subject

math.AP

math.AP: explore 29 source-linked works published from 2017 to 2026, with original documents and citations.

This collection is a preview while coverage and quality are evaluated.

Coverage and selection

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 2026-09-15. Counts describe this index, not the complete source archives.

arXiv2026

An intrinsic expansion approach to the Galerkin approximations for the Navier-Stokes equations (with an appendix by Chengzhang Fu)

We study the Galerkin approximation of the three-dimensional Navier-Stokes equations. In particular, we examine the convergence of these solutions in a sequence of finite dimensional spaces as the dimension goes to infinity. For any sequence of steady state or, respectively, time dependent Galerkin solutions that converges to a solution of the Navier-Stokes equations, we obtain a subsequence with an intrinsic asymptotic expansion in appropriate nested function spaces. Consequently, an induced asymptotic expansion is obtained in a more standard spatial Sobolev or, respectively, spatiotemporal Sobolev-Lebesgue space. In the case of steady states, we establish certain relations among leading terms of this expansion.

math.AP↗

arXiv2026

Frequency-explicit convergence analysis of a multiscale finite element method for highly heterogeneous scattering problems

We analyze the numerical approximation of time-harmonic scattering by highly heterogeneous penetrable obstacles. These problems are especially challenging in the high-frequency regime, where the size of the scatterer $L$ is much larger than the wavelength, i.e., the wavenumber $k$ is such that $kL \gg 1$. Here, we further consider the situation where the scatterer contains different materials, with a characteristic size $\varepsilon$ such that $k\varepsilon \ll 1$. We propose a high-order multiscale finite element method, and provide an error analysis that is explicit in both $k$ and $\varepsilon$. Crucially, our error estimates suggest that using a high-order method should reduce the computational cost for large frequencies, which is corroborated by numerical examples.

math.NA↗

arXiv2026

Regularity of Second-Order Elliptic PDEs in Spectral Barron Spaces

We establish a regularity theorem for second-order elliptic PDEs on $\mathbb{R}^{d}$ in spectral Barron spaces. Under mild ellipticity and smallness assumptions, the solution gains two additional orders of Barron regularity. As a corollary, we identify a class of PDEs whose solutions can be approximated by two-layer neural networks with cosine activation functions, where the width of the neural network is independent of the spatial dimension.

math.AP↗

arXiv2026

Analysis of Moment Closures Using $φ$-Divergences for Rarefied Dynamics with Binary Collisions and Their Galerkin Discretizations

This work introduces a robust deterministic framework for approximating solutions of the Boltzmann equation with binary collisions by discretizing their dependence on time, position, and velocity using Galerkin methods. By employing a family of parametric Galerkin closures based on $φ$-divergences in velocity space, we derive rigorous hierarchies of moment equations that govern fluid dynamic variables. Addressing the limitation that these closures alone do not guarantee dissipation of a $φ$-divergence entropy for the true binary collision operator, we restore this property by formulating a compatible approximate collision operator tailored to each closure. This constructed operator intrinsically retains fundamental physical properties essential for high-fidelity flow simulations, including Galilean invariance, exact conservation of mass, momentum, and energy, and strict dissipation of a $φ$-divergence entropy. Furthermore, we show that the resulting closed moment systems are symmetric-dissipative, yielding Cauchy problems that are well-posed locally in time. To translate this mathematical foundation into an efficient computational tool, we discretize the position and time variables with an entropy-stable discontinuous Galerkin (DG) finite element method. The fully implicit, entropy-stable space-time approach enables time steps far beyond typical CFL-limited step sizes and the direct computation of steady states. The robustness and accuracy of the methodology are verified and validated through numerical simulations on the supersonic nozzle flow of argon, mass flow through a channel, and heat transfer between parallel walls, demonstrating agreement with analytical benchmarks, experimental measurements, and stochastic particle simulations.

math.NA↗

arXiv2026

Residual neural networks overcome the curse of dimensionality for semilinear heat equations

Rigorous results show that feedforward neural networks can overcome the curse of dimensionality in the numerical approximation of high-dimensional partial differential equations (PDEs), but comparatively little is known about residual neural networks (ResNets) in the nonlinear PDE setting. We prove that ResNets overcome the curse of dimensionality in the numerical approximation of solutions of semilinear heat equations with globally Lipschitz continuous, gradient-independent nonlinearities: under polynomial growth and network approximability hypotheses on the PDE data, there exist $η\in(0,\infty)$ and ResNets $Ψ_{d,\varepsilon}$, $d\in\mathbb{N}$, $\varepsilon\in(0,1]$, with at most $ηd^η\varepsilon^{-η}$ parameters whose realizations approximate the solution in dimension $d$ with an $L^2$-error of at most $\varepsilon$. The proof represents one deterministic realization of a multilevel Picard estimator by a ResNet whose shortcut connections transmit the spatial variable and a scalar accumulator, while the residual branches successively add the summands of the estimator. For ridge-sum initial conditions, admissible sigmoidal activations, and globally Lipschitz truncations of the nonlinearity, we obtain, for every $ξ>0$, the explicit bound $C_ξd^{4+ξ}\varepsilon^{-(3+ξ)}$ on the number of parameters.

math.NA↗

arXiv2026

On the Abundance of Critical Points of the t-SNE Energy

This paper considers the energy landscape of the t-SNE algorithm. While this algorithm has enjoyed broad adoption, the non-convexity of the associated energy has made it difficult to rigorously understand what the algorithm captures in many settings. In particular, a number of well-known numerical examples, several of which are reproduced in this article, suggest a complicated energy landscape with many local minimizers that do not respect the topology or clustering structure of the underlying data. This work seeks to provide first steps towards a rigorous explanation of these phenomena. Specifically, for a general family of energies, which include both the original t-SNE algorithm and recently identified large data limits, and for densities in feature space which obey a continuous symmetry, we construct infinite families of distinct critical points. These critical points are based upon identifying pairs of discrete symmetries, one in the original feature space and the other in the target embedding space, which are preserved under gradient dynamics. These critical configurations exhibit many characteristics, such as topology breaking and spurious clustering, which are often observed empirically. Finally, numerical and analytical examples are given throughout as a means of illustrating the approach.

cs.LG↗

arXiv2026

A Temperature-Coupled Cahn-Hilliard-Stokes-Heat Model for Thermally Driven Phase Separation

We study a diffuse-interface model for thermally driven phase separation in viscous incompressible mixtures. The system couples a convective Cahn-Hilliard equation for the order parameter with a Stokes subsystem for the velocity-pressure field and a heat equation for the temperature. Temperature enters the bulk free energy through a Landau-type coefficient, while the phase field affects the flow through concentration-dependent density and viscosity. The model serves as a proxy for temperature-triggered condensation-like phase separation; humidity, latent heat, vapor pressure, and capillary forcing are absorbed into the choice of the threshold temperature $Θ_S$. We motivate the chemical potential through a temperature-dependent Landau free energy and use a regularized auxiliary formulation to prove local-in-time existence of weak solutions. For the numerical analysis, we employ a first-order sequential finite-element discretization of a simplified quasi-static formulation. The heat equation is advanced by implicit diffusion, the variable-coefficient Stokes problem is treated by a Taylor-Hood discretization, and the Cahn-Hilliard bulk derivative is evaluated at the previous time level, so each algebraic subproblem is linear. An isothermal diffusive test confirms mass conservation to roundoff and exhibits monotone discrete-energy decay for the tested parameters. Time-step and mesh-refinement studies show first-order temporal and approximately second-order spatial behavior. The remaining computations provide qualitative, parameter-specific illustrations; no global discrete energy law is claimed for the non-isothermal sequential scheme.

math.AP↗

arXiv2026

Two Adjoint Perspectives on Fokker-Planck Optimization: A Microscopic-Macroscopic Correspondence

The Fokker-Planck equation admits both a macroscopic Eulerian description through probability densities and a microscopic Lagrangian description through stochastic trajectories. Consequently, optimization problems constrained by the Fokker-Planck equation can be formulated from either perspective. Surprisingly, the corresponding adjoint equations appear to be fundamentally different: the macroscopic adjoint is governed by the backward Kolmogorov equation, whereas the microscopic adjoint evolves pathwise along stochastic trajectories. In this note, we reconcile these two formulations by establishing their correspondence in the continuum setting. We further show that, although their discrete gradients no longer coincide after discretization, both provide consistent numerical approximations of the continuum gradient. Explicit convergence rates are established for both discretization strategies.

math.NA↗

arXiv2026

Neural operators approximate strongly continuous convex monotone semigroups

We approximate strongly continuous convex monotone semigroups by learning their Chernoff-type one-step operators with neural operators. First, we introduce the general class of so-called Chernoff-neural operators and show in a universal approximation theorem that they can approximate the Chernoff one-step operators arbitrarily well. By using stability estimates between weighted Hölder spaces, the one-step approximation error can be propagated through the iterations which yields universal approximation of the corresponding semigroup. Second, we introduce the more specialized class of envelope-neural operators for envelope semigroups which allows us to derive quantitative approximation rates. Finally, we illustrate the effectiveness of these neural operators in several numerical examples arising from non-linear partial differential equations, stochastic optimal control and stochastic processes under model uncertainty.

math.NA↗

arXiv2026

Continuous data assimilation in steady Navier-Stokes equations with unknown viscosity: robust and efficient solvers and fast parameter recovery

Recent advances in equation discovery methods such as SINDy have highlighted the growing interest in identifying governing parameters and models directly from data. In this work, we take a complementary approach grounded in analysis and numerical PDE methods: we recover an unknown viscosity in steady Navier-Stokes equations (NSE) from partial incompressible flow observations using continuous data assimilation (CDA). We propose a simple and efficient parameter recovery algorithm and also a nonlinear solver for CDA-NSE. Together, this creates a highly efficient technique for recovering an unknown viscosity from partial solution data. Our analysis establishes the well-posedness of steady CDA-NSE, quadratic convergence of the parameter recovery algorithm, and quadratic convergence of a CDA-Picard + CDA-Newton nonlinear solver. Numerical experiments illustrate that the methods are very effective in restoring parameters quickly, even with poor initial guesses.

math.NA↗

arXiv2026

Structural Compatibility and Uniform Stability of Temporally Degenerate Parabolic Systems

Modern feedback design for distributed parameter systems presupposes that the closed-loop dynamics define a well-posed evolution problem. This presupposition becomes nontrivial for temporally degenerate parabolic systems, where temporal degeneracy affects not only the analytical properties of the evolution equation but also the mathematical formulation of the feedback interconnection itself. It is shown that admissible feedback interconnections for temporally degenerate parabolic systems are completely characterized by an operator compatibility condition linking the singular reaction operator with the actuator and observation operators. This characterization removes the singular component of the closed-loop dynamics and reduces the degenerate evolution equation to a regular evolution equation. Building upon this regularized formulation, a critical--residual decomposition yields a uniform exponential stability certificate, which is subsequently extended to the original infinite-dimensional evolution through a finite-to-infinite lifting theorem. A constructive static output feedback synthesis is finally obtained as a consequence of these results. Numerical experiments illustrate the regularization mechanism, validate the stability certificate, and confirm the finite-to-infinite lifting.

math.OC↗

arXiv2026

A Variational Method for Conformable Fractional Equations Using Rank-One Updates

We make a complete variational treatment of rank-one Proper Generalised Decomposition for separable fractional partial differential equations with conformable derivatives. The setting is Hilbertian, the energy is induced by a symmetric coercive bilinear form, and the residual is placed in the dual space. A greedy rank-one update is obtained by maximizing an energy Rayleigh quotient over the rank-one manifold, followed by an exact line search. An exact one step energy decrease identity is proved, together with geometric decay of the energy error under a weak greedy condition that measures how well the search captures the Riesz representer of the residual. The alternating least squares realization is analyzed at the level of operators, including well posedness of the alternating subproblems, a characterization of stationary points, and monotonicity of the Rayleigh quotient along the inner iteration. Discretizations based on weighted finite elements and on Grünwald type schemes are described in detail, including assembly, boundary conditions, complexity, and memory. Two model problems, a stationary fractional Poisson problem and a space time fractional diffusion problem, are treated from the continuous level down to matrices.

math.NA↗

arXiv2026

Rigorous Error Certification for Neural PDE Solvers: From Empirical Residuals to Solution Guarantees

Uncertainty quantification for partial differential equations is traditionally grounded in discretization theory, where solution error is controlled via mesh/grid refinement. Physics-informed neural networks fundamentally depart from this paradigm: they approximate solutions by minimizing residual losses at collocation points, introducing new sources of error arising from optimization, sampling, representation, and overfitting. As a result, the generalization error in the solution space remains an open problem. Our main theoretical contribution establishes generalization bounds that connect residual control to solution-space error. We prove that when neural approximations lie in a compact subset of the solution space, vanishing residual error guarantees convergence to the true solution. We derive deterministic and probabilistic convergence results and provide certified generalization bounds translating residual, boundary, and initial errors into explicit solution error guarantees.

cs.LG↗

arXiv2026

Computational Oncology of Chemotaxis-Driven Tumour--Immune Spatial Patterning and Stability

We develop a reaction--diffusion--chemotaxis model for spatial tumour--immune--chemokine dynamics that couples logistic tumour growth, immune-mediated killing, chemokine-dependent immune recruitment, chemotactic migration, and signal production. For the nondimensional system, we establish local classical solvability, nonnegativity, a uniform tumour-density bound, and global mass estimates for the immune and chemokine components. The tumour-free equilibrium is stable precisely when the baseline immune-control index satisfies $σ_0/δ>1$, whereas positive homogeneous coexistence is characterized by a scalar nonlinear equation. Linearization in the Neumann Laplacian eigenbasis yields a mode-dependent cubic dispersion relation, showing that chemotaxis does not alter the tumour-invasion threshold but can destabilize homogeneous coexistence through a finite-wavelength oscillatory instability above a critical sensitivity $ξ_c$. A conservative finite-volume discretization with upwind chemotactic fluxes and implicit backward differentiation formula time integration is used to test these predictions. Numerical experiments recover the analytical equilibria and growth rates, identify the dominant unstable mode, reproduce the transition to spatial heterogeneity, and quantify the effects of immune recruitment, decay, and diffusion on the stability boundary. Grid-refinement, mass-balance, residual, and nonnegativity diagnostics support the computational reliability of the results.

math.AP↗

arXiv2026

Sharp Mixed Spectral Barron Regularity of Coulombic Many-Electron Wave Functions

We establish sharp mixed spectral Barron regularity for eigenfunctions of molecular Coulomb Hamiltonians. The mixed norm is a Fourier $L^1$ norm with one isotropic weight and coordinate-product weights, and therefore detects regularity invisible to the isotropic Barron scale. For a nonempty set $I$ of electron indices on which the wave function is antisymmetric, we derive an explicit admissible region for the isotropic order $s$ and the coordinate orders $α,β$. This region is optimal as a uniform statement over the class of clamped-nuclei Coulomb Hamiltonians. For fixed-spin components with two occupied spin blocks, it reduces to $s+α+β<1$; in the fully spin-polarized class it reduces to $s+α<1$. In particular, if $\mathcal I_σ$ denotes the family of occupied same-spin blocks determined by $σ$, then every fixed-spin spatial component $ψ_σ$ satisfies, for every $0\leqα<1$, \[ \left(\sum_{I\in\mathcal I_σ}\prod_{i\in I}\langleξ_i\rangle^α\right)\widehat{ψ_σ}\in L^1(\mathbb{R}^{3N}). \] For a fully spin-polarized state, $\mathcal I_σ=\{\{1,\ldots,N\}\}$.

math.AP↗

arXiv2026

Projection-based low-rank assembly in IgA

Isogeometric Analysis (IgA) uses the same spline functions to represent the computational domain and to approximate the solution. This allows exact geometry descriptions, but the resulting mass and stiffness matrices are expensive to assemble and to store, especially in three dimensions. We present a projection-based low-rank approach for assembling the mass and stiffness tensors of orientation-preserving tensor-product B-spline geometries. For the mass tensor, we exploit the polynomial structure of the determinant of the Jacobian of the geometry map and represent it in reduced spline product spaces by univariate coefficient transfer operators; with exact quadrature and without truncation, the resulting low-rank tensor is an exact reformulation of the standard Galerkin mass tensor. For the stiffness tensor, we split the rational weight function into a polynomial numerator, again represented in reduced spline product spaces, and the reciprocal determinant, which is in general not a spline function and is therefore approximated by an $L^2$-projection onto a tensor-product spline space. Both constructions are carried out entirely in the tensor-train (TT) format, with the projection system solved by the alternating minimal energy (AMEn) method, so that full high-order coefficient tensors are never formed and the multidimensional integrals reduce to univariate integrals and contracted products. The method is implemented in MATLAB using GeoPDEs and the TT-Toolbox. Numerical experiments show that it is competitive with full assembly and with the interpolation-based low-rank method, and that it applies in two situations in which interpolation is problematic: a singular interpolation system and nearly singular geometries. The construction is restricted to orientation-preserving tensor-product B-spline geometries and does not cover NURBS.

math.NA↗

arXiv2026

The $α$-Limit Problem: Convergence of a Linear Degenerate Interface Transmission Problem

We study the singular limit of a family of linear degenerate interface transmission problems arising from a regularization procedure in the newly proposed Two-Parameter Diffuse Domain Method (DDM2p). For $α>0$, the regularized problem admits a strictly convex variational formulation on $H^{1}(Ω)$. In the limit $α\to0$, the problem degenerates to a weakly coupled interface system with a nonstandard energy structure. To characterize the limit, we introduce a closed Hilbert subspace $\mathcal{H}\subset H^{1}(Ω)$, defined through an auxiliary Helmholtz problem on an annular subdomain $Ω_2\subset Ω$, and identify the limiting energy functional $\mathcal{E}_{0}$ on $\mathcal{H}$. We prove that the regularized energies $\mathcal{E}_α$ $Γ$-converge to $\mathcal{E}_{0}$ in the strong $L^{2}(Ω)$ topology, using the standard framework. Consequently, minimizers of $\mathcal{E}_α$ converge to the unique minimizer of $\mathcal{E}_{0}$, which is shown to be equivalent to the solution of the limiting interface problem. We further prove strong convergence $u_α\to u_{0}$ in $H^{1}(Ω)$ and establish an $O(α)$ convergence rate. Numerical experiments in one spatial dimension confirm the predicted first-order convergence rate and suggest that this rate is sharp.

math.AP↗

arXiv2026

Shape Holomorphy and Sparse Approximation of the Maxwell Electric Field Integral Operator

Uncertainty quantification for time-harmonic Maxwell scattering by obstacles of uncertain shape needs more than holomorphic dependence of the scattered field: for a boundary element method it is the boundary integral operator family itself that must depend holomorphically on the shape parameters. Two obstructions stand in the way. The natural energy space of the electric field integral equation, $\boldsymbol H^{-1/2}_{\mathrm{div}_Γ}(Γ)$, depends on the geometry, and the available operator-valued shape-holomorphy theory for weakly singular kernels is set in $L^2$, which does not reach it. We remove both. A surface contravariant Piola transformation identifies the geometry-dependent Maxwell trace spaces with a fixed reference space, and in the pulled-back variational formulation the surface Jacobians cancel exactly. The principal analytical ingredient is then a uniform fractional mapping theorem $H^{-1/2}\to H^{1/2}$ for the complex-deformed scalar single-layer family on uniformly $C^{1,1}$ surfaces, obtained by realizing the Laplace principal part as the trace of a complex-coefficient Newton problem on a fixed ambient space. The pulled-back operators are consequently $(\bm b,p,\eps)$-holomorphic for $\bm b\in\ell^p(\N)$, $0<p<1$, and pointwise exclusion of interior electric resonances over the compact real parameter set yields uniform invertibility. Legendre coefficients are therefore $\ell^p$ summable, so the operator family, the surface current and the far field all admit sparse polynomial approximations at dimension-independent best $N$-term rates. These statements are for the operator family itself in its energy-space operator norm, not only for individual solutions.

math.NA↗

Compare source metadata on this page

Work	Published	Source identifier	Source
An intrinsic expansion approach to the Galerkin approximations for the Navier-Stokes equations (with an appendix by Chengzhang Fu)	2026-09-04	2602.16064	arxiv
Frequency-explicit convergence analysis of a multiscale finite element method for highly heterogeneous scattering problems	2026-09-04	2609.04930	arxiv
Regularity of Second-Order Elliptic PDEs in Spectral Barron Spaces	2026-09-03	2602.19381	arxiv
Analysis of Moment Closures Using $φ$-Divergences for Rarefied Dynamics with Binary Collisions and Their Galerkin Discretizations	2026-09-03	2608.03640	arxiv
Residual neural networks overcome the curse of dimensionality for semilinear heat equations	2026-09-03	2609.03626	arxiv
On the Abundance of Critical Points of the t-SNE Energy	2026-09-03	2609.04379	arxiv
A Temperature-Coupled Cahn-Hilliard-Stokes-Heat Model for Thermally Driven Phase Separation	2026-09-02	2604.21200	arxiv
Two Adjoint Perspectives on Fokker-Planck Optimization: A Microscopic-Macroscopic Correspondence	2026-09-02	2609.02072	arxiv
Neural operators approximate strongly continuous convex monotone semigroups	2026-09-02	2609.02727	arxiv
Continuous data assimilation in steady Navier-Stokes equations with unknown viscosity: robust and efficient solvers and fast parameter recovery	2026-09-02	2609.02862	arxiv
Structural Compatibility and Uniform Stability of Temporally Degenerate Parabolic Systems	2026-09-02	2609.03136	arxiv
A Variational Method for Conformable Fractional Equations Using Rank-One Updates	2026-09-01	2510.03778	arxiv
Rigorous Error Certification for Neural PDE Solvers: From Empirical Residuals to Solution Guarantees	2026-09-01	2603.19165	arxiv
Computational Oncology of Chemotaxis-Driven Tumour--Immune Spatial Patterning and Stability	2026-09-01	2607.03813	arxiv
Sharp Mixed Spectral Barron Regularity of Coulombic Many-Electron Wave Functions	2026-09-01	2609.00872	arxiv
Projection-based low-rank assembly in IgA	2026-09-01	2609.01218	arxiv
The $α$-Limit Problem: Convergence of a Linear Degenerate Interface Transmission Problem	2026-09-01	2609.01237	arxiv
Shape Holomorphy and Sparse Approximation of the Maxwell Electric Field Integral Operator	2026-08-31	2609.00466	arxiv

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