mat.supr-con↗

arXiv2021

Design of a system for controlling a levitating sphere in superfluid $^3$He at extremely low temperatures

We present a new mechanical probe to study the properties of superfluid $^3$He at microkelvin temperatures down to 100$μ$K. The setup consists of a set of coils for levitating a superconducting sphere and controlling its motion in a wide variety of regimes. In particular, the realisation of motion of a levitating body at a uniform velocity presents both an experimental challenge and a promising direction into the study of the edge states in topological superfluid $^3$He-B. We include the theoretical study of the device stability and simulations to illustrate the capabilities of the control system.

cond-mat.supr-con↗