Papers

The computation of finite-thrust extremal arcs in the elliptic restricted three-body trajectory optimization problem is considered. Libration-point orbits are approximated to near-machine precision using a fast Fourier transform of the sampled values of the state vector at Chebyshev-Gauss-Lobatto points. Checkable optimality conditions are derived by applying Pontryagin's principle to minimum-time and time-constrained minimum-propellant problems. These necessary conditions include criteria for optimal departure and arrival points. For propellant consumption, a recently developed computational model is employed. This model is agnostic to the specific impulse of the propellant and varies as the inverse quadratic of a cosine term. Candidate optimal solutions are generated by combining the universal Birkhoff theory for trajectory optimization with the fast, guess-free spectral algorithm. The extremality of the Birkhoff-computed solution is validated against the Hamiltonian minimization condition and the transversality conditions. It is shown that the Birkhoff-theoretic spectral algorithm can generate verifiable extremals without any assistance or initialization from dynamical systems theory.

We propose a finite difference discrete fracture model for single-phase flow in fractured porous media. The method is derived from a unified energy principle and is implemented on Cartesian grids that need not conform to the fractures. It introduces no additional degrees of freedom, modifies the underlying finite difference scheme only locally, handles both highly conductive fractures and low-permeability barriers, and naturally preserves symmetry and positive definiteness of the discrete system. Barrier-induced pressure jumps and fracture-induced fluxes can be recovered through inexpensive local post-processing, yielding a sharper representation of the interface effects. Numerical experiments based on manufactured solutions and published $2$D and $3$D benchmark problems demonstrate the accuracy, effectiveness, and flexibility of the method for isolated fractures, complex networks, and anisotropic porous media.

In this paper, we propose and analyze a splitting mixed finite element method for a stochastic Keller–Segel system with logistic growth driven by multiplicative noise. By introducing an auxiliary variable representing the chemical gradient together with a time-lagged splitting strategy, the proposed method decouples the original coupled system into a sequence of simpler subproblems. Consequently, it eliminates the Ladyzhenskaya–Babuška–Brezzi stability constraint, permits the use of continuous piecewise linear finite element spaces for all unknowns, and avoids solving a fully coupled nonlinear system at each time step, thereby significantly reducing the computational cost. Combined with an implicit Euler time discretization, the proposed approach yields a fully discrete numerical scheme for the stochastic Keller–Segel system. Using a localization technique together with suitable stochastic stability arguments, we establish optimal strong error estimates for the fully discrete approximations and prove convergence in probability with explicit convergence rates. Numerical experiments verify the theoretical convergence rates and demonstrate that the proposed method successfully captures the global boundedness induced by the logistic growth term as well as the influence of multiplicative noise on chemotactic aggregation.

We investigate the analysis and phase ordering dynamics of the active Cahn–Hilliard equation, providing novel results beyond the current state of the art concerning the well-posedness and the characterization of its coarsening dynamics. We consider both regular polynomial and singular logarithmic potentials. In particular, we exploit a new method based on heteroclinic trajectories in the phase plane to characterize static kink profiles and spherical droplet states, recovering the exact values of key quantities related to static phase-separated configurations; moreover, we develop a theory accounting for surface tension modifications driven by activity and local interface curvature, which explains the power-law shift $L(t)\sim t^{\frac{1}{z}}$ from $z=3$ to $z=4$ induced by activity for the characteristic domain length conjectured in the literature. This shows that there is a transitory effect before the attainment of a finite saturation length. We also design an efficient numerical scheme, based on finite elements, to approximate the model, proving its well-posedness and stability both for regular and singular potentials. In dimensions $d=2,3$ with singular potential, the convergence analysis of the finite element approximation proves the local-in-time existence and uniqueness of weak solutions satisfying the physical constraint $\phi \in (-1, 1)$. In dimension $d=1$ with singular potential, we establish global-in-time well-posedness and regularity of weak solutions under a smallness condition on the activity parameter. Finally, we show numerical simulations for different test cases which prove that our numerical algorithm correctly reproduces the expected phase separation dynamics. Moreover, we show the results for coarsening dynamics at late times which present a power law shift from $z=3$ to $z=4$ prior to reaching late-time length saturation, which confirms our theoretical findings.

In this work, we formally investigate the dealiasing properties of split form discontinuous Galerkin (DG) discretizations for the one-dimensional Burgers problem. By generalizing the proof of Blaisdell et al. for spectral discretizations, we show that split form DG schemes achieve dealiasing in the weakly underresolved range through integration error cancellation on the conservative and non-conservative forms. As a corollary, we identify quadrature- and order-dependent pairs of splitting coefficients that eliminate the dominant component of the aliasing error. While these “optimized” splitting coefficients minimize integration errors on the numerical scheme, we show that the skew-symmetric aliasing pattern resulting from the entropy stable split is required to maintain long-term stability of the numerical solution. The proposed framework provides an alternative proof for the entropy stability of the DG discretization of the (1/3, 2/3) split formulation introduced by Gassner which clarifies the connection between DG aliasing errors and entropy stability. Finally, we also show that the entropy stable split is associated with lower aliasing errors in the weakly underresolved range compared to the conservative form.

We propose a mesh-based filtering approach to alleviate the severe timestep restrictions arising in explicit Runge–Kutta discontinuous Galerkin (RKDG) methods formulated in spherical-polar coordinates. The filter enables stable evolution on the original logically Cartesian mesh while using larger time steps associated with an auxiliary merged mesh constructed to eliminate the extreme cell anisotropies produced by converging coordinate lines near coordinate singularities. The filter is implemented as a sequence of post-processing operations applied within an $s$-stage RK time integrator, making it straightforward to incorporate into existing structured-mesh DG frameworks. We analyze the filter in one spatial dimension and prove that the filtered RKDG method is equivalent to evolving the RKDG discretization on a nonuniform mesh obtained by merging selected elements of the underlying uniform mesh. This equivalence implies that the filtered method inherits the accuracy and stability properties of the corresponding RKDG discretization on the merged mesh. We apply the mesh-based filter to an existing RKDG method for the Euler equations in spherical-polar coordinates and demonstrate, through selected two- and three-dimensional examples, its effectiveness in accelerating simulations through significantly larger stable timesteps.

We propose a method to accelerate the conjugate gradient method (CG) on parallel computing hardware by performing block conjugate gradient on one linear system. We aim to reduce the number of iterations until convergence by solving several copies of the system to explore multiple regions of the solution space at once. Although solving multiple systems requires more floating point operations than solving one, these computations are highly parallelizable, even without additional hardware resources. Our method does not replace preconditioning and can be used with any preconditioner. We developed linear regression models to predict iteration reduction and speedup as functions of the coefficient matrix's size, its number of nonzeros, and the solve time with CG. Our experiments show that our method can reduce solve time by up to 6 times in dense systems and up to 5 times in sparse systems that require long solve times with CG relative to their size.

We compare additive and multiplicative domain decomposition preconditioners, without coarse correction, for three non-conforming polynomial discretizations of Helmholtz problems: discontinuous Galerkin, embedded Trefftz discontinuous Galerkin, and hybrid discontinuous Galerkin methods. The preconditioners are studied within stationary and Krylov iterative solvers. All three discretizations lead to complex-symmetric systems and local subproblems with inherited impedance-type boundary conditions. These properties enable the use of conjugate gradients and allow the local matrices to be obtained directly by restriction of the global matrix. Numerical experiments in two and three dimensions demonstrate promising performance for large-scale Helmholtz problems.

Solar Radiation Pressure (SRP), the force exerted by photons emitted by the Sun, is one of the main non-gravitational perturbations affecting spacecraft trajectories, making its accurate modeling essential for high-fidelity orbit propagation. While physically-based ray-tracing models improve SRP accuracy, their computational cost becomes a limitation during numerical integration, where the SRP force must be evaluated repeatedly throughout the trajectory propagation. This paper investigates the integration of high-fidelity SRP models into orbit propagation through two complementary contributions: a Vulkan-based GPU implementation that accelerates direct SRP evaluation, and an extension of the SRP model to account for the dynamic orientation of solar panels. These contributions are evaluated independently through orbit propagation experiments, while a precomputed SRP interpolation strategy (SPAD) is included as an alternative approach for reducing computational cost through offline sampling and interpolation. Numerical validation shows that the Vulkan implementation preserves the accuracy of the original OpenGL-based method, showing relative differences below $5\times10^{-4}$ while achieving speed-ups of up to 9.4 for individual SRP computations and up to 15.2 for complete orbit propagation, particularly for geometrically complex spacecraft. The movable solar panel model shows that neglecting panel motion can produce significant long-term propagation errors, especially for spacecraft with large articulated solar panels, while introducing only a moderate computational overhead. Based on the experimental evaluation, this work concludes with practical guidelines for integrating high-fidelity SRP models into orbit propagation frameworks, identifying the scenarios under which interpolation-based methods are sufficient and when online high-fidelity SRP computation is justified.

We develop a constrained Onsager variational framework for parametric finite element approximations of Willmore and Helfrich flows. The weak curvature relation is used as a PDE constraint to express the first variation of the bending energy in curvature-vector form. Onsager's principle then combines this variation with dissipation based on the normal velocity, while an independent relaxed minimal-deformation-rate (relaxed-MDR) constraint selects the tangential velocity. The resulting mixed formulation admits a semidiscretization by continuous piecewise linear elements that satisfies an exact energy-dissipation law and exactly preserves area and volume when the corresponding constraints are imposed. The framework accommodates spatially varying spontaneous curvature and applies to both closed surfaces and open surfaces with fixed boundaries under Navier or clamped conditions. We also propose a linearly implicit fully discrete scheme, prove its unique solvability, and present numerical experiments demonstrating convergence, energy decay, constraint preservation, and effective mesh redistribution.

It is well established that an appropriate modification of test functions may lead to pressure-robust mixed methods for Stokes problems. However, for immersed finite element approximations of Stokes interface problems on unfitted meshes, it remains unclear whether the velocity error is independent of the pressure, since the velocity and the pressure are coupled in one of the interface conditions. In this paper, we provide a positive answer through a novel decomposition of the discontinuous pressure into a continuous component and a velocity-dependent discontinuous component. We demonstrate that the immersed Crouzeix–Raviart/$P_0$ element method achieves pressure robustness via an $H(\operatorname{div})$-conforming reconstruction of the test functions on the right-hand side. The stability and optimal error estimates of the proposed method are established with constants independent of the interface position relative to the mesh. Numerical experiments are presented to validate the theoretical findings.

This work concerns surrogate modeling for quantities of interest (QoI) arising from parametric non-linear partial differential equations (PDEs). More specifically, we consider extending the sparse-grids surrogate modeling approach to incorporate derivatives of the QoI with respect to the PDE parameters. We discuss why this operation is not straightforward and propose a hybrid approach in which a sparse-grid scheme provides the collocation points in the parameter domain and a suitable polynomial space, but the surrogate model is built with a least-squares approach. We showcase our approach on several numerical tests, and we discuss in particular how its performance crucially depends on the relative cost and accuracy of evaluating the derivatives of the QoI compared to evaluating the QoI itself.

Fluid-structure interaction problems involving three-dimensional fluids coupled with thin elastic plates arise in many engineering applications and present significant computational challenges due to the strong coupling across the fluid-structure interface. In this work, we develop a Schur complement domain decomposition method for a 3D fluid-2D plate interaction problem in which the fluid is modeled by the unsteady Stokes equations and the structure by a reformulated Kirchhoff plate model. Starting from a mixed finite element discretization with Lagrange multipliers enforcing the interface conditions, we derive an interface Schur complement formulation that decouples the fluid and plate subproblems while preserving the strong interface coupling without requiring subiterations. We then analyze the conditioning of the resulting Schur complement system matrix and show how its condition number deteriorates under mesh refinement. Motivated by this analysis, we introduce an interface-based preconditioner and establish theoretical bounds showing that the proposed preconditioner significantly improves the conditioning of the resulting system. Numerical experiments verify the expected spatial and temporal convergence rates, demonstrate significant reductions in condition numbers and GMRES iteration counts for the proposed preconditioner compared with both the unpreconditioned system and a block Jacobi preconditioner, and illustrate the robustness of the proposed method for challenging added-mass regimes.

We propose an asymptotic-preserving micro–macro method bridging a kinetic description of electrons and a low-frequency reduced model in which the electrons are a massless, quasi-neutral fluid obeying the Boltzmann relation. Two features distinguish the construction. First, the fluid and low-Mach limits are coupled, so that the low-Mach stiffness is handled on a macroscopic system, where implicit treatment is affordable, rather than on the kinetic equations. Second, an auxiliary variable rescales the stiff force balance, turning the singular low-Mach limit into a regular limit of the augmented system, which is shown to remain non-degenerate uniformly in the Debye length as well. This matters at the discrete level: with an iterative linear solver, the stiffness induced by the small Mach number amplifies the solver residual, so that a scheme designed to be asymptotic-preserving in its time discretization alone loses that property once the full solution chain is taken into account. The proposed scheme retains it, with no tightening of the solver tolerance as the Mach number vanishes, and admits a post-processing variant that decouples the auxiliary variable and reduces the size of the linear system. Numerical experiments spanning distinct parameter regimes confirm the analysis: standard semi-implicit schemes lose low-Mach-number equilibrium under residual amplification, whereas the proposed schemes preserve it down to round-off.

Let $A$ be Hermitian positive definite and let $f_m$ denote the Lanczos approximation to $f(A)b$. We prove that if $f(z)$ or $f(z) / z$ is Stieltjes, then the $A^\alpha$-norm error of the Lanczos approximation is within a factor $\tfrac{1}{2}(\kappa(A)^{E/2} + \kappa(A)^{-E/2})$ of the the best possible Krylov Subspace Method, where $\kappa(A)$ is the condition number of $A$ and $E = \max\{\alpha,1-\alpha\}$. Our result strengthens and generalizes the upper bound of [Schweitzer; SIMAX, 46.3 (2025)]. Moreover, we prove that the constant $\tfrac{1}{2}(\kappa(A)^{E/2} + \kappa(A)^{-E/2})$ is optimal.

In the framework of the virtual element method with shifted boundary type treatment of curved geometries, we propose a novel approach to the treatment of Neumann boundary conditions that does not require the approximate aligning of the physical and numerical normal directions and is therefore well suited to handle domain approximations obtained by agglomeration from underlying fine structured grids. For such an approach we give conditions under which we are able to prove optimal error estimates for arbitrary orders of the virtual element discretization.

This paper studies a general class of semi-discrete hybridizable discontinuous Galerkin (HDG) methods, including mixed methods, for parabolic problems in nonconvex polygonal and polyhedral domains. By developing local energy error estimates together with energy estimates for a regularized Green's function, we establish a unified framework to prove the maximum-norm stability of both the semigroup defined by the semi-discrete scheme and the corresponding discrete solutions. The stability analysis shows that the main challenges stem from the treatment of numerical flux variables and the inherent asymmetry of the discrete scheme. These challenges are intrinsic to numerical approaches formulated within the mixed framework for parabolic equations. The asymmetry prevents the direct application of the double kick-back argument, while the presence of flux variables requires special techniques to control their values at the initial time. We emphasize that the analytical tools developed here to address these challenges are sufficiently general to be adapted for maximum-norm stability investigations of a broader class of numerical schemes arising from mixed formulations of parabolic equations. Furthermore, by the stability results and their proofs, we derive the maximal regularity of the semi-discrete solution in $L^{\infty}((0,T);L^p(\Omega))$-norm and reduce the maximum-norm error estimates to those of the corresponding elliptic equations and the $L^2$-orthogonal projection. Since the derivation of the local energy error estimates does not rely on the lifting operator, which is only available for simplicial meshes, our results (excluding mixed methods) remain valid for polygonal/polyhedral meshes.

Rectified flows, also called flow matching or stochastic interpolants, are generative models that learn a time-dependent vector field steering a probability curve between two probability distributions, usually referred to as latent and target distributions. Reflow accelerates inference by iteratively straightening the trajectories induced by this vector field. We study the asymptotic behavior of this iteration and characterize its limit points. First, we define weak rectified couplings which always exist. Next, when rectified flow updates are alternated with minibatch optimal transport steps of fixed batch size, we show that any limit is $N$-cyclically monotone, where $N$ is the batch size. Such $N$-cyclically monotone couplings enjoy favorable structural and stability properties such as rectifiability and straightness. Finally, restricting velocities to gradient fields and assuming additional support conditions, we prove that reflow limits coincide with the optimal transport map between the endpoint distributions.

We build an unbiased Monte Carlo estimator of the integral of any $C^1$ function on a prequantized compact symplectic manifold against a smooth Riemannian volume form, taking for quadrature nodes the determinantal point process associated with an appropriate spectral projection of the Bochner-Schrödinger operator. We show that the estimator satisfies a central limit theorem, and the decay rate of the mean squared error reaches the optimal worst-case rate investigated by Bakhvalov in Euclidean spaces. These results extend previous results of Lemoine and Bardenet on Monte Carlo methods on compact complex manifolds.

The Walk-on-Spheres algorithm, introduced by M. E. Muller in 1956, is a well known Monte Carlo method that leverages Brownian exit distributions from spheres to solve the Laplace equation with Dirichlet boundary conditions. Its mesh-free nature, robustness on complex geometries, favorable scaling with dimension, and intrinsic parallelism distinguish it from mesh-based solvers. However, its efficient applicability has been essentially limited to operators that admit explicit probabilistic exit laws, excluding most variable-coefficient and nonlinear elliptic operators. We propose a general framework that aims to overcome this limitation by using the classical Dirichlet Laplacian and harmonic extension as universal building blocks. Rather than seeking a custom stochastic representation for each operator, we employ Walk-on-Spheres to precompute a reusable numerical operator toolbox that approximates the inverse Dirichlet Laplacian, the harmonic extension operator, and their gradients. These precomputed operators are then used to represent candidate solutions and to transform arbitrary Dirichlet boundary value problems into a finite-dimensional algebraic system/optimization problem for an unknown source term. Solving the resulting algebraic system/optimization problem and substituting back yields an approximate solution to the original PDE. Even more, for a general linear second order elliptic operator, the above mentioned precomputed toolbox can be directly used to obtain not just an approximation of a certain solution of the corresponding generalized Dirichlet problem, but an estimator of both the Green's integral operator and the elliptic measure operator. Numerical experiments on a range of benchmarks, including non-symmetric and anisotropic linear elliptic equations, semilinear and quasilinear problems, demonstrate the method's flexibility and efficiency.

This paper develops a high-order surface finite element method for elliptic equations posed on a smooth closed surface implicitly defined as the zero level set of a function in three dimensions. In contrast to classical surface finite element methods that start from a prescribed triangulation of the surface, the proposed method constructs the discrete surface space from the intersections between the exact surface and an ambient tetrahedral mesh. More precisely, an active tetrahedral shell is generated around the implicit surface, and each cut tetrahedron contributes either a triangular or a quadrilateral surface patch according to its intersection pattern with the zero level set. The finite element space is first defined on the resulting piecewise planar cut surface and is then lifted to the exact surface by local level-set-based parameterizations. The resulting method combines features of surface FEM and unfitted/trace methods. Like surface FEM, it produces a conforming finite element space on the exact surface after lifting; however, the geometry and finite element space are induced by intersections with background tetrahedra like unfitted/trace methods. We prove that the local lifting maps agree pointwise across common faces and assemble into a global homeomorphism. We also derive explicit formulas for tangent vectors, Gram matrices, mass and stiffness integrals, and prove stability and high-order derivative estimates for the lifting maps. The latter estimates are formulated in broken Sobolev norms and lead to a Cea-type energy-norm error bound for the proposed high-order lifted surface finite element method.

Higher-order finite element methods are effective for solving partial differential equations. However, applying them in complex curved domains is often difficult due to challenges in creating high-quality curvilinear meshes. Unfitted, or embedded, methods provide a valid alternative by avoiding complete mesh generation and curved element integration, but their accuracy can suffer from geometric errors introduced during embedding. In this work, we explore a new family of polynomial corrections obtained by solving a local minimization problem to improve the consistency of embedded boundary methods. This approach generalizes existing techniques such as the shifted boundary method (SBM) and the reconstruction for off-site data (ROD) method. Unlike the SBM, which depends on truncated Taylor expansions, the proposed family of polynomial corrections is derived from a constrained minimization problem, similar to the ROD method. We demonstrate that this approach yields better system conditioning than the original SBM and provides a flexible framework that can be applied pointwise, without solving the full ROD linear system for each boundary element. This paper presents four formulations based on different functionals and extends the family of minimization-based polynomial corrections to handle Neumann and Robin boundary conditions, demonstrating that an elegant formulation is possible in this context. Several numerical experiments for the Poisson problem are presented to show that the generalized polynomial corrections achieve high-order accuracy across all boundary conditions.

We present the formulation of a boundary value problem that models the coupled behavior of a three-dimensional diffusive solid with one-dimensional diffusive fibers embedded inside it. We introduce a variational statement of the problem that identifies the linked diffusive fields as energy minimizers under a coupling constraint. This saddle-point problem is proved to be well posed. Then, we introduce a finite element discretization of the proposed boundary value problem, and we prove the convergence of the finite element solution to the exact one. The most significant feature of this approximation is that the meshes of the bodies need not be conforming. Numerical examples confirm the theoretical results

This work analyzes a method for time integration of high-dimensional linear Schrödinger-type problems based on hierarchical tensor approximations. In particular, this method provides a balance between error bounds and associated approximation ranks, using a scheme for iterative refinement with soft thresholding of tensors. The practical performance of the method is illustrated by numerical tests on coupled oscillators.

Classical Barron spaces are function spaces specifically designed for shallow neural networks mostly with ReLU, $\mathrm{ReLU}^k$ (RePU) or Lipschitz continuous activation functions. In the present work, we introduce a generalized Barron space \( B_{\sigma}^{\varphi} \) for shallow neural networks with a generic activation function possessing certain smoothness properties. The subscript \( \sigma \) denotes the activation function, while the superscript \( \varphi \) controls the smoothness of the generalized Barron spaces, defined via a \( \varphi \)-weighted integral norm imposed on neural network parameter measures. Under certain assumptions on \( \varphi \) and \( \sigma \), we show that \( B_{\sigma}^{\varphi} \) can be continuously embedded into Sobolev spaces. We also explore the relationships among various function spaces for shallow neural networks, demonstrating that our definition encompasses most conventional ones. As applications of the proposed generalized Barron spaces, we derive approximation rates within these spaces and establish error bounds for numerical differentiation with regularization penalized by the newly introduced generalized Barron norm. Numerical examples confirm that the proposed spaces allow the construction of neural networks with varying degrees of smoothness while using the same activation function.

In this paper, we extend a weak adversary neural network pushforward method for solving time dependent Boltzmann equation and a weak formulation of the collision operator is proposed where an invertible neural pushforward mapping is used to generating samples given by the distribution governed by the Boltzmann equation. The training of the pushforward mapping is learnt by enforcing the weak form of the Boltzmann equation. Numerical results have demonstrated the effectiveness of the proposed method.

We develop a rigorous theory of discrete residual least-squares approximation for elliptic spectral equations $\mathfrak L_\beta u=f$ using linearized ReLU$^k$ neural networks on the sphere, where $\mathfrak L_\beta$ is a positive elliptic spectral multiplier of order $\beta$. Given a parameter set $\Theta_n=\{\theta_{j}^*\}_{j=1}^n\subset\mathbb S^d$, we approximate $u$ in the linearized network space $L_n^k(\Theta_n)$ by the discrete residual on the collocation points $\{\eta_i^*\}_{i=1}^m$ \begin{equation*} u_{n,m}\in\arg\min_{v_n\in L_n^k(\Theta_n)}\frac1m\sum_{i=1}^m\left(f(\eta_i^*)-\mathfrak L_\beta v_n(\eta_i^*)\right)^2. \end{equation*} With $k>\frac{d-1}{2}+\beta$, for antipodally quasi-uniform network parameter sets and any quasi-uniform collocation points with $m\gtrsim n$, we prove that \begin{equation*} \|u-u_{n,m}\|_{\mathcal H^{\beta}(\mathbb S^d)}\eqsim\|f-\mathfrak L_\beta u_{n,m}\|_{\mathcal L^2(\mathbb S^d)}\lesssim n^{-\frac{r}{d}} \begin{cases} \|f\|_{\mathcal W^{r,p}(\mathbb S^d)},&\frac{d}{p}<r\leq \frac{d}{2},~p>2,\\ \|f\|_{\mathcal H^r(\mathbb S^d)},&r>\frac{d}{2}. \end{cases} \end{equation*} We also establish a high-probability residual estimate, up to a logarithmic factor and an arbitrarily small smoothness loss, for i.i.d.\ uniformly distributed collocation points. The key analytical ingredient is a Bernstein inequality for linearized ReLU$^k$ network spaces. If $\underline h$ denotes the antipodal separation distance of the network parameters, then \begin{equation*} \|v_n\|_{\mathcal H^r(\mathbb S^d)}\lesssim\underline h^{-(r-s)}\|v_n\|_{\mathcal H^s(\mathbb S^d)},\qquad 0\leq s<r<k+\tfrac12. \end{equation*}

The weighted half-period cosine space has often been employed in the theory of quasi-Monte Carlo methods for multivariate integration and approximation of non-periodic functions. For integer-order smoothness, its norm equivalence to certain weighted unanchored Sobolev spaces has been established in the literature. In this work, we extend this equivalence to fractional-order smoothness up to $2$. By introducing an explicit representation via Slobodeckij-type seminorms, we prove a norm equivalence between the half-period cosine spaces and the corresponding weighted unanchored Sobolev spaces. Our Sobolev norm representation clarifies how the fractional regularity dictates the presence or absence of boundary constraints and (non-)periodic structures. Furthermore, we investigate the limiting behavior of these fractional spaces as the smoothness parameter approaches integer boundaries, establishing a continuous bridge to the classical integer-order Sobolev spaces. These equivalence results enable us to transfer the near-optimal error bounds and tractability results for multivariate integration and function approximation from the half-period cosine settings to our newly introduced fractional Sobolev spaces.

Physics-Informed Kolmogorov-Arnold Networks have recently emerged as an effective class of neural solvers for partial differential equations, combining the expressive power of spline-based Kolmogorov-Arnold representations with physics-informed learning. However, existing PIKAN formulations are primarily developed for Cartesian domains and cannot naturally accommodate the geometric complexity introduced by curvilinear domains. In this work, we propose Coordinate-Transformed Physics-Informed Kolmogorov-Arnold Networks (CT-PIKAN), a geometry-aware framework for solving PDEs on arbitrarily shaped domains through coordinate transformation. A smooth mapping transforms the physical domain into a regular computational domain, while the transformed governing equations are enforced directly within the physics-informed loss. Unlike conventional transformed PINN approaches that require manually derived metric coefficients, CT-PIKAN employs automatic differentiation to evaluate Jacobians, metric tensors, and transformed differential operators directly from the coordinate mapping, eliminating analytical derivations and improving implementation flexibility. To establish the proposed framework, a data-free B-spline-based PIKAN is first constructed and validated on the two-dimensional advection equation. The CT-PIKAN methodology is subsequently assessed on representative elliptic, parabolic, and hyperbolic benchmark problems, including the Poisson, heat, and advection equations formulated in polar and wavy curvilinear coordinates. The proposed framework provides a general and extensible methodology for integrating differential geometry with physics-informed Kolmogorov-Arnold networks, enabling efficient and accurate PDE solutions on complex domains.

Entropy inequalities are fundamental to the well-posedness of hyperbolic conservation laws, providing the essential criterion for selecting the physically admissible solution among infinitely many weak solutions. Chen and Shu [J. Comput. Phys. 345 (2017)] proposed a unified framework for constructing high-order discontinuous Galerkin (DG) methods that satisfy entropy inequalities for any given entropy via specific numerical quadrature; however, their accompanying error analysis was limited to the truncation error level, leaving a critical gap in the rigorous convergence theory for these entropy-stable schemes. This paper closes that gap by establishing rigorous a priori error estimates for semi-discrete entropy-stable DG (ESDG) methods on general unstructured meshes for hyperbolic conservation laws. The analysis applies to both scalar equations and systems, and is built upon a finite-difference-type consistency-stability argument carried out directly at the nodal level. Under a polynomial-reconstruction hypothesis and an $L^\infty$ a priori bound, we prove an $O(h^k)$ error estimate in a quadrature-based norm, which is equivalent to the broken $L^2$ norm on the finite-dimensional reconstruction space. We further extend this framework to the entropy-stable oscillation-free DG (ESOFDG) method introduced by Liu, Lu, and Shu [SIAM J. Sci. Comput. 46 (2024)], demonstrating that the additional damping terms do not degrade the convergence order. Numerical experiments suggest that the observed convergence rates may exceed the theoretical prediction by up to half an order.