arXiv subjects

Agustín Somacal

Publications and source records attributed to Agustín Somacal.

5 recordsLinked to original sources

arXiv2026

Reduced Order Modeling and Applications to Inverse State Estimation

Solving parametric partial differential equations (PDEs) repeatedly for many parameter values,as needed in optimal control, uncertainty quantification, and inverse problems, is often prohibitive with classical discretizations. This chapter introduces Reduced Order Modeling (ROM), which builds compressed yet accurate representations of parametric solution sets for fast online evaluation. We formulate the approximation of parametric PDEs as a supervised learning task and review linear approximation, very effective for elliptic and parabolic problems, before turning to nonlinear methods needed when solutions exhibit discontinuities or steep gradients. We then show how to use reduced-order models to efficiently recover the state of a physical system from limited measurements, which is an inverse problem known as state estimation. Throughout, we favor results with theoretical guarantees while also discussing practical methods, and provide a companion Jupyter notebook implementing the main algorithms.

math.NA↗

arXiv2026

Natural gradient descent with momentum

We consider the problem of approximating a function by an element of a nonlinear manifold which admits a differentiable parametrization, typical examples being neural networks with differentiable activation functions or tensor networks. Natural gradient descent (NGD) for the optimization of a loss function can be seen as a preconditioned gradient descent where updates in the parameter space are driven by a functional perspective. In a spirit similar to Newton's method, a NGD step uses, instead of the Hessian, the Gram matrix of the generating system of the tangent space to the approximation manifold at the current iterate, with respect to a suitable metric. This corresponds to a locally optimal update in function space, following a projected gradient onto the tangent space to the manifold. Still, both gradient and natural gradient descent methods get stuck in local minima. Furthermore, when the model class is a nonlinear manifold or the loss function is not ideally conditioned (e.g., the KL-divergence for density estimation, or a norm of the residual of a partial differential equation in physics informed learning), even the natural gradient might yield non-optimal directions at each step. This work introduces a natural version of classical inertial dynamic methods like Heavy-Ball or Nesterov and show how it can improve the learning process when working with nonlinear model classes.

cs.LG↗

arXiv2024

High order recovery of geometric interfaces from cell-average data

We consider the problem of recovering characteristic functions $u:=χ_Ω$ from cell-average data on a coarse grid, and where $Ω$ is a compact set of $\mathbb{R}^d$. This task arises in very different contexts such as image processing, inverse problems, and the accurate treatment of interfaces in finite volume schemes. While linear recovery methods are known to perform poorly, nonlinear strategies based on local reconstructions of the jump interface $Γ:=\partialΩ$ by geometrically simpler interfaces may offer significant improvements. We study two main families of local reconstruction schemes, the first one based on nonlinear least-squares fitting, the second one based on the explicit computation of a polynomial-shaped curve fitting the data, which yields simpler numerical computations and high order geometric fitting. For each of them, we derive a general theoretical framework which allows us to control the recovery error by the error of best approximation up to a fixed multiplicative constant. Numerical tests in 2d illustrate the expected approximation order of these strategies. Several extensions are discussed, in particular the treatment of piecewise smooth interfaces with corners.

math.NA↗

arXiv2024

State estimation of urban air pollution with statistical, physical, and super-learning graph models

We consider the problem of real-time reconstruction of urban air pollution maps. The task is challenging due to the heterogeneous sources of available data, the scarcity of direct measurements, the presence of noise, and the large surfaces that need to be considered. In this work, we introduce different reconstruction methods based on posing the problem on city graphs. Our strategies can be classified as fully data-driven, physics-driven, or hybrid, and we combine them with super-learning models. The performance of the methods is tested in the case of the inner city of Paris, France.

cs.LG↗

arXiv2020

Uncovering differential equations from data with hidden variables

SINDy is a method for learning system of differential equations from data by solving a sparse linear regression optimization problem [Brunton et al., 2016]. In this article, we propose an extension of the SINDy method that learns systems of differential equations in cases where some of the variables are not observed. Our extension is based on regressing a higher order time derivative of a target variable onto a dictionary of functions that includes lower order time derivatives of the target variable. We evaluate our method by measuring the prediction accuracy of the learned dynamical systems on synthetic data and on a real data-set of temperature time series provided by the Réseau de Transport d'Électricité (RTE). Our method provides high quality short-term forecasts and it is orders of magnitude faster than competing methods for learning differential equations with latent variables.

stat.ML↗