Differential Constraints Research Articles

Supersymmetry of the Robinson-Trautman solution

The Robinson-Trautman solution in the Einstein-Maxwell-Λ system admits a shear-free and twist-free null geodesic congruence with a nonvanishing expansion. Restricting to the case where the Maxwell field is aligned, i.e., the spacetime is algebraically special, we provide an exhaustive classification of supersymmetric Robinson-Trautman spacetimes in the four-dimensional N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 gauged supergravity. The differential constraints that arise from the integrability conditions of the Killing spinor equation enable us to systematically reconstruct the metric. We derive the explicit form of the Killing spinor either by directly integrating the Killing spinor equation or by casting the solution into the canonical form of supersymmetric solutions given in [13]. In any case, the supersymmetric Robinson-Trautman solution generically exhibits a naked singularity.

Deformable surface reconstruction via Riemannian metric preservation

Estimating the pose of an object from a monocular image is a fundamental inverse problem in computer vision. Due to its ill-posed nature, solving this problem requires incorporating deformation priors. In practice, many materials do not perceptibly shrink or extend when manipulated, constituting a reliable and well-known prior. Mathematically, this translates to the preservation of the Riemannian metric. Neural networks offer the perfect playground to solve the surface reconstruction problem as they can approximate surfaces with arbitrary precision and allow the computation of differential geometry quantities. This paper presents an approach for inferring continuous deformable surfaces from a sequence of images, which is benchmarked against several techniques and achieves state-of-the-art performance without the need for offline training. Being a method that performs per-frame optimization, our method can refine its estimates, contrary to those based on performing a single inference step. Despite enforcing differential geometry constraints at each update, our approach is the fastest of all the tested optimization-based methods.

Minimax rates of convergence for sliced inverse regression with differential privacy

Sliced inverse regression (SIR) is a highly efficient paradigm used for the purpose of dimension reduction by replacing high-dimensional covariates with a limited number of linear combinations. This paper focuses on the implementation of the classical SIR approach integrated with a Gaussian differential privacy mechanism to estimate the central space while preserving privacy. We illustrate the tradeoff between statistical accuracy and privacy in sufficient dimension reduction problems under both the classical low- dimensional and modern high-dimensional settings. Additionally, we achieve the minimax rate of the proposed estimator with Gaussian differential privacy constraint and illustrate that this rate is also optimal for multiple index models with bounded dimension of the central space. Extensive numerical studies on synthetic data sets are conducted to assess the effectiveness of the proposed technique in finite sample scenarios, and a real data analysis is presented to showcase its practical application.

Growth and life history of freshwater chelydrid turtles (Testudines: Cryptodira): A bone histological approach.

The current study examines the growth pattern and lifestyle habits of the freshwater snapping turtles Chelydra and Macrochelys based on limb bone histology. Femora, humeri, and tibiae of 25 individuals selected from a range of ontogenetic stages were assessed to determine inter-element and intraskeletal histological variation. Osteohistological assessment of multiple elements is consistent with overall moderate growth rates as revealed by the dominance of parallel-fibered bone. However, the growth was cyclical as shown by deposition of multiple lines of arrested growths in the compacta. It appears that the bone tissue of C. serpentina is more variable through ontogeny with intermittent higher growth rates. M. temminckii appears to grow more slowly than C. serpentina possessing compact and thick cortices in accordance with their larger size. Overall, vascularization decreases through ontogeny with humeri and femora being well-vascularized in both species. Contrarily, epipodials are poorly vascularized, though simple longitudinal and radial canals are present, suggesting differences in growth patterns when compared with associated diaphyseal sections. The tibiae were found to be the least remodeled of the limb bones and therefore better suited for skeletochronology for snapping turtles. Intra-elementally, femora and humeri preserved higher cortical vascularity ventrally, suggestive of faster relative growth. We hypothesize that the differential growth pattern in limb bones of snapping turtles may relate to differential functional constraints, where forelimbs are operational in swimming while the hindlimbs provide stability.

Research and Application of Destriping Algorithm based on Weighted Block Sparse Representation to FengYun Satellite Imager Data

Abstract In remote sensing imaging systems, stripe noise is a pervasive issue primarily caused by the inconsistent response of multiple detectors. Stripe noise not only affects image quality but also severely hinders subsequent quantitative derived products and applications. Therefore, it is crucial to eliminate stripe noise while preserving detailed structure information in order to enhance image quality. Although existing destriping methods have achieved certain effects to some extent, they still face problems such as loss of image details, image blur, and ringing artifacts. To address these issues, this study proposes an image stripe correction algorithm based on weighted block sparse representation. This research applies techniques such as differential low-rank constraint and edge weight factor to remove stripe noise while retaining image detail information. The algorithm also uses the alternating direction method of multipliers (ADMM) to solve the MCP regularized least squares optimization problem model, improving the processing efficiency of the model. The results of this study have been applied and validated in imager data from the Medium Resolution Spectral Imager (MERSI-II) onboard FengYun-3D satellite, the Multi-channel Scanning Radiometer (AGRI) onboard FengYun-4A satellite and Precipitation Microwave Radiometer (MWRI–RM) onboard FengYun-3G. Compared to typical stripe correction methods, the proposed method achieves better stripe removal while preserving image detail information. The destriped image data can be used to generate high-quality quantitative products for various applications. Overall, by combining insights from prior research and innovative techniques, this study provides a more effective and robust solution to the stripe noise problem in remote sensing and weather forecast.

Substructures of the Weyl group and their physical applications

We study substructures of the Weyl group of conformal transformations of the metric of (pseudo)Riemannian manifolds. These substructures are identified by differential constraints on the conformal factors of the transformations which are chosen such that their composition is associative. Mathematically, apart from rare exceptions, they are partial associative groupoids, not groups, so they do not have an algebra of infinitesimal transformations, but this limitation can be partially circumvented using some of their properties cleverly. We classify and discuss the substructures with two-derivatives differential constraints, the most famous of which being known as the harmonic or restricted Weyl group in the physics literature, but we also show the existence of a lightcone constraint which realizes a proper subgroup of the Weyl group. We then show the physical implications that come from invariance under the two most important substructures, concentrating on classical properties of the energy-momentum tensor and a generalization of the quantum trace anomaly. We also elaborate further on the harmonic substructure, which can be interpreted as partial gauge fixing of full Weyl invariance using BRST methods. Finally, we discuss how to construct differential constraints of arbitrary higher-derivative order and present, as examples, generalizations involving scalar constraints with four and six derivatives.

Model Predictive Control with Variational Autoencoders for Signal Temporal Logic Specifications.

This paper presents a control strategy synthesis method for dynamical systems with differential constraints, emphasizing the prioritization of specific rules. Special attention is given to scenarios where not all rules can be simultaneously satisfied to complete a given task, necessitating decisions on the extent to which each rule is satisfied, including which rules must be upheld or disregarded. We propose a learning-based Model Predictive Control (MPC) method designed to address these challenges. Our approach integrates a learning method with a traditional control scheme, enabling the controller to emulate human expert behavior. Rules are represented as Signal Temporal Logic (STL) formulas. A robustness margin, quantifying the degree of rule satisfaction, is learned from expert demonstrations using a Conditional Variational Autoencoder (CVAE). This learned margin is then applied in the MPC process to guide the prioritization or exclusion of rules. In a track driving simulation, our method demonstrates the ability to generate behavior resembling that of human experts and effectively manage rule-based dilemmas.

Physics constrained pedestrian trajectory prediction with probability quantification

Accurately understanding and predicting the future trajectories of pedestrians around autonomous vehicles is a major challenge. Due to the uncertainty and diversity of pedestrian trajectories, current research focuses on the multimodality of trajectory prediction. Currently, multimodal pedestrian prediction methods mainly use deep generative networks to generate mutually independent trajectories through independent sampling strategy, thus focusing on modes with a large number of samples. Moreover, deep generative networks are black-box models that are entirely data-driven and cannot effectively explain trajectory generation. In addition, current research lacks an effective measure for each trajectory. Therefore, in our paper, we propose a physics constrained pedestrian trajectory prediction with probability quantification. We divide multimodal trajectory prediction into two phases: trajectory generation phase and trajectory selection phase. During the trajectory generation phase, rather than directly predicting trajectories, we incorporate a differential constraint module to ensure that the generated trajectories adhere to pedestrian motion laws. This approach enhances the interpretability of the model prediction process. Subsequently, we generate a set of candidate trajectory proposals with specific correlations through the correlated sampling module. During the trajectory selection phase, a probability selection module is proposed to establish explicit probability distributions for the candidate trajectory proposals and to select the proposal with the highest probability as the final output. Extensive experiments on a public real-world pedestrian trajectory dataset show that our proposed model exhibits significant advantages over existing models in multimodal trajectory prediction, which not only effectively reduces the error but also mitigates mode collapse. Moreover, we provide probabilities for each trajectory to better capture the model uncertainty and provide more comprehensive information for downstream decision-making systems.

A high contrast and resolution reconstruction algorithm in quantitative photoacoustic tomography

A framework for reconstruction of optical diffusion and absorption coefficients in quantitative photoacoustic tomography is presented. This framework is based on a Tikhonov-type functional with a regularization term promoting sparsity of the absorption coefficient and a prior involving a Kubelka–Munk absorption–diffusion relation that allows to obtain superior reconstructions. The reconstruction problem is formulated as the minimization of this functional subject to the differential constraint given by a photon-propagation model. The solution of this problem is obtained by a fast and robust sequential quadratic hamiltonian algorithm based on the Pontryagin maximum principle. Results of several numerical experiments demonstrate that the proposed computational strategy is able to obtain reconstructions of the optical coefficients with high contrast and resolution for a wide variety of objects.

Physics-informed deep learning and linear programming for efficient optimization of combined cycle power plants

Efficient operation of combined cycle power plants (CCPP) involves solving mixed-integer nonlinear optimization problems, constrained by thermodynamic differential equations, thermal constraints, and temporal dependencies between variables. The optimization problem aims to maximize profit for day-ahead operation by minimizing short-term and long-term costs, and accounting for volatility in ambient conditions and energy price. In operational terms, the challenge lies in developing an efficient and robust optimization algorithm capable of determining optimal values for thousands of temporal variables. It must also have the capacity to evaluate a multitude of scenarios encompassing various weather and market conditions. To address this challenge, this paper proposes a hybrid optimizer that utilizes deep learning models trained on historical operational data of a real CCPP, along with a surrogate physics-driven piecewise-linear model of CCPP, to determine the optimal values of thousands of control setpoints, with strong temporal co-dependencies, within a two-minute runtime required by industry constraints. A multi-resolution architecture is proposed to efficiently implement this optimizer. The algorithm sequentially solves the problem in coarse and fine resolutions to accelerate convergence. Final results are checked for feasibility using pre-trained deep learning models, ensuring accuracy and validity. The study shows that the proposed model provides comparable results to the exact solution while reducing computational costs, up to 95%. This research has practical implications for power plant system operators, as well as traders and dispatchers, enabling them to make optimal decisions for day-ahead contracting purposes and CCPP operation in a reasonable time, ultimately leading to more efficient and cost-effective power plant management. Furthermore, the methodology introduced in this work could potentially be extended to other industrial applications that require large-scale, complex optimizations, where the precise governing principles of a process can be approximated by piecewise-linear functions.

Differentially private distributed estimation and learning

We study distributed estimation and learning problems in a networked environment where agents exchange information to estimate unknown statistical properties of random variables from their privately observed samples. The agents can collectively estimate the unknown quantities by exchanging information about their private observations, but they also face privacy risks. Our novel algorithms extend the existing distributed estimation literature and enable the participating agents to estimate the expected value of a complete sufficient statistic from private signals acquired offline or online over time and to preserve the privacy of their signals and network neighborhoods. This is achieved through linear aggregation schemes with adjusted randomization schemes that add noise to the exchanged estimates subject to differential privacy constraints, both in an offline and online manner. We provide convergence rate analysis and tight finite-time convergence bounds. We show that the noise that minimizes the convergence time to the best estimates is the Laplace noise, with parameters corresponding to each agent’s sensitivity to their signal and network characteristics. Our algorithms are amenable to dynamic topologies and balancing privacy and accuracy trade-offs. Finally, to supplement and validate our theoretical results, we run experiments on real-world data from the US Power Grid Network and electric consumption data from German Households to estimate the average power consumption of power stations and households under all privacy regimes and show that our method outperforms existing first-order, privacy-aware, distributed optimization methods.

Double‐difference constrained reflection tomography in two‐dimensional elastic media

AbstractThe inverted velocity model obtained from the reflection tomography based on the angle domain common‐image gathers has a certain fuzziness. The inverted velocity model's stratigraphic interface is always not clear enough in areas with complex stratigraphic structure. In order to improve the accuracy and resolution of the inverted velocity model, a double‐difference constraint condition is added on the basis of minimizing the absolute travel‐time residual at the subsurface imaging points. This constraint makes the inverted velocity model local structure information more refined by minimizing the differential travel‐time residual at adjacent imaging points (i.e. closely spaced points within the same layer) and makes the variation of velocity model information within a certain range more accurate. The method in this paper is based on angle domain common‐image gathers, the tomography inversion equation is established by using the ray tracing method, and the conversion relationship between the traveltime residual and the residual curvature of the angle domain common‐image gathers. Then, by adding differential constraint and double‐differential constraint conditions and using the least squares QR decomposition method to solve the set of equations, the inverted velocity model can be obtained through multiple iterations, which provides a high‐precision velocity field for the migration and improves the accuracy of seismic imaging. Numerical experiments on both one typical model and a field data example demonstrate the effectiveness of the proposed double‐difference constrained elastic reflection tomography in generating high‐precision velocity models.

Rotary inverted pendulum control using neuro-adaptive robust generalized dynamic inversion

The objective of this research is to design a robust control system for trajectory tracking of the under-actuated Rotary Inverted Pendulum (RIP). The focus is on achieving semi-global asymptotic stability in the absence of knowledge about the RIP geometric and inertia parameters. The control design commences by formulating differential servo constraint dynamics (VCD) that encapsulates the control objectives for the RIP. The baseline generalized dynamic inversion (GDI) control law is derived by inverting the VCD for the control voltage input using the Moore–Penrose generalized inverse. To enhance robustness against modeling uncertainties and exogenous disturbances, a sliding mode control (SMC) element is integrated within the GDI control system, resulting in the robust GDI (RGDI) control system. Furthermore, an adaptive estimator that is based on Radial Basis Function Neural Networks (RBF-NN) is adopted to eliminate the dependency of the RGDI control system on the RIP mathematical model. The weighting matrices of the RBF-NN are updated with the aid of a Lyapunov control function. The proposed control system offers robust solution and guarantees semi-global asymptotic stability to the unstable equilibria of the RIP dynamics in the absence of a mathematical model. The proposed control system’s performance is evaluated through computer simulations and experimental tests on a real RIP system. Comparative analyses with traditional SMC and linear quadratic control methodologies are conducted to show the effectiveness of the proposed approach.

Interpretation of Second Law of Thermodynamics in Extended Procedures for the Exploitation of the Entropy Inequality: Korteweg Fluids and Strain-Gradient Elasticity as Examples.

In continuum physics the dissipation principle, first proposed by Coleman and Noll in 1963, regards second law of thermodynamics as a unilateral differential constraint on the constitutive equations. In 1996, Muschik and Ehrentraut provided a rigorous proof of such an approach under the assumption that, at an arbitrary instant, t0, in an arbitrary point, P0, of a continuous system, the entropy production is zero if, and only if, P0 is in thermodynamic equilibrium. In 2022, Cimmelli and Rogolino incorporated such an assumption in a more general formulation of the second law of thermodynamics. In this paper, we prove that the same conclusions hold if both the fundamental balance laws and their gradients are substituted into the entropy inequality. Such a methodology is applied to analyze the strain-gradient elasticity.

On generalized Nash equilibrium problems in infinite-dimensional spaces using Nikaido–Isoda type functionals

We present an analysis of generalized Nash equilibrium problems in infinite-dimensional spaces with possibly non-convex objective functions of the players. Such settings arise, for instance, in games that involve nonlinear partial differential equation constraints. Due to non-convexity, we work with equilibrium concepts that build on first order optimality conditions, especially Quasi-Nash Equilibria (QNE), i.e. first-order optimality conditions for (Generalized) Nash Equilibria, and Variational Equilibria (VE), i.e. first-order optimality conditions for Normalized Nash Equilibria. We prove existence of these types of equilibria and study characterizations of them via regularized (and localized) Nikaido-Isoda merit functions. We also develop continuity and (continuous) differentiability results for these merit functions under quite weak assumptions, using a generalization of Danskin's theorem. They provide a theoretical foundation for, e.g. using globalized descent methods for computing QNE or VE.

Utility-aware Privacy Perturbation for Training Data

Data perturbation under differential privacy constraint is an important approach of protecting data privacy. However, as the data dimensions increase, the privacy budget allocated to each dimension decreases and thus the amount of noise added increases, which eventually leads to lower data utility in training tasks. To protect the privacy of training data while enhancing data utility, we propose a Utility-aware training data Privacy Perturbation scheme based on attribute Partition and budget Allocation (UPPPA). UPPPA includes three procedures: the quantification of attribute privacy and attribute importance, attribute partition, and budget allocation. The quantification of attribute privacy and attribute importance based on information entropy and attribute correlation provide an arithmetic basis for attribute partition and budget allocation. During the attribute partition, all attributes of training data are classified into high and low classes to achieve privacy amplification and utility enhancement. During the budget allocation, a γ-privacy model is proposed to balance data privacy and data utility so as to provide privacy constraint and guide budget allocation. Three comprehensive sets of real-world data are applied to evaluate the performance of UPPPA. Experiments and privacy analysis show that our scheme can achieve the tradeoff between privacy and utility.

Statistical modelling under differential privacy constraints: a case study in fine-scale geographical analysis with Australian Bureau of Statistics TableBuilder data

ABSTRACT Consistent with the principles of differential privacy protection, the Australian Bureau of Statistics artificially perturbs all count data from the Australian Census prior to its release to researchers through the TableBuilder platform. This perturbation involves the addition of random noise to every non-zero cell count followed by the suppression of small values to zero. A consequent problem encountered by researchers working with geographical TableBuilder outputs is that the aggregate sums of cell counts over fine-scale units are often much lower than the direct query totals of their coarser enclosing units. Hierarchical Bayesian modelling is here proposed as a powerful methodology for achieving consistent multi-scale statistical reconstructions suitable for downstream spatial modelling. The utility of this framework is demonstrated for the toy problem of reconstructing Multiple Family Household counts by Mesh Block in the Perth Greater Capital City area.

Point-symmetry pseudogroup, Lie reductions and exact solutions of Boiti–Leon–Pempinelli system

We carry out extended symmetry analysis of the (1+2)-dimensional Boiti–Leon–Pempinelli system, which corrects, enhances and generalizes many results existing in the literature. The point-symmetry pseudogroup of this system is computed using an original megaideal-based version of the algebraic method. A number of meticulously selected differential constraints allow us to construct families of exact solutions of this system, which are significantly larger than all known ones. After classifying one- and two-dimensional subalgebras of the entire (infinite-dimensional) maximal Lie invariance algebra of this system, we study only its essential Lie reductions, which give solutions beyond the above solution families. Among reductions of the Boiti–Leon–Pempinelli system using differential constraints or Lie symmetries, we identify a number of famous partial and ordinary differential equations. We also show how all the constructed solution families can significantly be extended by Laplace and Darboux transformations.

Asymptotic analysis of single-slip crystal plasticity in the limit of vanishing thickness and rigid elasticity

Abstract We perform via Γ-convergence a 2d-1d dimension reduction analysis of a single-slip elastoplastic body in large deformations. Rigid plastic and elastoplastic regimes are considered. In particular, we show that limit deformations can essentially freely bend even if subjected to the most restrictive constraints corresponding to the elastically rigid single-slip regime. The primary challenge arises in the upper bound where the differential constraints render any bending without incurring an additional energy cost particularly difficult. We overcome this obstacle with suitable non-smooth constructions and prove that a Lavrentiev phenomenon occurs if we artificially restrict our model to smooth deformations. This issue is absent if the differential constraints are appropriately softened.

A Closed-Form, Pairwise Solution to Local Non-Rigid Structure-from-Motion.

A recent trend in Non-Rigid Structure-from-Motion (NRSfM) is to express local, differential constraints between pairs of images, from which the surface normal at any point can be obtained by solving a system of polynomial equations. While this approach is more successful than its counterparts relying on global constraints, the resulting methods face two main problems: First, most of the equationsystems they formulate are of high degree and must be solved using computationally expensive polynomial solvers. Some methods use polynomial reduction strategies to simplify the system, but this adds some phantom solutions. In any event, an additional mechanism is employed to pick the best solution, which adds to the computation without any guarantees on the reliability of the solution. Second, these methods formulate constraints between a pair of images. Even if there is enough motion between them, they may suffer from local degeneracies that make the resulting estimates unreliable without any warning mechanism. %Unfortunately, these systems are of high degree with up to five real solutions. Hence, a computationally expensive strategy is required to select a unique solution. Furthermore, they suffer from degeneracies that make the resulting estimates unreliable, without any mechanism to identify this situation. In this paper, we solve these problems for isometric/conformal NRSfM. We show that, under widely applicable assumptions, we can derive a new system of equationsin terms of the surface normals, whose two solutions can be obtained in closed-form and can easily be disambiguated locally. Our formalism also allows us to assess how reliable the estimated local normals are and to discard them if they are not. Our experiments show that our reconstructions, obtained from two or more views, are significantly more accurate than those of state-of-the-art methods, while also being faster. %In this paper, we show that, under widely applicable assumptions, we can derive a new system of equationsin terms of the surface normals, whose two solutions can be obtained in closed-form and can easily be disambiguated locally. Our formalism also allows us to assess how reliable the estimated local normals are and to discard them if they are not. Our experiments show that our reconstructions, obtained from two or more views, are significantly more accurate than those of state-of-the-art methods, while also being faster.