arXiv:2606.23727v1 Announce Type: cross Abstract: We establish a direct quantum-classical duality based on the degree-$2$ Sum-of-Squares (SoS) semidefinite programming cone: the matrix of two-qubit Pauli-$Z$ correlation functions obtained from \emph{any} quantum state $\rho$ is automatically a feasible point of the classical Goemans-Williamson (GW) relaxation. This observation provides a universal ``safety net'' for quantum optimization algorithms: applying GW random hyperplane rounding to the quantum-driven moment matrix yields a certified expected cut value $\mathbb{E}[\mathrm{Cut}] \ge \alpha_{\mathrm{GW}}\langle\mathcal{H}\rangle_\rho$, valid for every state produced by variational algorithms such as QAOA or the Variational Quantum Power Method (VQPM), regardless of convergence quality. We further show that the same moment matrix reveals the tensor-product structure of the underlying unitary circuit, enabling a polynomial-time, correlation-based circuit cutting procedure with rigorous error bounds. The framework is validated numerically on Max-Cut instances for variational quantum algorithms and on random states for circuit cutting, demonstrating that the cheap two-point correlation data are sufficient to locate near-optimal bipartitions and that the theoretical error bounds hold in practice.
Science Journals
arXiv:2606.24247v1 Announce Type: new Abstract: We report the enhanced sensitivity of the Rydberg microwave (MW) receiver by exploiting the Doppler effect in a vapor cell. A two-photon Rydberg ladder scheme is implemented via the co-propagation of probe and coupling lasers, which enhances the Doppler effect. When an MW field is applied, microwave dressing modifies the velocity-dependent resonance condition, enabling stronger contributions from atoms with non-zero velocities and leading to an enhancement of the EIT transmission. Based on this mechanism, we achieve a sensitivity of $35.1\ \mathrm{nV\ cm^{-1}\ Hz^{-1/2}}$ using the heterodyne technique, which is 1.5 times better than that obtained in the counter-propagating configuration. Meanwhile, the required local oscillator (LO) field is reduced by a factor of 17.6 compared with the counter-propagating configuration, which is advantageous for applications requiring minimal radiation and low power consumption. Moreover, the co-propagating configuration is more amenable to integration or portable sensing platforms because multiple laser fields can be delivered through a single optical fiber.
arXiv:2606.23745v1 Announce Type: cross Abstract: We present JEDEL, a framework for generating synthesis-ready DNA-encoded libraries (DELs) directly from three-dimensional pharmacophore representations of active ligands. JEDEL is the first model to map pharmacophore interaction patterns to actionable, scalable synthesis instructions, enabling the design of targeted libraries comprising potentially millions of molecules. Unlike existing generative approaches that produce virtual compounds requiring downstream synthesis planning, JEDEL operates within the space of purchasable building blocks and validated reactions, ensuring that every output is experimentally realizable by construction. JEDEL learns a predictive alignment between pharmacophore geometry and molecular structure and decodes this into combinatorial synthesis routes at scale. Across 18 protein targets, it generates focused libraries that outperform random and diversity-based baselines in predicted binding affinity, pharmacophore recovery, and sample efficiency, without target-specific retraining. JEDEL enables a shift from virtual molecule generation to experimentally deployable library design.
arXiv:2606.23827v1 Announce Type: cross Abstract: A data-driven method is developed for approximating value functions in deterministic optimal control problems with nonlinear control-affine dynamics. The Pontryagin Maximum Principle optimality system is solved from multiple initial conditions to generate training data consisting of values, gradients, and Hessians of the value function, where Hessian information is obtained from a matrix Riccati equation along optimal trajectories. These quantities augment a weighted least-squares regression over sparse polynomial bases on hyperbolic cross index sets, with gradients and Hessians contributing additional linear equations per sample and substantially reducing sample complexity compared to value-only regression. Feedback laws are recovered analytically from the learned value function. In high dimensions, a partial Hessian strategy controls the cost of data generation. The approach is validated on problems of increasing state dimension, where second-order data augmentation is shown to improve approximation accuracy and closed-loop performance, with up to an order-of-magnitude reduction in the number of training samples required relative to lower-order methods.
arXiv:2606.24283v1 Announce Type: new Abstract: Principal component analysis (PCA) is widely used to characterize collective protein motions from molecular dynamics (MD) simulations. While PCA identifies the dominant modes of structural fluctuation, it does not reveal whether different principal components (PCs) causally influence each other. Here, we investigate this question using a recently introduced causal-discovery framework [Del Tatto et al, PNAS 2024], which allows to infer putative causal asymmetries between high-dimensional time series. We apply this approach to long-timescale MD trajectories of two proteins. By analyzing relationships among PCs, we construct directed networks describing how PCs influence one another across time scales. These directional relationships, whose existence is a necessary condition for the presence of a causal link, are not captured by conventional covariance-based analyses and provide information that is complementary to PCA and Time-lagged Independent Component Analysis (TICA). Our results suggest that our causal inference approach can uncover previously hidden aspects of the dynamical organization of protein motions and offer a new perspective on this very popular class of collective variables.
arXiv:2606.23853v1 Announce Type: cross Abstract: This paper studies strict majority reasoning in finite electorates using so-called $\textit{social decision frames}$: finite sets of voters equipped with distinguished families of coalitions interpreted as those voting blocs evaluated to form a strict majority. A coherence criterion for qualitative majority judgments is identified and shown to give an exact characterization for representability of strict majorities by finitely additive measures. In addition, a minimal natural logic for reasoning about strict majorities is shown to be sound and complete. These developments motivate examination of associated combinatorial questions concerning incoherence in finite families of sets; partial results and a conjecture are given. Finally, the results of this paper are applied to correct a classical representation theorem for weak qualitative probability structures due to Patrick Suppes and to establish a May-type characterization for ordinary strict majority rule for social decision frames.
arXiv:2606.23854v1 Announce Type: cross Abstract: The Viking missions showcased multiple spaceflight technologies representing state-of-the-art capabilities: from digital line-scan imaging to the operation of complex onboard laboratories and software-controlled process autonomy. Since Viking, there have been extraordinary, and still accelerating, advancements in computing technology impacting science, society, and exploration. These developments have occurred in both hardware and software, resulting in increasingly capable devices, advanced programming tools, and algorithmic innovations. The subset of artificial intelligence known as machine learning has emerged as one of the most transformative of these developments, with major implications for space exploration and for improvements to the search for evidence of life beyond the Earth. Those improvements include the integration of data across different scales and increased sensitivity to complex features in data, as well as the generation of adaptive strategies for sampling environments. In this paper, the present and future nature of space exploration and astrobiological research is examined through the contextual lens of Viking, and through the history and possible future of artificial intelligence.
arXiv:2606.23888v1 Announce Type: cross Abstract: While Vision-Language Models (VLMs) show great promise in volumetric medical report generation, they frequently suffer from visual hallucinations and a lack of grounding in 3D CT data. Current Supervised Fine-Tuning (SFT) and Reinforcement Learning (RL) strategies typically optimize text fidelity alone, essentially rewarding correct diagnoses derived from language priors rather than genuine visual perception. To address this, we propose cross-view aligned Evidence-driven Multimodal Reinforcement Learning (Evidence-MRL, noted as E-MRL), a reliable RL reasoning framework that formulates the generation process as a Markov Decision Process of "diagnosis-localization-verification". Unlike standard approaches, our model is explicitly trained to identify a "key evidence slice" alongside the global diagnostic report, grounding its findings in verifiable visual evidence. Crucially, we introduce a novel cross-view consistency reward, which validates the semantic alignment between the golden-standard report and a local visual re-query of the selected key slice, providing additional rewards for correctly-localized reasoning. Experiments on large-scale 3D CT tumor datasets demonstrate that E-MRL significantly reduces hallucinations and improves diagnostic accuracy compared to SFT and RL baselines, offering a clinically interpretable solution for visually-grounded and tumor analysis.
arXiv:2606.23939v1 Announce Type: cross Abstract: Variable projection is a classical technique for separable nonlinear least-squares problems, in which variables that enter linearly are eliminated exactly, yielding a reduced nonlinear problem. By expressing this framework as a particular instance of a broader class of bilevel optimization problems, we develop a constrained variable-projection framework for data-science models, where the remaining variables are subject to convex constraints and the eliminated variables arise from a lower-level least-squares problem. In particular, by interpreting variable projection as a collapsed bilevel optimization problem, we derive exact reduced-gradient formulas compatible with automatic differentiation and propose a conditional-gradient algorithm for the resulting constrained reduced problem. We establish convergence guarantees under standard smoothness and compactness assumptions, and discuss extensions to structured lower-level variables. Numerical experiments on sparse autoencoding, dictionary learning, blind deconvolution, and few-shot learning suggest that the method can improve wall-clock efficiency and data efficiency relative to natural joint-optimization baselines.
arXiv:2606.23944v1 Announce Type: cross Abstract: State--space models provide a flexible framework for analyzing dynamical systems, yet they often rely on Gaussian assumptions that fail to capture heavy-tailed or outlier-prone measurement noise. We propose a robust estimation scheme for linear state--space models subject to compound-Gaussian noise, as encountered for instance in radio interferometry affected by radio-frequency interference (RFI). The method relies on a Stochastic Approximation Expectation--Maximization (SAEM) algorithm in which the standard E-step is replaced by Monte Carlo sampling of the latent states and noise texture through closed-form Gibbs updates, enabling tractable inference despite the heavy-tailed likelihood. Numerical experiments show that the proposed method significantly improves reconstruction fidelity and robustness to RFI, outperforming a Gaussian EM algorithm and even an oracle RTS smoother. These results highlight the benefits of heavy-tailed state--space modeling and SAEM-based inference in interference-dominated imaging scenarios.
arXiv:2606.23966v1 Announce Type: cross Abstract: Bremsstrahlung activation measurements were performed to study the production of $^{57}\mathrm{Ni}$, $^{56}\mathrm{Ni}$, $^{58}\mathrm{Co}$, $^{57}\mathrm{Co}$, $^{56}\mathrm{Co}$, and $^{55}\mathrm{Co}$ from natural nickel targets irradiated with photons generated by \SI{40}{MeV} electrons incident on a tantalum converter. Bremsstrahlung spectra were modeled using MCNP6.3\texttrademark{} and experimentally validated through activation of natural tin. Bremsstrahlung-averaged cross sections were extracted from end-of-irradiation activities, yielding $\langle\sigma\rangle = 8.983~\pm~0.028$~mb for $^{58}\mathrm{Ni}(\gamma,n)^{57}\mathrm{Ni}$, $0.248~\pm~0.025$~mb for $^{58}\mathrm{Ni}(\gamma,2n)^{56}\mathrm{Ni}$, $0.704~\pm~0.218$~mb for $^{nat}\mathrm{Ni}(\gamma,pxn)^{58}\mathrm{Co}$, $9.192~\pm~0.386$~mb for $^{58}\mathrm{Ni}(\gamma,p)^{57}\mathrm{Co}$, and $2.239~\pm~0.355$~mb for $^{58}\mathrm{Ni}(\gamma,pn)^{56}\mathrm{Co}$. A $90\%$ confidence-level upper limit of $\langle\sigma\rangle < 0.021$~mb is established for the $^{58}\mathrm{Ni}(\gamma,p2n)^{55}\mathrm{Co}$ channel. Comparison with JENDL-5 evaluations and prior studies indicates channel-dependent agreement, with residual discrepancies observed for selected charged-particle emission reactions. In particular, the measured $^{58}\mathrm{Ni}(\gamma,p)^{57}\mathrm{Co}$ cross section exceeds the JENDL-5 prediction by approximately a factor of two, whereas TALYS-2.2 calculations reproduce the experimental value, suggesting an underestimation of the $(\gamma,p)$ channel strength in JENDL-5 under bremsstrahlung conditions. For the $^{58}\mathrm{Ni}(\gamma,pn)^{56}\mathrm{Co}$ channel, both TALYS-2.2 and JENDL-5 predictions are in reasonable agreement, while the present measurement is higher by approximately a factor of two, though with larger associated uncertainty.
arXiv:2606.24361v1 Announce Type: new Abstract: Sign language models are typically trained on datasets captured under constrained conditions, with limited viewpoint, background, and signer-identity diversity, leading to poor robustness under real-world distribution shifts. We introduce SignNet-1M, a large-scale augmented dataset spanning ASL, CSL, and German Sign Language (DGS). SignNet-1M synthesizes realistic variations along three axes: (i) novel-view rendering (rotation and zoom) via 3D Gaussian Splatting (3DGS), (ii) scene/identity editing via diffusion models for background replacement and signer substitution while preserving sign motion and linguistic content, and (iii) post-rendering augmentations that emulate capture and compression artifacts (e.g., pose/temporal perturbations and video-level corruptions) to better match in-the-wild recordings. Beyond data release, we provide a unified benchmark suite across downstream tasks (e.g., translation and recognition) and ablations that isolate each augmentation component. Experiments across backbones show that training with SignNet-1M consistently improves generalization under cross-view, cross-background, cross-identity, and post-rendering shifts, while maintaining strong in-distribution performance. The dataset, full augmentation pipeline, and benchmark are available at https://signnet.chatsign.ai/.
arXiv:2606.24422v1 Announce Type: new Abstract: We introduce EgoSAT, the first comprehensive benchmark for egocentric video reasoning in streaming settings, designed to evaluate the capabilities of modern vision-language models (VLMs). The benchmark targets streaming interaction understanding, where video frames arrive sequentially and models must continuously interpret evolving visual context. EgoSAT unifies several previously distinct tasks within a single streaming framework. In this formulation, queries about completed events correspond to retrospective reasoning, queries about ongoing activities require online understanding, and queries about future actions involve prospective anticipation. This unified setting requires models to reason about the past, present, and future while operating under the constraint that only previously observed frames are available. EgoSAT contains 1,997 unique videos spanning 165 hours of egocentric footage and around 4,800 high-quality question-answer pairs, carefully designed to probe reasoning across varying temporal contexts. Using this benchmark, we evaluate a diverse set of both open-weight and closed-weight VLMs, providing a systematic assessment of their ability for streaming interaction understanding. By distinguishing answerability and conducting diagnostics on confidence of models, we find existing models not only struggle with prospective and retrospective modeling, but also exhibit severe mis-calibration: confidence often fails to track inherent answerability, leading to dangerous "confidently wrong" behaviors. Project page: https://leiyj23.github.io/EgoSAT/
arXiv:2606.23691v1 Announce Type: new Abstract: Second order derivatives of model outputs with respect to input parameters are key to several applications in ice sheet modelling. For example, the ability to compute Hessian-vector products broadens the list of available optimisation methods, and facilitates certain kinds of parametric uncertainty quantification. Some modern ice sheet models are built on frameworks supporting algorithmic differentiation (AD), allowing for the computation of higher order derivatives with relative ease. However, many of our most widely-used models are not. A natural alternative might be to follow common practise in first order gradient computation and construct an approximate second-order adjoint model at the PDE level, which neglects the nonlinear dependence of ice viscosity on velocity. Here, we present such a model for the shallow-stream approximation allowing one to compute approximate second-order derivatives, and compare with full second-order derivates found using AD. We find that this produces Hessian-vector products that are superficially similar to those computed via AD. However, an analysis of the spectral decomposition of the Hessians calculated in each way reveals that the subspaces spanned by their eigenvectors diverge after the leading 4 modes, though divergence does not accelerate after this. We conclude that the utility of the approximate Hessian is case-dependent, and a full Hessian, likely computed using AD, should be used where high fidelity is required above very low rank.
arXiv:2606.23693v1 Announce Type: new Abstract: Text-to-SQL enables users to query databases using natural language by generating executable SQL queries. Recent methods have increasingly adopted Large Language Models based reinforcement learning (RL) to leverage execution feedback for training. However, existing RL methods assign uniform query-level rewards to all clauses in a SQL query, treating correct and incorrect clauses equally. This coarse-grained reward design leads to insufficient learning signals for correct SQL generation. To address this issue, we propose EXPO-SQL (EXecution-based clause-level Policy Optimization for Text-to-SQL) which provides fine-grained supervision through clause-level rewards. To assign clause-level rewards, our method identifies erroneous clauses by analyzing execution results, including error messages and clause-wise incremental execution. Experiments on widely-used Text-to-SQL benchmarks demonstrate that EXPO-SQL significantly outperforms existing supervised fine-tuning, prompting, and RL-based methods through fine-grained clause-level learning. Our code is available at https://github. com/jhn25/EXPO-SQL.
arXiv:2606.24500v1 Announce Type: new Abstract: Reachability analysis for dynamical systems seeks to compute a set containing all reachable states at a given time. Compared to ordinary differential equations (ODEs), the analysis of nonlinear reaction--diffusion PDEs with parametric uncertainties remains largely underexplored, due to the infinite-dimensional state space and the variety of solutions under different parameters. We address this through a three-step procedure: 1) Finite Element Methods (FEM)s to discretise the space and generate a finite-dimensional FEM-based model, 2) Proper Orthogonal Decomposition (POD) to build a Reduced-Order Model (ROM), and 3) set-based reachability-analysis methods applied to the ROM. We propose a framework that enables us to derive explicit upper bounds on the approximation errors introduced at each stage of the pipeline. In particular, we quantify the discrepancy between trajectories of the original PDE and those of the FEM-based discretization, as well as the error between the FEM-based model and the reduced-order model. Importantly, these bounds are shown to hold uniformly over the considered set of parameters. By combining these error estimates, we obtain an over-approximation of the reachable set of the original PDE. The approach is illustrated on the Allen--Cahn equation and a logistic growth PDE.
arXiv:2606.24013v1 Announce Type: cross Abstract: We study public-good provision when a planner observes agents' preferences only through a fixed local-privacy channel that randomizes each report before it reaches the planner. We characterize the optimal reduced-form allocation: the project is implemented when an aggregate posterior score is positive, where each agent's score combines the posterior expected valuation and posterior virtual value. Privacy enters through these posterior objects, muting the responsiveness of provision to private preferences and, under weak monotone likelihood ratios, potentially generating pooling. We then distinguish the optimal reduced-form allocation from its implementation through signal-measurable transfers: the required transfers solve a Fredholm integral equation whose solution is unique under completeness when it exists, while existence requires a separate range condition. Maximum reduced-form revenue exhibits three population regimes: it is asymptotically linear, of square-root order, or exponentially small according as the lower endpoint of the valuation distribution is positive, zero, or negative. Finally, welfare comparisons depend on the privacy calibration. At a common noise scale, Laplace Blackwell-dominates logistic noise, while under a common tight $\mu$-GDP calibration the ordering reverses for the maximally separated binary endpoint experiment. Thus the preferred privacy channel depends on the standard used to hold privacy fixed.
arXiv:2606.24016v1 Announce Type: cross Abstract: We study how fixed-volume spanning-tree extremality changes when product-grid boundary factors are free, periodic, or mixed. In two dimensions, extremality depends sharply on the boundary type. The free/free and periodic/periodic products both obey a closest-to-square principle: among fixed-area rectangles, $P_r\square P_s$ and $C_r\square C_s$ are maximized by the closest-to-square admissible side lengths. The mixed free/periodic cylinder $P_r\square C_s$ is different: closest-to-square fails, and in the divisor-rich case the optimizing cyclic circumference has scale $N^{1/3}$ when the area is $N=rs$. In arbitrary dimension we prove pairwise balancing theorems for pure free products and pure periodic products, and then strengthen them by a heat-trace Schur-concavity theorem in logarithmic side lengths. At perfect-power volume this gives the unique maximizers $P_n^{\square d}$ and, for $n\ge3$, $C_n^{\square d}$. These product-grid comparisons motivate perfect-power conjectures for connected induced lattice subgraphs and periodic analogues.
arXiv:2606.24618v1 Announce Type: new Abstract: In ontology-based data access (OBDA), multiple data sources are integrated via mappings to an ontology. We consider an OBDA setting based on existential rules and the certain answer semantics. We address the recent issue of query abstraction, which consists of abstracting data queries by translating them to the ontology layer. Since a perfect abstraction may not exist, the notions of minimally complete and maximally sound abstractions have been introduced. We study abstractions within an extension of UCQs with a limited form of inequality and a special predicate marking database constants. While this extension does not lead to an increased complexity of the problems of interest, it is able to express minimally complete abstractions, hence perfect abstractions when they exist. We also characterize maximally sound abstractions by making a new connection with the notion of maximum recovery stemming from data exchange.
arXiv:2606.24042v1 Announce Type: new Abstract: Recommender systems often induce filter bubbles and semantic homogenization by monolithically optimizing for immediate user engagement. Standard single-objective models, including traditional Deep Q-Networks, are ill-equipped to navigate the trade-offs between platform retention and critical societal values like information diversity and provider fairness. To address these limitations, we introduce a multi-objective reinforcement learning framework that formalizes recommendation as a semantic multi-objective Markov decision process. By integrating high-fidelity semantic embeddings with a Pareto-DQN agent, our architecture treats engagement, diversity, and fairness as distinct, non-aggregable reward signals, avoiding the pitfalls of static reward scalarization. Empirical evaluations on the MovieLens small dataset shows that our hypervolume based action selection disrupts the feedback loops responsible for semantic collapse. By sustaining high state-trajectory variance, the Pareto-DQN effectively maps the Pareto frontier, achieving gains in auxiliary societal objectives with only marginal impacts on engagement. This work provides a path toward intrinsically aligned, responsible recommender systems.
arXiv:2606.24420v1 Announce Type: new Abstract: In high-stakes document processing pipelines, including financial reconciliation, compliance verification, and procurement automation, an LLM extraction that is silently wrong is more dangerous than one that is visibly absent. The central challenge is not extraction accuracy alone but reliable confidence estimation: knowing, field by field, whether an extraction can be trusted for automation or deferred to human review. Token-level log-probabilities, verbalized confidence, and multi-sample self-consistency all collapse toward all-positive behaviour at practical thresholds, offering no reliable separation between trustworthy and untrustworthy extractions. We present ExtractConf, a cross-domain, field-agnostic confidence engine that grounds confidence estimation in two structurally different readings of the same document. A field-guided Hunter call extracts each field under schema-slot completion pressure; a document-guided Mapper call scans holistically and surfaces values grounded in document content. This asymmetry yields different failure modes: Hunter hallucinates values for absent fields, while Mapper misses visually non-salient ones. Their disagreement is independently informative. ExtractConf fuses cross-call disagreement, LLM-internal uncertainty, OCR, image quality, and spatial layout into a classifier requiring no domain-specific rules or retraining. On DocILE (55-field invoices, 26% failure rate), it achieves 0.928 ROC AUC and reduces selective prediction risk by 70% over logprob-mean. At 80% coverage, accuracy reaches 99.1%, enabling a practical human-in-the-loop workflow. Zero-shot transfer to CORD receipts achieves 0.858 AUC; lightweight Lasso recalibration reduces ECE by 89% and Brier by 43%, confirming the signals generalise across document domains.
arXiv:2606.24082v1 Announce Type: cross Abstract: Large audio-language models (LALMs) can reason about audio, yet it remains unclear whether they can perform comparative judgments between two speech signals along emotional, environmental, linguistic, prosodic, and interpersonal dimensions. We study this question in the context of speech emotion recognition (SER), where the model determines which utterance exhibits higher arousal, valence, or dominance. We introduce a reasoning-guided ordinal SER framework that conditions an LALM on paired speech inputs. The model is trained using reasoning traces generated from both semantic audio descriptions and acoustic evidence derived from GeMAPS features, enabling interpretable comparative decisions. Beyond direct supervision, we also employ direct preference optimization to encourage stronger separation for emotional differences. Experiments show that the proposed framework improves preference prediction while requiring only 5% of the training data used by conventional ordinal SER systems.
arXiv:2606.24631v1 Announce Type: new Abstract: The development of trajectory planning strategies for autonomous ground vehicles (AGVs) represents a prevailing research interest within the domain of intelligent transportation systems. This paper introduces a trajectory planning framework tailored for multi-floor scenarios. The framework consists of two main modules: the task planning module and the trajectory planning module. The task planning module involves a strategic selection phase, where a task planning strategy based on generalized voronoi diagrams (GVD) and multi-objective algorithms is proposed to select the floor exits for each floor. The trajectory planning module utilizes optimization-based methods to generate high-quality trajectories, and a warm-started hierarchical planning framework is designed to ensure rapid convergence. Additionally, for handling complex obstacle constraints, a correlation constraint calculation method is designed for reducing obstacle constraints in trajectory planning. Finally, the feasibility and effectiveness of the proposed framework are verified through simulations.
arXiv:2606.24106v1 Announce Type: cross Abstract: Variational principles specify the dynamics of a physical system via the extremization of associated functional data. Onsager's variational principle (OVP), which characterizes dissipation-dominated phenomena such as phase separation, admits an unconditionally energy-stable time discretization through the minimization of a Rayleighan functional combining free energy and dissipative effects. The present work considers the case where one or more parts of this functional are empirically approximated or otherwise uncertain. To address this, a novel variational discretization of OVP is introduced which recovers previous work as a special case, and a learning problem is formulated which identifies uncertain terms in the free energy and dissipation potential from observable data. It is shown that the resulting OVP-based models connect directly to previous work in terms of proximal methods, Sobolev and Wasserstein gradient flows, while remaining provably energy-stable under arbitrarily long rollouts. The approach is illustrated on examples including Allen-Cahn, Fokker-Planck, and Cahn-Hilliard system models, where the effects of bulk free-energy densities, nonlocal potentials, and nonstandard boundary conditions are effectively learned with model classes consisting of polynomials, shallow neural networks, and spectral convolution kernels.
arXiv:2606.24229v1 Announce Type: cross Abstract: We study metric entropy and uniform sampling for classes of signals on ${\mathbb Z}_N$ with prescribed Fourier ratio. The Fourier ratio measures how spread out the Fourier transform of a signal is, interpolating between sparse spectral support and nearly uniform spectral distribution. Our main result gives upper and lower bounds for the metric entropy of a Fourier-ratio layer of size $r.$ At any sufficiently small fixed covering scale, these bounds match in their dependence on $r$ and $N$ and show that $FR(f)^2$ acts as an effective dimension parameter governing the size of the class. We use the entropy estimate to obtain uniform bounds for empirical approximation over Fourier-ratio classes. We also establish a phase-orbit packing result. If a single signal has a flat spectral block of size $k,$ then phase perturbations of that signal generate an exponentially large family with the same Fourier ratio and positive $\ell^2$ separation. Together, these results show that the Fourier ratio governs not only approximation properties of individual signals, but also the geometric size and uniform sampling behavior of entire signal classes.