Papers

We study a non-isothermal model for incompressible magnetoviscoelastic fluids in $\mathbb{R}^N$, $N=2,3$, coupling the incompressible Navier-Stokes equations with the evolution equations for the deformation tensor, temperature, and magnetization. The material coefficients are allowed to depend on the temperature, and no artificial diffusion is imposed on the deformation tensor, leading to a coupled hyperbolic-parabolic system. We establish local well-posedness of strong solutions for sufficiently regular initial data and prove global existence for small perturbations of a constant equilibrium. The global analysis relies on a suitable auxiliary variable combining the velocity and the inverse deformation tensor, which reveals a hidden dissipative structure and allows us to derive uniform higher-order energy estimates for the coupled system.

We prove two dimension-free estimates in Gaussian space. The first is an optimal Meyer-type inequality for the Gaussian divergence: for $p\geq2$, \[ \|\delta V\|_{L^p(\gamma_d)} \leq \sqrt p\,|\mathbb E V| +Cp\,\|DV\|_{L^p(\gamma_d;\mathrm{HS}_d)}\, . \] Here $\mathrm{HS}_d$ is the space of $d\times d$ matrices with its Hilbert-Schmidt norm. Its main ingredient is a dimension-free weak-type $(1,1)$ bound for the second-order transform $D^2\mathcal N^{-1}$, where $\mathcal N := -\Delta+x\cdot D$ is the Ornstein-Uhlenbeck operator. We also include a direct change-of-variables proof of the weaker bounded derivative divergence inequality in the appendix. The second result is a Gaussian Bernstein-Markov inequality. If $P$ is a polynomial, $p\geq2$, and $m_p := (\mathbb E|g|^p)^{1/p}$ for a standard Gaussian $g$, then \[ \|DP\|_{L^p(\gamma_d;\mathbb R^d)} \leq\frac{2\sqrt{2e\, \mathrm{deg}(P)}}{m_p}\,\|P\|_{L^p(\gamma_d)}\, . \] The factor $2$ is unnecessary when $p$ is even, or $P$ is even or odd. Thus, the constant is of order $\sqrt{\mathrm{deg}(P)/p}$ throughout the range $p\geq2$.

We propose sliced Orlicz-Wasserstein (SOW) distance which is a generalization of sliced Wasserstein (SW) distance. SOW replaces the $L^p$ norm in SW with a Luxemburg norm cost induced by an Orlicz function $\phi$. First, we prove that SOW distance is a metric on the space of measures with finite Orlicz norm, and show that it recovers the SW distance when the Orlicz function is $\phi(x)=x^p$. Next, we derive the topological properties of the SOW distance. In particular, we show that convergence under SOW implies weak convergence, and the converse is true under the compact support condition. We then present the theoretical results for estimating the SOW distance. We derive sample complexity for both the distance itself and the powered functional of the distance, and prove their minimax optimality. In addition, we discuss the computational algorithm for approximating the SOW distance by Monte-Carlo estimation and bisection search, as well as the associated approximation error and computational complexity analysis. Our experimental results reveal the superior computational efficiency of SOW compared with Orlicz-Wasserstein (OW) distance. Also, in the experiments, we demonstrate the favorable flexibility of SOW distance over SW in detecting differences between distributions by comparing their performance in two-sample tests and evaluating generative models on image datasets.

We investigate a class of two-dimensional hybrid non-autonomous periodically forced neuron models known as the Meng-Huguet-Rinzel neuron model, with a dynamic threshold and reset mechanism. We focus on the interplay between the continuous subthreshold dynamics and discrete spike-induced resets. The model consists of a linear voltage equation coupled with a threshold variable governed by a nonlinear function of the membrane potential (voltage), and incorporates periodic external forcing in the form of either pulse or rectified sinusoidal currents. We analyze the existence and uniqueness of periodic solutions and prove that the non-hybrid version of the system possesses a unique globally attracting periodic orbit. For the hybrid system, we show that there exists at most one periodic orbit without resets, while numerical simulations indicate the possibility of coexistence of a reset-free periodic orbit and periodic spiking attractors. We also prove that some natural polygons contained in the phase space of the hybrid system form compact positively invariant globally attracting sets of this system. These results provide a rigorous mathematical framework for the analysis of dynamic threshold mechanisms in neuron models and contribute to the theoretical understanding of transitions between subthreshold oscillations and spiking behavior under a time-periodic forcing.

A subspace design is a collection of subspaces $H_1,\ldots,H_n$ of $\mathbb{F}_q^k$ with the property that no low-dimensional subspace $W$ intersects the collection "too much". Subspace designs and related objects in linear-algebraic pseudorandomness have found a broad range of applications, ranging from list decoding, to derandomizing algorithms. We construct explicit strong subspace designs over every finite field. In the extremal case where the co-dimension $t$ of each $H_i$ is equal to the dimension of $W$, for every constant field size our construction attains $n=\Omega(k)$ and matches the probabilistic intersection bound up to a constant factor. All previous constructions required the field size to grow with $t$ (or $k$). Our subspace designs also imply new construction of rank condensers over arbitrary finite fields. This result is the first to achieve an optimal dependence on $k$ while maintaining both a constant output entropy rate and a constant field size. As an application, we construct lossless rank extractors for linear sources of rank $r$, for all $r < q$, with parameters matching those of Guo, Raj, Shangguan and Zhang (FOCS '26), thereby generalizing their result to prime fields and smaller field sizes. Our construction is based on an algebraic-geometric version of the Parvaresh-Vardy codes (Parvaresh-Vardy FOCS '05, Guruswami ECCC '05), extending the framework underlying the condensers of Guruswami, Umans and Vadhan (JACM '09). We view our construction as a linear-algebraic analysis - tailored to affine sources - of the GUV construction, generalized to functions over algebraic curves. More specifically, inspired by Ta-Shma and Umans (CCC 12') we develop a two-level evaluation scheme, where we first evaluate a function on a curve at extension-field points, and then evaluate a corresponding affine-linear polynomial to obtain outputs over the base field.

Over-the-air computation has emerged as a scalable and efficient solution for deploying federated learning algorithms in wireless networks by exploiting waveform superposition for simultaneous model aggregation. Most existing work struggles with heterogeneous fading channels. These approaches either enforce unbiased updates from all devices or allow partial device contributions, requiring careful tuning of the convergence bound to mitigate bias under specific fading models. However, the former significantly amplifies receiver noise due to the weakest channel, whereas the latter is sensitive to fading model mismatch and converges only to a biased objective. To tackle these challenges, we propose FedOAG, which employs algorithmic components to automatically satisfy energy constraints via gradient normalization and evenly mix devices' updates through implicit gossiping. Importantly, FedOAG does not require transmission from all devices, nor does it rely on a specific fading model or knowledge of time-varying statistical channel distributions. We show that FedOAG converges to a stationary point of an unbiased non-convex objective at the best possible rate $O(1/\sqrt{T})$ for any stochastic first-order method. We corroborate our analysis with numerical experiments over dynamic wireless conditions on real-world datasets.

The authors consider several variants of the notion of nilpotency in pre-Lie algebras and investigate the properties of nilpotent ideals.

We study stable solutions of the Allen-Cahn equation with homogeneous Neumann boundary condition on Riemannian manifolds with boundary in the regime where the transition layers meet the boundary orthogonally. Under an a priori bound for the enhanced second fundamental form in the transition region, we establish uniform second-order Hölder estimates and quantitative mean curvature decay for the nodal hypersurfaces up to the boundary in ambient dimensions at most 10. The proof adapts the technique pioneered by Wang-Wei, adding a further correction determined by the geometry of the ambient boundary. Its leading contribution to mean curvature cancels on the nodal set but survives on non-zero level sets in the transition region, whose mean curvature need not decay uniformly. This phenomenon has no interior analogue and is exhibited by stable solutions constructed by Kowalczyk.

We deformalize an LLM-generated proof within the formalism of Homotopy Type Theory that an iterated pushout construction for $\operatorname{SP}^3(X)$ is a set whenever $X$ is a set. The proof expands on a similar proof by Buchholtz for $\operatorname{SP}^2(X)$, employing a similar encode-decode strategy and the same formal machinery.

In this paper, we study the infinite reciprocal cubic sum $\sum_{k=n}^{\infty}\frac{1}{B_k^3}$ of balancing numbers. We obtain explicit bounds for this sum and use them to determine the integer part of its reciprocal.

We consider singular Liouville systems on closed Riemann surfaces with a common sign-changing coefficient. Direct continuation from a positive coefficient is obstructed by the unavoidable appearance of degenerate zeroes along such a homotopy. We instead identify an intrinsic ray in the parameter space on which the system admits synchronized solutions and reduces exactly to a scalar sign-changing mean-field equation. Combining this reduction with the variational existence theory of De Marchis–López-Soriano–Ruiz and the nodal singularity compactness theorem of Wu, we obtain existence for every noncritical parameter on this ray. Positive singular sources are allowed on the nodal curve. In particular, when the surface has nonpositive Euler characteristic and every negative component of the coefficient is a disk, the topological hypothesis required by the scalar min–max construction is automatic.

If $M$ is a closed oriented manifold of even dimension, then its $n$th symmetric product $M(n)=M^n/S_n$ is a rational homology manifold for any $n \geq 0$ and hence has a canonically defined $L$-class in $H^*(M(n);\mathbb{Q})$. A complicated formula for these classes was given many years ago in the second author's thesis. We obtain a simpler result by studying the dual classes in homology, showing that their generating function (after embedding each $H_*(M(n); \mathbb{Q})$ into the Hopf algebra $H_*(M(\infty); \mathbb{Q})$) is the product of an elementary factor depending only on the Euler characteristic of $M$ and the exponential of a series consisting of odd Adams twists of the Poincaré dual of the $L$-class of $M$. We also use this to obtain both a simpler proof and a simpler expression for the cohomological $L$-classes than the previous one.

Let $\mathcal{F}^{m}=\{A\subset[m]\}$, endowed with the symmetric difference metric and the Cardinality Reverse-Lexicographic order. That way, we compute the homotopy type $\operatorname{\v{C}ech}\left(\mathcal{F}_{\preceq a}^m; \frac{r}{2}=1\right)$ by relating it to consecutively adding vertices of $\mathbb{I}^m$, the unit hypercube of $m$th dimension, to point $0$. Further, for higher $r$, we propose a conjecture regarding contractibility of a subcomplex implying a formula for the exact number of wedges of $\mathbb{S}^ 3$'s and $\mathbb{S}^ 4$'s in the $\operatorname{\v{C}ech}$ complex.

We prove that every finite-dimensional commutative algebra can be embedded as an ideal of a finite-dimensional evolution algebra. We then determine the smallest possible dimension of an evolution algebra admitting such an embedding and study some structural properties preserved by the construction. Over fields of characteristic different from two, we relate this minimal dimension to simultaneous Waring decompositions of the quadratic map $x\mapsto x^2$. In particular, for unital commutative algebras, the smallest possible dimension of an evolution algebra containing the given algebra coincides with the Waring rank of its squaring map. As an application, we determine this minimal dimension for the truncated polynomial algebras $\mathbb{F}[t]/(t^n)$. Finally, we study embeddings for which additional structure is preserved, including automorphisms and multiplicative bases.

We compute $\E[(S_T-K)^+]$ by Monte Carlo for a scalar stochastic-volatility model with a fast mean-reverting factor of time scale $\eps$, for $\eps$ ranging from $1$ down to $10^{-3}$. A conditional (mixing) estimator gives finite variance, whereas the direct estimator has infinite variance for this model. The volatility factor is simulated with its exact Ornstein–Uhlenbeck transition. As $\eps\to0$ the price converges, at rate $O(\eps)$, to the Black–Scholes price with the averaged volatility $\bar\sigma$, and the implied-volatility smile flattens to $\bar\sigma$. Finally, we test a martingale control variate built on the Black–Scholes delta with volatility $\bar\sigma$. If the martingale is driven by the true volatility $\sigma(Y_t)$, the variance is reduced by a factor that grows like $1/\eps$, about $160$ at $\eps=10^{-3}$. If it is driven by the constant $\bar\sigma$, the variance is essentially not reduced. Last, we calibrate the model with $\rho\neq0$ to S&P~500 implied volatilities by full simulation of $S$. At the same number of paths the control variate reduces the variance by a factor $1.7$–$16$ (median $5.6$) and gives more accurate calibrated parameters; at equal CPU time it pays off only if the delta is rebalanced on a coarser grid than the time step. For a basket of four indices (state dimension $8$) the gain is larger (median $8$) and the relative cost smaller, so that the control variate is about $4$–$5$ times more efficient than plain Monte Carlo at equal CPU time.

We study Viehweg's weak positivity for vector bundles on quasi-projective schemes over noetherian rings, including mixed characteristic. We prove that weak positivity is preserved under tensor products, symmetric powers, divided powers, and exterior powers. We introduce weakly flat bundles, generalizing numerically flat bundles, and we use them to construct an S-fundamental group scheme for normal varieties admitting a small projective compactification. We compare weak flatness with strong numerical flatness and establish analogues of the Demailly-Peternell-Schneider theorem. For smooth complex varieties admitting a small compactification, we prove that the semisimple objects in the category of weakly flat bundles are precisely the unitary flat bundles. We also show that a vector bundle equipped with an integrable algebraic connection and an invariant filtration with unitary flat quotients is weakly flat. As applications, we characterize quotients of abelian varieties in terms of weak positivity of some standard vector bundles.

This paper presents a path-following controller for a quadrotor system to guarantee safe maneuvers, in terms of forward path invariance, in the presence of cyber-physical attacks. We assume that an adversarial agent can control any one of the rotors through a false data injection (FDI) type of attack. A feedback controller is designed using transverse feedback linearization which guarantees that the system follows a class of smooth curves under FDI attacks. Our proposed controller is computationally efficient, with a closed-form analytical expression, that not only mitigates the effect of bounded malicious signal but also ensures mission success. We provide theoretical guarantees of forward path invariance under FDI attacks with realistic assumptions and demonstrate the effectiveness of our approach through simulation.

Let $B$ be a finite-dimensional algebra, $M\in\modu B$, and $A=B[M]$ the one-point extension. We give a stable two-step extension criterion for the new simple $A$-module to have derived delooping level at most one. For a concrete algebra $B$ with a self-injective Nakayama quotient $R$, we determine the derived and classical delooping levels of $B[M]$ for every nonzero $M\in\modu R$. In particular, we obtain an explicit algebra $A$ satisfying \[ \Findim(A^{\mathrm{op}})=1<2=\ddell A=\dell A, \] which gives a negative answer to a question of Guo and Igusa.

In the game of flow over time, infinitesimal flow particles aim to travel from a source to a sink in a network as quickly as possible. Under the Vickrey bottleneck model, the congestion effects on network edges are captured through FIFO queues, which arise when the inflow into an edge exceeds its capacity. This work addresses two open conjectures about the game: 1) the characterization of networks that are immune to Braess's paradox, and 2) the monotonicity relationship between the network inflow rate and the overall flow makespan. We show that a single class of network topologies, called chains of bipolar pseudo-arborescences (BPAs), fully resolves the first conjecture and yields partial progress on the second. Chains of BPAs constitute precisely the cases left open by Macko et al.~(2013) in their study of Braess's paradox for flow over time. By structurally characterizing the dynamic evolution of Nash flows over time on such networks, we prove that removing edges from these networks never decreases the maximum equilibrium latency. Combined with the results of Macko et al., this establishes a necessary and sufficient condition: a network does not admit Braess's paradox for flow over time if and only if it is a chain of BPAs, thereby resolving their conjecture. Furthermore, we confirm the monotonicity conjecture of Correa et al.~(2021) for all chains of BPAs. This result strictly generalizes the previously known monotonicity for chains of parallel paths under uniform inflows.

Vietoris-Rips filtrations, which are standard in topological data analysis, are flag complexes, and a simplex tree stores these without any attaching data. In this paper we ask what survives of this economy when a flag complex $K$ is divided by a subcomplex $A$, each connected component of $A$ being crushed to a point. Such a quotient is a CW complex whose cells are the simplices of $K\setminus A$, but their attaching maps are no longer implicit. We show that for flag $K$ the face order of the quotient is strictly graded exactly when $A$ is flag, and that the surviving labelled cells are determined by those of dimension at most 3. For $m$-flag pairs the threshold is $2m+1$, and it drops to $m+2$ when $K$ is flag. The prescribed cells form a regular CW decomposition only when $A$ is full in $K$. These results justify the QF-tree: a cell table that stores, for each surviving simplex, its ordered list of $d+1$ facets with collapsed facets flagged, indexed by a trie of quotient-vertex words. For bounded dimension its size is linear in the number of surviving simplices plus the retained provenance, and we derive and verify a simple formula for the collapsed fraction above which it is smaller than the homotopy-equivalent cone model. Because a collapse changes the attaching data only on the closed star of $A$, the QF-tree can also be applied locally inside a simplex tree. For a ball-shaped $A$ in the sampled Vietoris–Rips regime the closed star is a thin shell, and the median compact budget is below the cone model at every sampled radius. An accompanying library, modelled on Gudhi, implements the QF-tree, its local variant, an editable layer with local quotient updates, gluing, disc attachment, induced maps, cup products, fundamental-group presentations and zigzag persistence, and provided experiments separate the cost of maintaining a quotient from the cost of the algebra computed on it.

Domain Adaptation (DA) often lacks worst-case guarantees, while Distributionally Robust Optimization (DRO) based only on source data centers its ambiguity set at the source law and ignores available target structure. To bridge this gap, we introduce a doubly-anchored DRO framework whose ambiguity set is the intersection of $\phi$-divergence balls centered at the source law and a source-completed target reference law, the latter pairing the target covariate law with the source conditional law. We derive dual-induced adversarial bridge geometries for symmetric and asymmetric divergence pairings, notably introducing a Kullback–Leibler/squared-Hellinger (KL/HD) bridge. This asymmetric formulation yields a Lambert-$W$ geometry in which source-side exponential risk tilting and target-side Hellinger stabilization enter through distinct terms, attenuating, but not bounding, the effect of large likelihood ratios. Furthermore, without imposing covariate shift, we establish finite-sample generalization bounds for the minimizer of a structural, loss-agnostic density-bridge risk under bounded-overlap conditions; these bounds do not apply directly to the loss-aware DRO min–max estimator. We translate our framework into a bridge-weighted Nadaraya–Watson estimator, proving uniform consistency for the source regression function and pointwise asymptotic normality, with target recovery when the source and target regression functions coincide, as under covariate shift. Finally, an empirical evaluation on a domain-shifted Fashion-MNIST dataset illustrates the finite-sample stability of the asymmetric KL/HD bridge under severe synthetic target-covariate corruption.

AdaGrad and AdaGrad-Norm are widely used adaptive methods, but their precise behavior in stochastic convex optimization remains less understood. We first show that AdaGrad-Norm and AdaGrad do not admit any universal last-iterate rate, even under sub-Gaussian noise and bounded iterates. We then prove that bounded variance alone is too weak: even with bounded iterates, it cannot yield high-probability average-iterate rates, and without bounded iterates it may not even guarantee convergence in expectation. Besides, we construct tight $\Omega(\log T/\sqrt T)$ lower bounds for both AdaGrad-Norm and AdaGrad under sub-Gaussian noise, showing that $\log T$ in existing average-iterate upper bounds is unavoidable. Finally, we show that this $\log T$ loss disappears once bounded-iterate condition is imposed: under a general ABC condition, both methods achieve rates ${O}(1/\sqrt{T})$.

We study distributionally robust reinforcement learning (DR-RL) in the average-reward setting under weak communication. Our main result provides finite-sample guarantees for estimating the robust optimal average reward and learning a near-optimal policy, covering both SA-rectangular and S-rectangular structures with divergence-based and distance-based uncertainty sets. Specifically, for Kullback–Leibler and $f_k$-divergence balls, we establish explicit radius conditions under which the robust average-reward Bellman equation admits a constant-gain solution, while for total variation and Wasserstein balls, any positive radius suffices without requiring the nominal MDP to be weakly communicating. Our algorithm is prior-knowledge-free and achieves sample complexities of $\widetilde O(|\mathcal{S}||\mathcal{A}|p_{\wedge}^{-1}\operatorname{Span}^{2}(u_{\delta}^{\ast})\epsilon^{-2})$ for estimating the robust optimal average reward and $\widetilde O(|\mathcal{S}||\mathcal{A}|p_{\wedge}^{-2}\operatorname{Span}^{2}(u_{\delta}^{\ast})\epsilon^{-2})$ for learning an $\epsilon$-optimal policy. Here, $p_{\wedge}$ is the smallest positive nominal transition probability and $u_{\delta}^{\ast}$ is a robust optimal bias function. We further provide an almost-tight explicit upper bound on $\operatorname{Span}(u_{\delta}^{\ast})$. Finally, we validate the predicted $n^{-1/2}$ convergence rate through numerical experiments.

This paper compares two candidate notions of morphism in the setting of fully local topological field theory. A first description can be formulated in terms of defects of codimension $k$ separating a pair of defects of codimension $(k-1)$. Lurie's cobordism hypothesis with singularities identifies such defects with suitably dualizable $k$-morphisms in the target category. A second description can be formulated in terms of symmetric monoidal (op)lax natural transformations between the truncations of the source and target theories. We show that these two descriptions agree. The comparison rests on a criterion for when a $k$-morphism determines a codimension-$k$ defect, expressed as a one-sided iterated adjunctibility condition. Assuming the ordinary cobordism hypothesis, this comparison provides a reformulation of the cobordism hypothesis for defects in terms of systems of oplax natural transformations. We also apply the criterion to the higher Morita category, where it yields a reducibility theorem for modules over $E_n$-algebras: a module over an $(n+1)$-dualizable $E_n$-algebra $A$ is $n$-dualizable over $A$ if and only if its underlying $E_{n-1}$-algebra is $n$-dualizable.

Simon's conjecture (1994) asserts that any pure $d$-dimensional shellable complex on $n$ vertices can be extended to the $d$-skeleton of the simplex on $n$ vertices, one facet at a time, while maintaining shellability. Bolognini and Sentinelli (2026) recently disproved it for every $d \geq 3$. We show that the conjecture becomes true once $d-2$ new vertices are allowed, and that the same holds with $k$-decomposability in place of shellability for any $k \geq 1$. We also show that the number $d-2$ cannot be lowered. Inflating the counterexample of Bolognini and Sentinelli gives, for every $d \geq 3$, a $1$-decomposable complex that cannot be extended in this way with only $d-3$ new vertices, even while merely maintaining shellability.

We study the three-dimensional incompressible Navier–Stokes equations with a componentwise nonlinear damping term of the form $\alpha\sum_{k=1}^3 u_k^{2m+1}e_k$ in Fourier and Fourier– Gevrey spaces based on $\mathcal{X}^0(\mathbb{R}^3)$. We first establish local-in-time existence and uniqueness of solutions and obtain the corresponding unique maximal solutions. We then derive blow-up criteria in the Fourier–Gevrey framework, including an integral blow-up criterion in a weaker exponential weight. Moreover, we establish a quantitative lower bound for the solution near a possible finite maximal existence time. By iterating the loss of exponential weight, we finally obtain a corresponding lower bound in the unweighted Fourier space $\mathcal{X}^0(\mathbb{R}^3)$.

For a class of regular estimators, Hájek, in his celebrated “Convolution Theorem,” showed that the asymptotic distribution of a regular estimator is the convolution of the distribution of an efficient estimator and some residual distribution. This result constitutes the foundation of the concept of asymptotic efficiency of regular estimators. In this paper, we provide a second-order version of that classical result. Introducing a class of second-order regular estimators with a valid Edgeworth expansion, we derive their asymptotic distribution under contiguous alternatives and show that it is the convolution of the second-order efficient distribution and some second-order residual distribution. This constitutes a second-order extension of Hájek's convolution theorem. Based on this, we introduce a concept of second-order robustness for second-order regular estimators. For a class of general Bayes estimators and minimum contrast estimators in time series models, this second-order robustness is used (i) in the characterization of second-order robust priors, (ii) in a comparison between the second-order robustness of maximum likelihood and Whittle estimators.

We study plug-in estimators of the entropy rate and the transfer entropy rate for stationary finite-alphabet chains with unbounded memory. Under a coalescent coupling-from-the-past assumption, we establish non-asymptotic concentration inequalities around the expectations of the estimators. Under additional uniform non-nullness and exponential beta-mixing assumptions, we control the corresponding biases and obtain concentration around the entropy and transfer entropy rates, together with almost-sure consistency. The results apply to chains governed by g-functions that need not be globally continuous.

We identify the limit free energy of all mean-field multi-species spin glasses defined on products of spheres. In particular, we do not make any convexity assumption on the covariance function of the Hamiltonian, and we allow for the presence of external fields. We also obtain a one-species reduction of the formula in the special case of balanced models.

Drifting models enable one-step generation by amortizing distribution transport into training, but this efficiency places greater demands on the transport field estimated at each update. In large-scale training, the field is computed from finite mini-batches of generated and real samples, which provide only imperfect approximations to the underlying distributions. Balanced optimal transport enforces exact mass matching within every mini-batch, making the estimated field sensitive to the particular composition of the real-data batch. We find that generated and real samples should be treated asymmetrically: letting the mass assigned to real samples adapt while keeping every generated sample fully transported improves generation across six feature-space metrics in controlled ablations, and is more robust to the relaxation strength than relaxing both marginals simultaneously, which falls below balanced transport under stronger relaxation. Motivated by this observation, we propose Unbalanced Optimal Transport Gradient Flow (UOT-GF), which keeps the generated-sample marginal fixed and relaxes only the real-data marginal. Under identical settings at DiT-B/2 on ImageNet-256, UOT-GF improves Fréchet Inception Distance (FID) from 1.53 to 1.46 over the balanced W-Flow baseline; scaling the same recipe yields 1.34 and 1.22 FID at L/2 and XL/2, the best FID among the one-step models we compare. We further derive the induced UOT transport force, establish a kinetic Vlasov–Fokker–Planck formulation whose overdamped zero-temperature limit recovers the drifting dynamics, and characterize non-target stationary states together with sufficient conditions for convergence.