Papers

This paper investigates semantic-aware neural joint source-channel coding (JSCC) for robust video transmission over block erasure channels. We propose a neural video compression framework exploring both spatial-domain and feature-domain designs. In the spatial domain, video frames are partitioned into blocks, enabling localized erasure handling and fine-grained robustness control via uniform erasure and two-level, semantic-guided non-uniform erasure strategies. In the feature domain, latent features are partitioned, enabling missing features to be semantically recovered while maintaining overall spatial consistency. Comprehensive experiments quantify reconstruction quality under varying uniform and non-uniform erasure probabilities. Our results show that spatial-domain JSCC excels at handling random localized losses, whereas feature-domain JSCC provides superior robustness to distributed erasures and maintains fidelity under low-loss scenarios. The analysis highlights the trade-offs between spatial continuity and semantic redundancy, offering insights for designing robust, task-aware video communication systems.

We consider the relative entropy between the vacuum and a coherent state in the Rindler wedge for an interacting $\lambda \phi^4$ theory to first order in $\lambda$. We construct the perturbatively interacting Weyl algebra of the wedge, and employ Tomita-Takesaki modular theory and the Araki-Uhlmann formula to compute the relative entropy. We verify that the relative entropy reduces to the classical (interacting) boost Noether charge, analogously to the free theory, and that the Bekenstein bound holds.

We show that the separation profile of a hyperbolic planar graph and, more generally, a hyperbolic apex-minor-free graph, grows at most logarithmically, answering a question of Benjamini, Schramm, and Timár in the affirmative.

Chiral algebras and topological defect lines (TDLs) represent two complementary notions of symmetry in 2d conformal field theories. In this paper, we introduce chiral tube algebras to unify and extend these two notions. Chiral tube algebras generalize chiral algebras in two ways. First, they extend the action of chiral algebras beyond the local Hilbert space to include defect Hilbert spaces twisted by TDLs. Second, they allow for non-local chiral currents attached by TDLs and thus can map between different defect Hilbert spaces, analogous to the tube algebras of TDLs. Since local chiral currents can become non-local after finite gauging, chiral tube algebras provide a natural framework for describing the image of chiral algebras under such gauging. We illustrate this framework through a variety of examples that generalize familiar chiral algebras, including Kac-Moody algebras, $\mathcal{W}$ algebras, superconformal algebras, and their orbifolds/bosonizations. We construct their irreducible modules, which are isomorphic to twisted modules of the corresponding chiral algebras, and use them to organize local and defect Hilbert spaces. In a subsequent paper, we will study chiral tube algebras generated by non-local chiral currents with fractional spins, which have no counterparts in chiral algebras.

We introduce categories of strings to give an explicit hom-object formula for colimits of enriched categories. As corollaries, we recover the necklace construction by Dugger and Spivak and describe colimits in $\mathrm{Cat}$ and of Lawvere metric spaces.

Bubeck, Li and Nagaraj conjectured that, for generic data, any two-layer neural network with $m$ neurons that fits $n$ noisy labels must have Lipschitz constant at least of order $\sqrt{n/m}$, with no restriction on the size of the weights. Bubeck and Sellke proved a universal version of this law for Lipschitz-parameterized classes, but under a polynomial bound on the parameters; at depth three that boundedness hypothesis is genuinely necessary. The two-layer unbounded-weight case requires a different argument. We prove the conjectured law, up to one logarithmic factor, for every continuous piecewise-linear activation, in particular for ReLU networks. For data drawn uniformly from $\mathbb{S}^{d-1}$, $d\ge3$, or from $N(0,I_d/d)$, labels in $[-1,1]$ with noise level $\sigma^2>0$, and any width-$m$ two-layer network with arbitrary real weights, biases and affine skip connection, fitting the data $\varepsilon$ below the noise floor forces $\mathrm{Lip}(f)\ge c\,\varepsilon\sqrt{n/(\bar m\log(C\bar m nd/\varepsilon))}$, $\bar m=(K-1)m+1$, with high probability. A realized-kink-count version holds on the same event: every realized two-layer piecewise-linear function with $k(f)\le n$ distinct kink hyperplanes obeys the bound with $\bar m$ replaced by $k(f)+1$, irrespective of how many redundant hidden units parameterize it. The proof replaces parameter-space covering, impossible for unbounded weights, by a function-space covering. The central deterministic ingredient is a rigidity lemma: on $B_2$, and on $\mathbb{S}^{d-1}$ for $d\ge3$, the coefficient of each canonical kink is controlled by the Lipschitz constant of the realized function, because kinks on distinct hyperplanes cannot cancel at generic points. Rigidity genuinely fails at $d=2$, and an explicit two-layer ReLU interpolant with $O(1)$ Lipschitz constant at width $2n$ matches the law at the overparameterized endpoint.

Friedland et al. [PRL 129, 110402 (2022)] proposed and studied a quantum analogue of the $p$-Wasserstein distance based on quantum cost matrices and quantum couplings. They conjectured that, despite being only a semidistance in general, this quantity is a true distance for a particular quantum cost matrix and for cost matrices in a small neighborhood of it. We disprove these conjectures by exhibiting an explicit family of triples of states for which the triangle inequality fails.

Modern machine learning (ML) increasingly relies on complex models whose behavior is difficult to characterize beyond empirical performance metrics. Across a wide range of tasks, including prediction, generation, and decision-making, models with similar empirical performance can exhibit markedly different properties in terms of their transparency, interpretability, robustness, fairness, privacy, and certifiability. This survey highlights how optimization- and certification-oriented reasoning can provide a useful framework for reasoning about such differences, supporting tasks ranging from model training and selection to auditing and certification. We review and synthesize recent advances at the intersection of combinatorial optimization (CO) and trustworthy ML, covering both training and post-training tasks, including interpretable model learning, explanation generation, robustness analysis, fairness auditing, model compression, and privacy attacks and protections. Across these domains, CO formulations offer additional capabilities over purely heuristic approaches, e.g., gradient-based ones, notably global guarantees, formal certificates, and explicit treatment of trade-offs. While scalability remains an important challenge, continued progress in solvers and hybrid algorithms suggests a growing role for CO in the design and deployment of trustworthy ML systems.

This paper addresses the matrix Waring problem for matrices over finite principal local rings. Let $\mathcal{O}_{\ell}$ be a finite principal local ring of length $\ell$ with the maximal ideal $\mathfrak{m}$ and the residue field $\mathbb{F}_q = \mathcal{O}_\ell/\mathfrak{m}$. When $-1$ is a $k$-th power in $\mathbb{F}_q$ and the characteristic of $\mathbb{F}_q$ does not divide $k$, we show that for sufficiently large $q$, any matrix in $M_n(\mathcal{O}_\ell)$ can be expressed as a sum of two $k$-th powers. Furthermore, we establish that these two conditions are strictly necessary for the result to hold in general.

We approach discrete and differential Faà di Bruno formulas from a Möbius inversion angle. On the Boolean cube, Newton's discrete Taylor formula and the definition of iterated forward differences form a zeta–Möbius dual pair, and composing two Taylor expansions and inverting once yields a closed discrete Faà di Bruno formula at a fixed basepoint: for arbitrary maps $f, g$ between abelian groups, $$ \Delta(f \circ g;\,x;\,u_1,\dots,u_k) = \sum_{H \in \mathrm{Cov}(k)} \Delta(f;\,g(x);\,(\Delta(g;x;u_T))_{T\in H}), $$ where $\mathrm{Cov}(k)$ denotes the coverings of $[k]$ by nonempty subsets. Grouping repeated directions gives binomial versions on multi-index grids, and iterating gives formulas for $m$-fold composites, with integer covering coefficients governed by explicit cross and level recursions, a discrete analogue of the Constantine–Savits formulas. The relationship between coverings and partitions appearing in classical Faà di Bruno formulas is exhibited in an algebraic setting. The discrete formulas are Taylor expansions over the function algebra of the Boolean cube, whose idempotent generators absorb overlapping products; in the differential analogue nilpotent generators annihilate overlaps and only partitions remain. We demonstrate how these algebraic identities can be lifted to the analytical setting of $C^n$ maps between Banach spaces, recovering the multivariate Faà di Bruno formula of Constantine–Savits and extending it to composites of several maps. Boolean finite differences, binomial grid formulas, infinitesimal Taylor algebras, and Fréchet derivatives thus appear as four realizations of one Möbius-dual Faà di Bruno formula, connected by a flat family.

We specialize Agarwal's multi-level collective spontaneous-emission formalism to the four-level case by formulating it in the fully symmetric \SU(4) representation of $N$ identical atoms. In the irreducible representation $(N,0,0)$, the occupation-number basis forms a tetrahedral weight lattice on which the six embedded $\mathfrak{su}(2)$ transition subalgebras act as ladder operators. From these algebraic factors we obtain a compact Pauli-type population-rate equation and a closed-form expression for the total emitted intensity that apply to any combination of open dipole channels. The formalism is then specialized to the seven dipole-allowed four-level topologies – tripod, inverted tripod, Y, inverted Y, double-$\Lambda$, closed cascade, and diamond – and the resulting rate equations are solved numerically for atom numbers up to $N=50$. In every case the emitted intensity develops a delayed cooperative burst whose peak height obeys a power law $I_{\mathrm{peak}}=aN^{p}$ with topology-dependent parameters $(a,p)$; the fitted exponents lie in the range $1.81\lesssim p\lesssim 1.92$, indicating a superlinear. The \SU(4) tetrahedral flow and the seven configuration-dependent transients together provide a unified geometric picture of multi-channel collective dissipation in four-level atomic ensembles.

The induced Erdős–Pósa property in graphs relates the maximum number of pairwise anti-adjacent copies of an object with the minimum number of neighborhoods required to hit all copies. In this paper, the objects we consider are long cycles and long thetas, both as induced minors. Let $C_t$ denote the cycle with $t$ vertices and let $\Theta_t$ be the graph consisting of three internally disjoint and anti-adjacent paths, each with $t$ internal vertices, connecting the same pair of distinct vertices. We show that for every fixed $t$, both $C_t$ and $\Theta_t$ have the induced Erdős–Pósa property with respect to the induced minor relation. More precisely, for every integer $k$ and every graph $G$, one of the following two outcomes occurs: (i) $G$ contains $k$ pairwise vertex-disjoint and anti-adjacent copies of $C_t$ (resp., $\Theta_t$) as induced minors, or (ii) there is a set $X \subseteq V(G)$ of size $\mathcal{O}(tk \log k)$ such that the set $N[X]$, consisting of $X$ and its neighbors, hits all $C_t$ (resp., all $\Theta_t$) induced minors in $G$. This resolves in a strong form a special case of a conjecture of Ahn, Gollin, Huynh, and Kwon [SODA 2025]. From these results we derive that graphs that exclude $k$ disjoint copies of $\Theta_t$ as an induced minor admit balanced separators consisting of the neighborhood of $\mathcal{O}(tk \log k)$ vertices. This in turn resolves a special case of a conjecture of Gartland and Lokshtanov and, combined with known techniques, yields a QPTAS for Maximum Weight Independent Set and a number of its generalizations.

We study minimum-norm interpolation (MNI) in overparameterized linear regression with isotropic Gaussian covariates, in settings where the MNI has no closed-form formula. Whereas most prior work relied on Gaussian comparison tools such as the convex Gaussian min–max theorem (CGMT), our approach uses tools from high-dimensional geometry and probability. First, when the norm is in isotropic position, we obtain an “offset” bound that controls the amount by which the MNI shrinks the ground truth. Second, we show that the “intrinsic” variance of the $\ell_1$-MNI is at most $O(\tfrac{1}{n\log(d/n)^2})$, using a variant of Talagrand's $L_1$–$L_2$ inequality due to Cordero-Erausquin and Ledoux [2012], together with a classical result of Gluskin [1988]. We recover the sharp mean-squared error (MSE) bound for the $\ell_1$-MNI obtained by Wang et al. [2022], using the work of Fleury [2012] on the symmetric Gaussian polytope, which is defined via \[ P_{n,d} := \mathrm{conv}\{\pm X_i\}_{i=1}^{d} \text{ where } X_i \overset{\mathrm{i.i.d.}}{\sim} N(0,\mathrm{I}_{n \times n}), \] rather than CGMT. Our methods also imply improvements on previous results in high-dimensional geometry that may be of independent interest. First, we show that with overwhelming probability, the ratio between the isotropic constant of $P_{n,d}$ and that of the Euclidean ball in $\mathbb{R}^n$ is at most $1+O((\log(d/n))^{-2})$, improving a result of Klartag and Kozma [2009]. We also establish a refined weighted thin-shell estimate on $P_{n,d}$, and provide an elementary proof of the main theorem of Fleury [2012].

The Baker–Campbell–Hausdorff (BCH) formula plays a critical role in many branches of mathematics and physics. It expresses the logarithm of the product of exponentials of non-commuting operators as an infinite series of nested commutators of the operators involved. The Zassenhaus formula is the dual of the BCH formula: the exponential of a sum of operators is written as an infinite product of exponentials involving the operators and their commutators. In practical computations, however, one typically has to truncate the expansions, and so understanding the error committed by the resulting approximations and eventually providing suitable bounds for this error is of paramount interest. In this work we present a general strategy to derive rigorous error bounds and explicit error constants for the BCH and Zassenhaus formulas when the operators involved are skew-adjoint, as is the case for quantum evolution problems.

The spectral condition number is a widely adopted measure of worst-case cost for quantum linear system solvers. Yet it can significantly overestimate the actual runtime for a typical problem instance. We present two quantum algorithms that produce the normalized solution $|x\rangle$ of linear system $Ax=| b \rangle$ to accuracy $\epsilon$ with complexity independent of the condition number $\kappa=\lVert A^{-1}\rVert$. We focus on the standard input model where $A$ is accessed through a block encoding and $| b \rangle$ is prepared by a unitary. But we also introduce an affine dilation model that encodes $A$ and $| b \rangle$ jointly, allowing further refinements of the query complexity. Our truncation-based solver makes an optimal number of queries to $| b \rangle$ and $\operatorname{\mathbf{O}}\left(\kappa_{\mathrm{eff}}\operatorname{polylog}\left(\frac{\kappa_{\mathrm{eff}}}{\epsilon}\right)\right)$ queries to $A$. We prove a family of upper bounds on the effective condition number, including $\kappa_{\mathrm{eff}}\leq\frac{\lVert(A^\dagger A)^{-t/2}|x\rangle\rVert^{1/t}}{\epsilon^{1/t}}$ for positive even integer $t$ and $\kappa_{\mathrm{eff}}\leq\frac{\lVert A^{-1\dagger}(A^\dagger A)^{-(t-1)/2}|x\rangle\rVert^{1/t}}{\epsilon^{1/t}}$ for positive odd $t$, overcoming the $\kappa$-barrier. Our filtering-based solver is extremely simple with a favorable runtime prefactor. In particular, the solver has query complexity $3\frac{\lVert A^{-1\dagger}|x\rangle\rVert}{\epsilon}$ to leading order when the solution norm is known. We then present a similarly simple solution norm estimator with the same asymptotic cost up to logarithmic factors. Our quantum linear system solvers thus substantially improve a recent algorithm of Li, enabling faster quantum linear system solving beyond the condition number.

How close to singularity can an $n \times n$ unimodular matrix be? For ternary cases as $n$ increases, exact expressions are unlikely, but upon fixing $n=4$ and assessing $(2k+1)$-ary cases as $k$ increases, we make significant progress; similarly for $(k+1)$-ary cases of $4\times 4$ nonnegative unimodular matrices.

Many machine learning models are defined for inputs of different sizes, such as point clouds containing different numbers of points, sequences of tokens of different lengths, and graphs on different numbers of nodes. Such models are trained on finitely many examples of necessarily limited sizes. How well do these models generalize from inputs of small size to larger inputs of size not seen during training? Furthermore, evaluating such models on large inputs is often expensive. How can we sketch large inputs to obtain smaller ones on which the model takes similar values? At the heart of both questions is the need to compare inputs of different sizes and to approximate large inputs by small ones. We present a unified approach to address these questions by using random sampling maps to compare inputs of different sizes. The sampling maps we consider are generalizations of sampling with replacement, random binning, and species sampling. We characterize the application domains in which each type of sampling is appropriate in terms of the symmetries and relations between problem instances of different sizes in the domain. Our framework yields explicit generalization and sketching rates for function classes continuous with respect to a chosen notion of sampling, encompassing large families of functions defined on sequences, graphs, and tensors of different sizes. Specific examples include moment polynomials on measures, homomorphism densities and numbers of graphs, permutation-invariant transformers, and graph neural networks.

We study first-order locally convex Lie algebroids in the setting of Bastiani calculus. The first-order condition is automatic in finite rank, but is an additional regularity hypothesis for general locally convex vector bundles. We define sheaves of scalar-valued and vector-valued Lie algebroid forms as fiberwise continuous alternating maps with smooth local representatives. Morphisms are defined by requiring the induced pullback on inverse-image sheaves of scalar-valued forms to commute with the Lie algebroid differentials. We prove that these algebroids form a category, study representations and cohomology sheaves, and show that the Lie algebroids and morphisms of locally convex Lie groupoids belong to this category. We also give two applications. For a compact manifold $K$, the current algebroid $C^\infty(K,A)\to C^\infty(K,M)$ of a first-order Banach Lie algebroid $A\to M$ is a first-order Fréchet Lie algebroid, provided $M$ admits local addition. Finally, we prove a Banach Lie II theorem: a morphism from the Lie algebroid of a locally convex Lie groupoid to that of a Banach-Lie groupoid, both with Hausdorff source fibers, integrates uniquely to a groupoid morphism when the source groupoid is source-connected and source-simply connected.

Schubert calculus on complex Grassmannians can be played by means of differential operators acting on Schur polynomials or Vertex Operators acting on exterior algebras. In this paper we develop this point of view systematically and complement it with a parallel exterior-algebra formalism, leading to what we call, respectively, the bosonic and the fermionic Schubert calculus. The two alluded realizations are related by (a finite type version of) the boson–fermion correspondence, thereby providing a unified framework connecting Schubert calculus and integrals on the Grassmannian, symmetric functions, exterior algebras and the representation theory of symmetric groups.

Classifier-free guidance (CFG) is the standard way to strengthen class-conditioning in diffusion and flow-matching samplers, yet at large guidance it oversaturates and destabilizes, symptoms practitioners suppress with more steps or limited-interval schedules. We analyze CFG through an asymptotic-preserving, numerical-analysis lens. Building on a recent result that the deterministic DDIM step is the unique fitted operator for the unguided terminal layer, exact on the final small-sigma stretch of sampling, we show that guidance re-stiffens exactly the discriminative subspace to an anomalous exponent 1+w. DDIM is therefore no longer fitted there, and on coarse meshes its guided residual diverges as sigma_min goes to zero. We prove a guided clock barrier with three ordered step-size thresholds, and read one-step oversaturation as its endpoint: a solver artifact on the calibration model rather than the continuous guided law. The same analysis yields a one-coefficient, zero-extra-NFE repair: replace CFG's w(r-1) by r^(1+w)-r on the guidance direction. On the calibration model's discriminative crossover, this removes CFG's sigma_min-divergent blow-up and is first-order accurate against the exact guided flow as sigma_min goes to zero. On learned CIFAR-10 checkpoints, and as a cross-domain smoke test on Stable Diffusion 1.5 DDIM, it acts as a high-guidance stabilizer at no extra cost rather than a universal quality knob: it cuts residual amplification and saturation, gives 9/9 point-FID wins over CFG on the tested grid, and preserves classifier-proxy target accuracy in the hard-cell blocks. We report the limits alongside: it is not a universal image-quality win, and against a dense vanilla-CFG reference it is not a uniformly better integrator of that field.

We describe the set of palindromic Poincaré polynomials in $B_n$. As a key ingredient in this classification, we prove that $A_n$ embeds into $B_n$ as a lower interval in Bruhat order. Consequently, all Poincare polynomials $P_w(q)$ and all Kazhdan–Lusztig polynomials $P_{x,y}(q)$ of $A_n$ occur in $B_n$.

The paper establishes extra invariant in magnetohydrodynamics (MHD) of rotating and stratified fluid layer. This invariant is conserved adiabatically, i.e. approximately over long time. The existence of the invariant is interesting by itself, as such invariants are extremely rare. However, in addition, this invariant appears to be connected to the famous dynamo phenomenon.

We study the Gelfand-Tsetlin realization of generic Verma modules for the affine Lie algebra $\widehat{\mathfrak{gl}}_n$ by viewing them as thin modules over the affine Yangian $Y(\widehat{\mathfrak{sl}}_n)$. By results of arXiv:0812.4656, these modules admit a basis indexed by periodic Gelfand-Tsetlin patterns with explicit formulas for the Yangian action, and we identify them with the evaluation modules introduced by Kodera arXiv:1806.09884. Our main result describes the specialization from generic highest weights to dominant highest weights (not necessarily integral). We call the resulting construction athinization: an irreducible $\widehat{\mathfrak{gl}}_n$-module, which is not thin as a module over the affine Kac-Moody algebra, is realized as a thin module over the larger (and ”more affine”) algebra $Y(\widehat{\mathfrak{sl}}_n)$. Combinatorially, this realization is obtained by restricting the generic periodic Gelfand-Tsetlin basis to a distinguished subset of permitted patterns. We prove that the span of these patterns carries a well-defined affine Yangian action. In particular, this construction yields explicit Gelfand-Tsetlin-type bases for admissible representations of $\widehat{\mathfrak{gl}}_n$ in the sense of Kac-Wakimoto, providing a new combinatorial realization of these modules. We compare the formulas for characters coming from this combinatorics with those for minimal models of $W$-algebras of the type $A_n$ via the principal specialization. Further, we obtain analogous results for representations of $U_q\widehat{\mathfrak{gl}}_n$ via their realization as thin modules over the quantum toroidal algebra of $\mathfrak{gl}_n$.

We propose a new abc-type conjecture. We motivate the conjecture and illustrate its relevance through several applications. Our main result concerns the function $$ W(x,y) := \sum_{j = 1}^{y}\omega(x+j) \quad (y \in \mathbb{N},\ x \in \mathbb{Z}_{\ge 0}) $$ where $\omega(n)$ denotes the number of distinct prime divisors of $n$. The new conjecture implies that, for each fixed $y \in \mathbb{N}$, $$ \limsup_{x \to \infty} \frac{W(x,y)\log\log x}{\log x} = 1. $$

We prove rates of decay at infinity for solutions to variable-coefficient Schrödinger equations of the form $-\text{div}(A \nabla u) + W \cdot \nabla u + V u = \lambda u$ in cylinders, $\mathbb{T}^d \times \mathbb{R}^m$. We assume that $W$ and $V$ are bounded and that $\lambda \in \mathbb{C}$. Our rates depend on the decay of $|\nabla A|$ at infinity. In particular, we prove a range of quantitative unique continuation-type results at infinity when $|\nabla A(\theta, x)| \le C (1 + |x|)^{-\tau}$ for $\tau \in [0,1]$. By adapting the methods in [KLP25], we construct explicit solutions to demonstrate the sharpness of our estimates for each such $\tau$.

For integers $l>0$ and $m\geqslant0$, we introduce the numbers $$S_l^{(m)}(n) = \sum_{k_1,\ldots,k_l\in\mathbb N\atop k_1+\cdots+k_l = n} \binom n{k_1,\ldots,k_l}^m \ \ (n=0,1,2,\ldots),$$ and prove that for any prime $p$ not dividing $l+1$ we have the congruence $$\sum_{n=1}^{p-1}\frac{(-1)^{mn}}{n^{m-1}}S_l^{(m)}(n)\equiv0\pmod p.$$ We also obtain a $q$-analogue of this result. For the Domb numbers given by $$D(n)=\sum_{k=0}^n\binom nk^2\binom{2k}k\binom{2(n-k)}{n-k}=S_4^{(2)}(n)\ \ (n=0,1,2,\ldots),$$ we confirm a previous conjecture which states that $$\sum_{n=1}^{p-1}\frac{D(n)}n\equiv\left(\frac p3\right)\frac 25pB_{p-2}\left(\frac13\right)\pmod{p^2}$$ for any prime $p$, where $(\frac p3)$ is the Legendre symbol, and $B_{p-2}(x)$ is the Bernoulli polynomial of degree $p-2$.

This work presents a novel approach for adapting neural network architecture along the depth based on a posteriori error estimation. By formulating neural network training as a continuous-time optimal control problem, we derive rigorous error estimates that quantify how approximation error distributes across network layers. This error decomposition enables a principled depth adaptation strategy: new layers are inserted at locations of maximum estimated error, allowing the network to efficiently capture complex, nonlinear variations in the underlying problem. Our framework introduces a novel network architecture that treats weights and biases as piecewise linear functions varying across layers, with the error estimator bounding the discrepancy between this discrete representation and the true continuous optimal control solution. The approach leverages dual weighted residual methodology from finite element analysis to derive computable upper bounds on the functional error. A key theoretical contribution is the derivation of explicit error bounds that decompose the total approximation error into interval-wise contributions, providing a rigorous basis for targeted architecture refinement. We demonstrate the effectiveness of our method on scientific datasets, including learning the observable-to-parameter map for the Navier-Stokes equation. Numerical results reveal that our approach consistently outperforms existing architecture adaptation methods in terms of generalization performance.

This work investigates a Hénon-type equation with the cap-shaped weight $V_{\alpha}(|x|)=(4|x|(1-|x|))^\alpha$, which concentrates on the sphere of radius $1/2$ as $\alpha\to\infty$. Particular attention is devoted to ground state solutions and the phenomenon of symmetry breaking in the large-$\alpha$ regime. A complete description of the asymptotic behavior of ground state radial solutions is obtained as $\alpha\to\infty$. The analytical results are further supported by numerical simulations of the radial ground states.

In this note, we introduce an enhanced notion of probabilistic well-posedness for dispersive PDEs with random initial data by imposing stability at the origin as the amplitude of randomization tends to $0$. We then use this notion to re-interpret recent works on "beyond variance blowup" for dispersive PDEs, by the authors with their collaborators (2025, 2026), as probabilistic ill-posedness results. By drawing an analogy to the failure of $C^k$-smoothness of a solution map in the deterministic setting, we interpret variance blowup results as mild probabilistic ill-posedness.

We study well-posedness issues of the stochastic Korteweg-de Vries equation (SKdV) with an additive noise, posed on the real line. By using the Fourier restriction norm method adapted to the Fourier-Lebesgue space in time, we first prove global well-posedness of SKdV in $L^2(\mathbb R)$ without assuming the homogenous Sobolev regularity, which was imposed in a work by de Bouard, Debussche, and Tsutsumi (1999). Then, by adapting the argument by Zhou (1997) to the stochastic setting, we prove optimal pathwise unconditional uniqueness for SKdV in $L^2(\mathbb R)$. In the appendix, we present a short argument for proving boundedness of the multiplication by a sharp cutoff function in the Fourier-Lebesgue and Sobolev spaces, which is of interest in its own right.