arXiv subjects

Gonzalo G. de Diego

Publications and source records attributed to Gonzalo G. de Diego.

8 recordsLinked to original sources

arXiv2026

Well-posed by Design: Learning Constitutive Laws from Velocity Data using Convex Neural Network Potentials

Learning constitutive laws of complex fluids from velocity data (i.e. indirect observations) is a PDE-constrained inverse problem in which expressive neural parameterizations risk breaking the well-posedness of the forward physics model. We address this tension by learning, rather than the constitutive law itself, the dissipation potential: a scalar function whose convexity, frame-indifference, and dissipativity propagate into the underlying continuum mechanics and guarantee the structural properties needed for a well-posed and generalizable forward PDE. We introduce ICNNE, an input-convex neural architecture that exactly enforces convexity, frame-indifference, and evenness in the second strain-rate invariant via symmetrization, alleviating the numerical instabilities that occur at small strain rates in standard formulations. By weakly enforcing zero gradients in the origin of the potential via penalization, ICNNE also satisfies the dissipativity property. The loss objective and its gradient are computed using a combination of finite element and neural network methods, coupling the Firedrake and PyTorch libraries. We evaluate on four problems spanning compressible and incompressible regimes: compressible Navier-Stokes, Herschel-Bulkley yield-stress flow, Hibler's viscous-plastic sea-ice model, and discrete-element-method data with no known constitutive law. We show that the learned potentials (i) recover ground-truth physics where it is known, (ii) transfer accurately to geometries unseen during training, and (iii) succeed in regimes where unstructured methods diverge. These results suggest that embedding mathematical structure into the parameterization, rather than into the loss, is a robust path to learning physics from indirect data.

physics.flu-dyn↗

arXiv2026

Learning parameter-dependent shear viscosity from data, with application to sea and land ice

Complex physical systems which exhibit fluid-like behavior are often modeled as non-Newtonian fluids. A crucial element of a non-Newtonian model is the rheology, which relates inner stresses with strain-rates. We propose a framework for inferring rheological models from data that represents the fluid's effective viscosity with a neural network. By writing the rheological law in terms of tensor invariants and tailoring the network's properties, the inferred model satisfies key physical and mathematical properties, such as isotropic frame-indifference and existence of a convex potential of dissipation. Within this framework, we propose two approaches to learning a fluid's rheology: 1) a standard regression that fits the rheological model to stress data and 2) a PDE-constrained optimization method that infers rheological models from velocity data. For the latter approach, we combine finite element and machine learning libraries. We demonstrate the accuracy and robustness of our method on land and sea ice rheologies which also depend on external parameters. For land ice, we infer the temperature-dependent Glen's law and, for sea ice, the concentration-dependent shear component of the viscous-plastic model. For these two models, we explore the effects of large data errors. Finally, we infer an unknown concentration-dependent model that reproduces Lagrangian ice floe simulation data. Our method discovers a rheology that generalizes well outside of the training dataset and exhibits both shear-thickening and thinning behaviors depending on the concentrations.

math.NA↗

arXiv2026

Non-Newtonian viscous fluid models with learned rheology accurately reproduce Lagrangian sea ice simulations

Polar sea ice is crucial to Earth's climate system. Its dynamics also affect coastal communities, wildlife, and global shipping. Sea ice is typically modeled as a continuum fluid using a model proposed almost 50 years ago, which is moderately accurate for packed ice, but loses its predictive accuracy outside of the central ice pack. Discrete element methods (DEMs), which are commonly used for modeling granular media, offer an alternative by resolving the behavior of individual ice floes, including collisions, frictional contact, fracture, and ridging. However, DEMs are generally too costly for large-scale simulations. To address this, we present a framework for inferring rheological behavior from DEM velocity data. We characterize isotropic constitutive laws as scalar functions of the principal invariants of the strain-rate tensor. These functions are parameterized by neural networks trained on DEM data. By combining machine learning and finite element methods, we incorporate the governing partial differential equation (PDE) into the training, requiring to solve a PDE-constrained optimization problem for the network parameters. We focus on unidirectional parallel shear flows, which allow us to infer the effective shear viscosity. We find that, over a wide range of ice concentrations, the velocity fields observed in a complex sea ice DEM can be captured by a nonlinear rheology. Depending on the ice concentration, a shear-thinning or a shear-thickening behavior is observed. Moreover, the effective shear viscosity is found to increase by several orders of magnitude with changes as small as 5% in the sea ice concentration. We show that the learned rheology generalizes to different forcing scenarios, time-dependent problems, and settings in which compressibility is not a dominant factor.

physics.flu-dyn↗

arXiv2024

Modeling sea ice in the marginal ice zone as a dense granular flow with rheology inferred from discrete element model data

The marginal ice zone (MIZ) represents the periphery of the sea ice cover. In this region, the macroscale behavior of the sea ice results from collisions and enduring contact between ice floes. This configuration closely resembles that of dense granular flows, which have been modeled successfully with the $μ(I)$ rheology. Here, we present a continuum model based on the $μ(I)$ rheology which treats sea ice as a compressible fluid, with the local sea ice concentration given by a dilatancy function $Φ(I)$. We infer expressions for $μ(I)$ and $Φ(I)$ by nonlinear regression using data produced with a discrete element method (DEM) which considers polygonal-shaped ice floes. We do this by driving the sea ice with a one-dimensional shearing ocean current. The resulting continuum model is a nonlinear system of equations with the sea ice velocity, local concentration, and pressure as unknowns. The rheology is given by the sum of a plastic and a viscous term. In the context of a periodic patch of ocean, which is effectively a one dimensional problem, and under steady conditions, we prove this system to be well-posed, present a numerical algorithm for solving it, and compare its solutions to those of the DEM. These comparisons demonstrate the continuum model's ability to capture most of the DEM's results accurately. The continuum model is particularly accurate for ocean currents faster than 0.25 m/s; however, for low concentrations and slow ocean currents, the continuum model is less effective in capturing the DEM results. In the latter case, the lack of accuracy of the continuum model is found to be accompanied by the breakdown of a balance between the average shear stress and the integrated ocean drag extracted from the DEM.

physics.flu-dyn↗

arXiv2022

On the finite element approximation of a semicoercive Stokes variational inequality arising in glaciology

Stokes variational inequalities arise in the formulation of glaciological problems involving contact. We consider the problem of a two-dimensional marine ice sheet with a grounding line, although the analysis presented here is extendable to other contact problems in glaciology, such as that of subglacial cavitation. The analysis of this problem and its discretisation is complicated by the nonlinear rheology commonly used for modelling ice, the enforcement of a friction boundary condition given by a power law, and the presence of rigid modes in the velocity space, which render the variational inequality semicoercive. In this work, we consider a mixed formulation of this variational inequality involving a Lagrange multiplier and provide an analysis of its finite element approximation. Error estimates in the presence of rigid modes are obtained by means of a specially-built projection operator onto the subspace of rigid modes and a Korn-type inequality. These proofs rely on the fact that the subspace of rigid modes is at most one-dimensional. Numerical results are reported to validate the error estimates.

math.NA↗

arXiv2022

Numerical approximation of viscous contact problems applied to glacial sliding

Viscous contact problems describe the time evolution of fluid flows in contact with a surface from which they can detach and reattach. These problems are of particular importance in glaciology, where they arise in the study of grounding lines and subglacial cavities. In this work, we propose a novel numerical method for solving viscous contact problems based on a mixed formulation with Lagrange multipliers of a variational inequality involving the Stokes equation. The advection equation for evolving the geometry of the domain occupied by the fluid is then solved via a specially-built upwinding scheme, leading to a robust and accurate algorithm for viscous contact problems. We first verify the method by comparing the numerical results to analytical results obtained by a linearised method. Then, we use this numerical scheme to reconstruct friction laws for glacial sliding with cavitation. Finally, we compute the evolution of cavities from a steady state under oscillating water pressures. The results depend strongly on the location of the initial steady state along the friction law. In particular, we find that if the steady state is located on the downsloping or rate-weakening part of the friction law, the cavity evolves towards the upsloping section, indicating that the downsloping part is unstable.

math.NA↗

arXiv2021

Convergence analysis of the scaled boundary finite element method for the Laplace equation

The scaled boundary finite element method (SBFEM) is a relatively recent boundary element method that allows the approximation of solutions to PDEs without the need of a fundamental solution. A theoretical framework for the convergence analysis of SBFEM is proposed here. This is achieved by defining a space of semi-discrete functions and constructing an interpolation operator onto this space. We prove error estimates for this interpolation operator and show that optimal convergence to the solution can be obtained in SBFEM. These theoretical results are backed by a numerical example.

math.NA↗

arXiv2020

A conservative, physically compatible, dual-field discretization for turbidity currents: with application to the lock-exchange problem

In this work we present a structure preserving discretization for turbidity currents based on a mass-, energy-, enstrophy-, and vorticity-conserving formulation for 2D incompressible flows. This discretization exploits a dual-field formulation for the time evolution of the velocity and vorticity fields together with the transport equation for the particles. Due to its staggered time integration the resulting system of equations is quasi-linear, eliminating the need to solve for a fully nonlinear system of equations. It is shown that this discretization preserves the energy balance equation up to a bounded residual due to the staggering in time of the velocity and vorticity. This leads to a numerical scheme that does not introduce artificial energy dissipation. A comparison with literature results is presented showing that this approach can retrieve the dynamics of the system with a much smaller number of degrees of freedom.

math.NA↗