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cs.NA

cs.NA: explore 425 source-linked works published from 2021 to 2026, with original documents and citations.

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Includes records with this source-supplied label or an explicit phrase match in their metadata. Matches indicate a mention, not proof that a paper uses a method or tests a material. Source versions are consolidated by DOI.

Sources: arxiv. Collection updated 2026-09-16. Counts describe this index, not the complete source archives.

arXiv2026

Random attractors and almost-sure stability under discretization of a stochastic autoparametric system

For a stochastic autoparametric block-and-pendulum system, the long-time dynamics exhibit two fundamental features: the almost-sure stability of the single mode solution, characterized by its Lyapunov exponent, and the global asymptotic dynamics when this single mode solution loses stability. This naturally raises the question of whether these dynamical features are preserved under discretization, since such preservation is essential for the resulting discrete system to faithfully capture the qualitative behavior of the continuous system. To address this question, we first establish the existence of a random attractor for the continuous system subject to multiplicative stochastic excitation, providing a rigorous characterization of the global asymptotic dynamics. We then propose a numerical discretization that induces a discrete random dynamical system and prove the convergence of its random attractor to the continuous one as the step size tends to zero. In addition, we show that the numerical Lyapunov exponent of the single mode solution has the same sign as its continuous counterpart for sufficiently small step sizes, thus preserving the corresponding almost-sure stability or instability classification. These results demonstrate that the proposed discretization captures both the global asymptotic dynamics and the stability characteristics of the underlying stochastic autoparametric system.

math.DS↗

arXiv2026

A Tensor Neural Network Method for High-Order Homogenization of Locally Periodic Elliptic Problems

We develop a high-order tensor neural network (TNN) method for locally periodic elliptic multiscale problems of the form $-\nabla\cdot(A(x,x/\varepsilon)\nabla u_\varepsilon)=f$. Because the coefficient depends on both the slow variable $x$ and the fast periodic variable $y=x/\varepsilon$, the high-order cell problems and macroscopic corrector equations are more involved than in the classical case $A=A(y)$, and the correctors depend parametrically on $x$. We derive a computable high-order two-scale expansion and prove an $H^1$ convergence estimate for the partial expansion in boundary-layer-free settings, including periodic domains and ideal boundary-matching configurations. The proof uses the recursive compatibility structure of the corrector hierarchy and a zero-mean oscillation estimate in $H^{-1}$. We then construct a TNN framework for the high-dimensional corrector problems. Its tensor-product structure permits deterministic one-dimensional quadrature for the cell problems, homogenized coefficients, macroscopic source terms, and loss functions, avoiding Monte Carlo integration error. The numerical realization assumes that the coefficient entries and assembled data admit finite or controlled tensor-product representations; this computational assumption is separate from the general matrix-valued coefficient class used in the analysis. Experiments with scalar locally periodic coefficients show accurate high-order correctors. The $H^1$ semi-norm errors are consistent with the proved estimate, while point-normalized $L^2$ errors display the nominal high-order behavior predicted by the formal expansion.

math.NA↗

arXiv2026

Entropy-Stable and Physical-Constraint-Preserving DGSEM for Symmetry-Reduced General-Relativistic Hydrodynamics on Stationary Spacetimes

We develop an entropy-stable and physical-constraint-preserving discontinuous Galerkin spectral element method for symmetry-reduced general-relativistic hydrodynamics on prescribed stationary spacetimes. Using a local orthonormal transformation, the fluid variables are expressed in a form for which the relativistic hydrodynamic algebra and the admissible set are independent of the spatial metric, while the spacetime geometry enters through stationary coefficients. This separation allows entropy-conservative special-relativistic fluxes to be combined with a compatible discretization of the geometric source terms. On affine tensor-product meshes, the resulting DGSEM is conservative and satisfies a semidiscrete entropy inequality, while the transformed variables provide a convex framework for physical-constraint preservation. For practical stabilization, we use a geometry-only causal speed that is sufficient for both classical local Lax--Friedrichs entropy dissipation and the physical-constraint-preserving Lax--Friedrichs splitting. The fully discrete method combines this stabilization with SSP Runge--Kutta time stepping, oscillation elimination, and conservative local-orthonormal-state scaling. Numerical experiments cover smooth and strongly shocked special-relativistic flows, an axisymmetric jet, stationary Michel accretion, Schwarzschild Bondi--Hoyle flow, and four Kerr accretion cases. The results demonstrate the designed high-order accuracy in smooth regimes and robust performance for demanding relativistic flows on curved stationary backgrounds.

math.NA↗

arXiv2026

Nechvile-Transformed Spacecraft Dynamics and Propellant Computation in the 3-Body Problem

The uncontrolled equations of motion in the Nechvile frame for the restricted three-body problem have been well-known since at least the 1960s. It would seem that adding an external force to these equations is quite trivial: simply add an external force per mass term to the acceleration equations. Here we show that the last statement is not true. In fact, we show that the additive generic external force must be multiplied by the inverse of $(1 + e \cosθ)^3$ where $e$ is the relative eccentricity of the primaries and $θ$ is the true anomaly of the rotating frame located at the barycenter. Furthermore, when this result is combined with the mass flow rate equation, it generates several surprising results due to the mismatch between the resulting quadratic term and the cubic term in the equations of motion. This leads to a corresponding modification of the rocket equation itself. A Birkhoff-theoretic solution to an illustrative cislunar space mission problem shows propellent savings of 80% with the use of the correct cost functional. The popular quadratic cost utilizes more than $2X$ the minimum propellant consumption.

math.OC↗

arXiv2026

Dynamic subscales and unconditional semi-discrete stability for non-residual VMS approximations of generalised Newtonian flows

We analyse a non-residual variational multiscale finite element formulation with dynamic subscales for incompressible generalised Newtonian Navier--Stokes flows. The apparent viscosity is assumed to be bounded and Lipschitz continuous with respect to the shear rate, covering several regularised rheological laws. The method separates, through orthogonal projections, the unresolved pressure-divergence, convective, and pressure-gradient contributions. For the linearised semi-discrete problem, we prove well-posedness and an unconditional stability estimate in an anisotropic VMS norm. The estimate controls viscous dissipation, subscale energies, discrete divergence, and a coupled acceleration-convection-pressure balance. For the non-linear formulation, a fixed-point argument yields existence and optimal-order a priori bounds under suitable regularity and smallness assumptions. The analysis also provides weak-in-time control of the discrete pressure-convection balance. A key point is that the dynamic pressure-related subscale provides the time-derivative contribution needed to control the projected pressure-gradient error, a mechanism unavailable in the corresponding quasi-static setting.

math.NA↗

arXiv2026

A posteriori certification of PDE approximations with particular application to neural networks

We propose rigorous and efficiently computable lower and upper a posteriori error bounds for given approximations to PDEs on a given domain, which might be geometrically complex. This is done by embedding or enveloping the original domain towards geometrically simpler domains, enabling the use of fast numerical solvers. To this end, we extend and restrict the residual and provide efficient methods to compute those Hahn-Banach extensions. Then, we efficiently compute their Riesz representations on the geometrically simpler domains and obtain the desired a posteriori bounds for which we prove that they are sharp. The resulting bounds control the error in the natural norm induced by a well-posed variational formulation, require only minimal regularity assumptions, and thus remain applicable on complex geometries. The framework is detailed for elliptic as well as parabolic problems. Numerical experiments demonstrate the good quantitative behavior of the derived upper and lower error bounds. A central motivation for this paper comes from physics-informed and related neural-network approximations of PDEs, which are naturally mesh-free and can be evaluated pointwise on complex or parameter-dependent geometries. Nevertheless, the framework applies to any approximation for which the variational residual can be evaluated.

math.NA↗

arXiv2026

On the Subsample Size of Quantile-Based Randomized Kaczmarz

Quantile-based randomized Kaczmarz (QRK) was recently introduced to efficiently solve sparsely corrupted linear systems $\mathbf{A} \mathbf{x}^*+\mathbfε = \mathbf{b}$ [SIAM J. Matrix Anal. Appl., 43(2), 605-637], where $\mathbf{A}\in \mathbb{R}^{m\times n}$ and $\mathbfε$ is an arbitrary $(βm)$-sparse corruption. However, all existing theoretical guarantees for QRK require quantiles to be computed using all $m$ samples (or a subsample of the same order), thus negating the computational advantage of Kaczmarz-type methods. This paper overcomes the bottleneck. We analyze a subsampling QRK, which computes quantiles from $D$ uniformly chosen samples at each iteration. Under some standard scaling assumptions on the coefficient matrix, we show that QRK with subsample size $D\ge\frac{C\log (T)}{\log(1/β)}$ linearly converges over the first $T$ iterations with high probability, where $C$ is some absolute constant. This subsample size is a substantial reduction from $O(m)$ in prior results. For instance, it translates into $O(\log(n))$ even if an approximation error of $\exp(-n^2)$ is desired. Intriguingly, our subsample size is also tight up to a multiplicative constant: if $D\le \frac{c\log(T)}{\log(1/β)}$ for some constant $c$, the error of the $T$-th iterate could be arbitrarily large with high probability. Numerical results are provided to corroborate our theory.

math.NA↗

arXiv2026

Examining the robustness of Physics-Informed Neural Networks to noise for Inverse Problems

Approximating solutions to partial differential equations (PDEs) is fundamental for the modeling of dynamical systems in science and engineering. Physics-informed neural networks (PINNs) are a recent machine learning-based approach, for which many properties and limitations remain unknown. PINNs are widely accepted as less computationally efficient and accurate than traditional methods for solving PDEs, such as the finite element method. However, PINNs are commonly claimed to show promise in solving inverse problems and handling noisy or incomplete data. We compare the performance of PINNs in solving inverse problems with that of a traditional approach using the finite element method combined with a numerical optimizer. The models are tested on viscosity identification in 1D Burgers' equation and in 2D/3D Taylor-Green Vortex, in all cases with additive Gaussian noise applied to training and validation data. We find that while PINNs may require less human effort and specialized knowledge, they are outperformed by the traditional approach. For example, for 2D Taylor-Green Vortex with $σ$=1 noise, the baseline has a mean prediction RMSE of 0.0013 compared to 0.01 for the best PINN variation. However, PINNs scale better than the baseline with the computational complexity of the problem. We identify failures during training to be addressed if the PINN performance on noisy inverse problems is to become more competitive.

physics.comp-ph↗

arXiv2026

Mathematical and numerical analysis of quantum signal processing

Quantum signal processing (QSP) provides a representation of scalar polynomials of degree $d$ as products of matrices in $\mathrm{SU}(2)$, parameterized by $(d+1)$ real numbers known as phase factors. QSP is the mathematical foundation of quantum singular value transformation (QSVT), which is often regarded as one of the most important quantum algorithms of the past decade, with a wide range of applications in scientific computing, from Hamiltonian simulation to solving linear systems of equations and eigenvalue problems. In this article we survey recent advances in the mathematical and numerical analysis of QSP. In particular, we focus on its generalization beyond polynomials, the computational complexity of algorithms for phase factor evaluation, and the numerical stability of such algorithms. The resolution to some of these problems relies on an unexpected interplay between QSP, nonlinear Fourier analysis on $\mathrm{SU}(2)$, fast polynomial multiplications, and Gaussian elimination for matrices with displacement structure.

quant-ph↗

arXiv2026

The Adaptive Solution of High-Frequency Helmholtz Equations via Multi-Grade Deep Learning

The Helmholtz equation is fundamental for modeling wave propagation in acoustics, electromagnetics, and geophysics; however, high-frequency regimes remain notoriously difficult due to the severe numerical ``pollution effect.'' We propose FD-MGDL, an adaptive framework that synergizes finite difference discretizations with Multi-Grade Deep Learning (MGDL) to efficiently resolve high-frequency wavefields. Unlike standard physics-informed neural networks (PINNs), which frequently suffer from spectral bias and heavy automatic differentiation overhead, FD-MGDL employs a progressive, grade-wise training strategy that incrementally incorporates shallow sub-networks to refine residual errors. By leveraging ReLU activations in refinement grades, the framework reformulates the highly non-convex global optimization problem into a sequence of tractable convex subproblems, dramatically improving training stability and convergence reliability. Extensive numerical experiments in two and three dimensions with wavenumbers up to $κ=200$ demonstrate that FD-MGDL significantly outperforms single-grade networks and standard benchmark neural solvers in both accuracy and computational efficiency. When applied to an inhomogeneous concave velocity model, the proposed method accurately captures wave focusing and caustic formations, markedly outperforming classical five-point finite difference schemes in resolving sharp phase transitions and peak amplitudes. These results establish FD-MGDL as a robust, scalable, and mathematically grounded paradigm for high-frequency wave simulation in complex media.

math.NA↗

arXiv2026

A Reduced Magnetic Vector Potential Approach with Higher-Order Splines

This work presents a high-order isogeometric formulation for magnetoquasistatic eddy-current problems based on a decomposition into Biot-Savart-driven source fields and finite-element reaction fields. Building upon a recently proposed surface-only Biot-Savart evaluation, we generalize the reduced magnetic vector potential framework to the quasistatic regime and introduce a consistent high-order spline discretization. The resulting method avoids coil meshing, supports arbitrary winding paths, and enables high-order field approximation within a reduced computational domain. Beyond establishing optimal convergence rates, the numerical investigation identifies the requirements necessary to recover high-order accuracy in practice, including geometric regularity of the enclosing interface, accurate kernel quadrature, and compatible trace spaces for the source-reaction coupling.

math.NA↗

arXiv2026

Pressure-robustness for the axisymmetric Stokes problem by velocity reconstruction

This paper studies pressure-robustness for the axisymmetric Stokes problem. The transformation to cylindrical coordinates requires that the radially weighted velocity is divergence-free in the classical sense. Consequently, traditional divergence-free finite element methods from the Cartesian setting -- even if inf-sup stable -- are in general not divergence-free in the axisymmetric formulation. We therefore explore the approach that restores pressure-robustness via reconstruction operators for a low-order Bernardi--Raugel discretization. We show that an application of standard interpolation operators from the Cartesian setting to radially weighted test functions works in principle, but it lacks properties needed to derive optimal consistency error estimates. To address this, we introduce a reconstruction operator into a finite element space spanned by Raviart--Thomas functions that are modified such that they vanish on the rotation axis. This vanishing-on-axis property is the key to obtain optimal consistency error estimates. Numerical examples demonstrate the overall feasibility of the approach and include cases where the vanishing-on-axis property yields significantly better results.

math.NA↗

arXiv2026

Deflation-PINNs: Learning Multiple Solutions for PDEs and Landau-de Gennes

Nonlinear Partial Differential Equations (PDEs) are ubiquitous in mathematical physics and engineering. Although Physics-Informed Neural Networks (PINNs) have emerged as a powerful tool for solving PDE problems, they typically struggle to identify multiple distinct solutions, since they are designed to find one solution at a time. To address this limitation, we introduce Deflation-PINNs, a novel framework that integrates a deflation loss with an architecture based on PINNs and Deep Operator Networks (DeepONets). By incorporating a deflation term into the loss function, our method systematically forces the Deflation-PINN to seek and converge upon distinct finitely many solution branches. We provide theoretical results on the approximation capabilities of our model and demonstrate the efficacy of Deflation-PINNs through numerical experiments on the Landau-de Gennes model of liquid crystals, a system renowned for its complex energy landscape and multiple equilibrium states, and on an Allen--Cahn benchmark whose solution set is provably known. Our results show that Deflation-PINNs can successfully identify and characterize multiple distinct crystal structures: a single unsupervised run recovers all six stable states of the benchmark, each branch certified to lie in the basin of attraction of a different equilibrium, and the discovered branches are refined to percent-level accuracy by a purely neural Deflation--Deep-Ritz stage and to the accuracy of a mesh-converged reference by a classical solver that they initialize.

math.NA↗

arXiv2026

Parameterization-driven arbitrary Lagrangian-Eulerian method for large-deformation isogeometric fluid-structure interaction

Body-fitted arbitrary Lagrangian-Eulerian (ALE) methods provide a sharp representation of the fluid-structure interface but rely on mesh-update strategies that incrementally deform a reference configuration. To address this issue, we reformulate the ALE mesh-motion problem in the isogeometric setting as a sequence of independent domain parameterization problems. At each time step, a multi-patch spline parameterization of the fluid domain is constructed from the current interface geometry. Three technical components realize this framework: (i) a barrier-function-based spline parameterization that enforces a strictly positive Jacobian at every time step; (ii) a tangential-slip reparameterization that handles unbounded cumulative rotations of closed domains, where no fixed boundary-to-parameter correspondence is admissible; and (iii) a constant-preserving quasi-interpolation operator for solution transfer between consecutive parameterizations, ensuring that the discrete geometric conservation law holds algebraically. We validate the method on three two-dimensional FSI benchmarks, covering standard and large-rotation regimes, and on a three-dimensional rotor problem. On a rotating-square benchmark, the tangential-slip strategy enables simulations under sustained rotation far beyond the range accessible to classical mesh-update schemes--a regime that is fundamentally inaccessible to any mesh-deformation formulation, not merely numerically difficult. A three-dimensional rotor example further demonstrates that the framework extends naturally to volumetric spline parameterizations. Finally, we show that the per-step spline parameterizations can be used directly within a standard finite element solver.

math.NA↗

arXiv2026

A superlinear arclength endgame for corank-one singular polynomial systems

In Numerical Algebraic Geometry (NAG) isolated solutions of polynomial systems are usually computed by tracking a solution curve defined by a homotopy equation. The tracking problem becomes especially challenging close to a singular root (the ``endgame'' regime). Existing approaches include power series endgames, Cauchy endgames, and various methods that regularize the system via dual-space-based {\em deflation}. We introduce a new ``Arclength Endgame'' for corank-1 systems which combines the idea of the classical {\em pseudo-arclength continuation method} with the estimation of the Puiseux series of the curve. We formally prove that it has an R-superlinear rate of convergence in some neighborhood of the root. The method uses only evaluations of the system and its Jacobian. In contrast, previous homotopy techniques with proven superlinear convergence (such as deflation) require computing additional derivatives of the system, or need additional nondegeneracy assumptions involving higher-order derivatives of the system at the singular root. A key step in our approach (as well as in the standard power series endgame) is estimating the ratio $k_1/c$ of the Puiseux series of the curve. We present a new method for that, and discuss its advantages over previous methods.

math.NA↗

arXiv2026

Optimal enforcement of natural conditions on smooth boundaries with piecewise polynomials upon fitted straight-edged meshes

A few decades ago some possible remedies to an inaccurate enforcement of Neumann or Robin conditions prescribed on the boundary of a smooth domain, owing to its approximation by the union of straight-edged triangles or tetrahedra in a fitted mesh, were addressed. These studies, due to authors such as Barrett and Elliott (1988), advocated the use of isoparametric finite elements with a single curvilinear edge or face fitting the true boundary at two or three vertexes and also at additional points on those curves or curved surfaces, so as to define non polynomial piecewise solutions with the optimal order of approximation. In this work we adopt a different approach, whose main feature is the use of a fitted mesh consisting only of straight-edged elements, upon which the solution is a polynomial. Lost optimal orders of convergence, by virtue of the domain's approximation by a polytope, are recovered by means of the addition of terms to the bilinear form. These account for natural boundary conditions of the same type as the true ones, though to be prescribed on the approximating boundary instead. The new technique is applied to the case of triangular Lagrange finite elements, for which we give a rigorous reliability study in the solution of reaction-diffusion equations. Numerical experimentation is supplied in support of the theoretical results.

math.NA↗

arXiv2026

Energy-Based Physics-Informed Form Finding for Clustered Tensegrity Structures

Tensegrity form-finding and physical property prediction are fundamental problems in structural mechanics, which aim to determine equilibrium configurations and internal force distributions. These problems are challenging due to strong nonlinearity arising from the coupling between geometry and forces, and the need to satisfy equilibrium, stability, and structural constraints. This paper proposes an energy-based learning approach for clustered tensegrity form finding and physical property prediction. The proposed approach incorporates total potential energy minimization and constitutive relations into the training objective, enabling the prediction of equilibrium nodal configurations and the reconstruction of physical quantities such as member forces and force densities. By integrating energy-based physical losses directly into the learning process, the method promotes physical consistency while combining data-driven learning with physics-based constraints. Numerical experiments on tensegrity prism and lander structures demonstrate accurate prediction of equilibrium configurations and internal forces across different training-data ratios, indicating the potential of the proposed approach for nonlinear tensegrity form finding and structural analysis.

cs.LG↗

arXiv2026

Data driven non-equilibrium moist phase exchanges for atmospheric convection within a discontinuous Galerkin model of the compressible Euler equations

A discrete formulation of the multiphase moist compressible Euler equations is presented for which the energy and tracer variance conserving dynamics is coupled to the non-equilibrium moist phase exchanges in a thermodynamically consistent manner. A neural network is then trained to learn the mass exchanges between vapour, liquid and ice phases in atmospheric convection. The network is trained on convection resolving output from a regional configuration of the LFRic model with a multi-moment microphysics parameterisation (CASIM). The loss function for learning these phase exchanges is formulated under the assumptions of thermal and mechanical equilibrium (same temperature and pressure for all phases), and chemical dis-equilibrium (different Gibbs free energies for all phases). The network outputs determine the exchanges of vapour, liquid and ice so as to conserve mass, and the resulting change in entropy is determined from the network outputs so as to conserve energy. The neural network is implemented in a thermodynamically consistent manner within the discontinuous Galerkin moist compressible Euler model in order to simulate the formation of three-phase clouds for convection at sub-km resolution. The results are compared to those from a physics based representation of three-phase moist processes at thermodynamic equilibrium, as well as to an idealised configuration of LFRic model.

math.NA↗

Compare source metadata on this page

Work	Published	Source identifier	Source
Random attractors and almost-sure stability under discretization of a stochastic autoparametric system	2026-08-29	2608.29149	arxiv
A Tensor Neural Network Method for High-Order Homogenization of Locally Periodic Elliptic Problems	2026-08-29	2608.29176	arxiv
Entropy-Stable and Physical-Constraint-Preserving DGSEM for Symmetry-Reduced General-Relativistic Hydrodynamics on Stationary Spacetimes	2026-08-29	2608.29229	arxiv
Nechvile-Transformed Spacecraft Dynamics and Propellant Computation in the 3-Body Problem	2026-08-29	2608.29271	arxiv
Dynamic subscales and unconditional semi-discrete stability for non-residual VMS approximations of generalised Newtonian flows	2026-08-29	2608.29457	arxiv
A posteriori certification of PDE approximations with particular application to neural networks	2026-08-28	2502.20336	arxiv
On the Subsample Size of Quantile-Based Randomized Kaczmarz	2026-08-28	2507.15185	arxiv
Examining the robustness of Physics-Informed Neural Networks to noise for Inverse Problems	2026-08-28	2509.20191	arxiv
Mathematical and numerical analysis of quantum signal processing	2026-08-28	2510.00443	arxiv
The Adaptive Solution of High-Frequency Helmholtz Equations via Multi-Grade Deep Learning	2026-08-28	2602.20719	arxiv
A Reduced Magnetic Vector Potential Approach with Higher-Order Splines	2026-08-28	2602.22997	arxiv
Pressure-robustness for the axisymmetric Stokes problem by velocity reconstruction	2026-08-28	2603.01030	arxiv
Deflation-PINNs: Learning Multiple Solutions for PDEs and Landau-de Gennes	2026-08-28	2603.27936	arxiv
Parameterization-driven arbitrary Lagrangian-Eulerian method for large-deformation isogeometric fluid-structure interaction	2026-08-28	2604.27537	arxiv
A superlinear arclength endgame for corank-one singular polynomial systems	2026-10-02	2607.06329	arxiv
Optimal enforcement of natural conditions on smooth boundaries with piecewise polynomials upon fitted straight-edged meshes	2026-08-28	2607.07963	arxiv
Energy-Based Physics-Informed Form Finding for Clustered Tensegrity Structures	2026-08-28	2607.12888	arxiv
Data driven non-equilibrium moist phase exchanges for atmospheric convection within a discontinuous Galerkin model of the compressible Euler equations	2026-08-28	2607.13360	arxiv

These are bibliographic comparisons, not experimental rankings. Follow the original document for methods and conditions.