Deep learning methods for stochastic Galerkin approximations of random domain problems

Fabio Musco; Andrea Barth

Home/Archive/arXiv

arXiv · 2609.32508

Deep learning methods for stochastic Galerkin approximations of random domain problems

Fabio Musco·Andrea Barth

View original source ↗Read source document ↗Export metadata ↓

Abstract

This work considers strong and weak stochastic Galerkin approximations of random domain problems for the elliptic random partial differential equation (PDE). The random domain problems are realized by sufficiently regular random domain mapping, allowing a transformation on deterministic reference domain. A traditional numerical method for solving the resulting high-dimensional coupled stochastic Galerkin systems is replaced by deep learning techniques. We compare a physics-informed neural network approach, based on the strong stochastic Galerkin residual, with a Deep Ritz approach, based on the weak stochastic Galerkin system, reformulated as Ritz energy minimization. The neural networks serve as surrogates for the deterministic spectral coefficients of the respective stochastic Galerkin solution. The stochastic parameters are not used as neural network input parameters and the stochastic domain is not sampled during training. The efficiency of the methods is demonstrated on a randomly stretched interval and a randomly deformed annulus in two spatial dimensions.

Explore related subjects

math.NA cs.NA

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Fabio Musco, Andrea Barth. 2026-09-26. Deep learning methods for stochastic Galerkin approximations of random domain problems. https://arxiv.org/abs/2609.32508

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On Trimming Tensor-structured Measurements and Efficient Low-rank Tensor Recovery

In this paper, we take a step towards developing efficient hard thresholding methods for low-rank tensor recovery from memory-efficient linear measurements with tensorial structure. Theoretical guarantees for many standard iterative low-rank recovery methods, such as iterative hard thresholding (IHT), are based on model assumptions on the measurement operator, like the restricted isometry property (RIP). However, tensor-structured random linear maps -- while memory-efficient and convenient to apply -- lack good restricted isometry properties; that is, they do not preserve the norms of low-rank tensors sufficiently well. To address this, we propose local trimming techniques that provably restore point-wise geometry-preservation properties of tensor-structured maps, making them comparable to those of unstructured sub-Gaussian measurements. Then, we propose two novel versions of tensor IHT algorithms: an adaptive gradient trimming algorithm and a randomized Kaczmarz-based IHT algorithm, that efficiently recover low-rank tensors from linear measurements. We provide initial theoretical guarantees for the proposed methods and present numerical experiments on real and synthetic data, highlighting their efficiency over the original TensorIHT for low HOSVD and CP-rank tensors.

math.NA↗

arXiv2026

Corrected Trapezoidal Rules for Near-Singular Surface Integrals Applied to 3D Ellipsoids in Stokes Flow

Interfacial Stokes flow can be efficiently computed using the Boundary Integral Equation method. In 3D, the fluid velocity at a target point is given by a 2D surface integral over all interfaces, thus reducing the dimension of the problem. A core challenge is that for target points near, but not on, an interface, the surface integral is near-singular and standard quadratures lose accuracy. This paper presents a method to accurately compute the near-singular integrals arising in elliptic boundary value problems in 3D. It is based on a local series approximation of the integrand about a base point on the surface, obtained by orthogonal projection of the target point onto the surface. The elementary functions in the resulting series approximation can be integrated to high accuracy in a neighborhood of the base point using a recursive algorithm. The remaining integral is evaluated numerically using a standard quadrature rule, chosen here to be the 4th order Trapezoidal rule. The method is reduced to the standard quadrature plus a correction, and is uniformly of 4th order. The method is applied to resolve Stokes flow past several ellipsoidal rigid bodies. We compare the error in the velocity near the bodies, and in the time and displacement of particles traveling around the bodies, computed with and without the corrections.

math.NA↗

arXiv2026

A coupled HDG discretization for the interaction between acoustic and elastic waves

We propose and analyze an HDG scheme for the Laplace-domain interaction between a transient acoustic wave and a bounded elastic solid embedded in an unbounded fluid medium. The elastic and acoustic domains are coupled through transmission conditions derived from the continuity of the normal stress and of the normal component of the velocities at the interface. The analysis of the HDG discretization of the coupled weak formulation is the main focus of the article. Two mixed variables (the stress tensor and the velocity of the acoustic wave) are included, while the symmetry of the stress tensor is imposed weakly by considering the antisymmetric part of the strain tensor (the spin or vorticity tensor) as an additional unknown. Convergence of the method is demonstrated and theoretical rates are obtained; numerical results suggesting optimal order of convergence and superconvergence of the traces are presented.

math.NA↗