Nonlinear Geometric Phase Gradient Metasurfaces beyond the Dipole Approximation.
Abstract not provided.
Gennaro, Sylvain Damien↗
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Implicit shock tracking for unsteady flows by the method of lines
Here, a recently developed high-order implicit shock tracking (HOIST) framework for resolving discontinuous solutions of inviscid, steady conservation laws is extended to the unsteady case. Central to the framework is an optimization problem which simultaneously computes a discontinuity-aligned mesh and the corresponding high-order approximation to the flow, which provides nonlinear stabilization and a high-order approximation to the solution. This work extends the implicit shock tracking framework to the case of unsteady conservation laws using a method of lines discretization via a diagonally implicit Runge-Kutta method by “solving a steady problem at each timestep”. We formulate and solve an optimization problem that produces a feature-aligned mesh and solution at each Runge-Kutta stage of each timestep, and advance this solution in time by standard Runge-Kutta update formulas. A Rankine-Hugoniot based prediction of the shock location together with a high-order, untangling mesh smoothing procedure provides a high-quality initial guess for the optimization problem at each time, which results in rapid convergence of the sequential quadratic programing (SQP) optimization solver. This method is shown to deliver highly accurate solutions on coarse, high-order discretizations without nonlinear stabilization and recover the design accuracy of the Runge-Kutta scheme. We demonstrate this framework on a series of inviscid, unsteady conservation laws in both one- and two- dimensions. We also verify that our method is able to recover the design order of accuracy of our time integrator in the presence of a strong discontinuity.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Deep Koopman learning of nonlinear time-varying systems
Here this paper presents a data-driven approach to approximate the dynamics of a nonlinear time-varying system (NTVS) by a linear time-varying system (LTVS), which results from the Koopman operator and deep neural networks. Analysis of the approximation error between states of the NTVS and the resulting LTVS is presented. Simulations on a representative NTVS show that the proposed method achieves small approximation errors, even when the system changes rapidly. Furthermore, simulations in an example of quadcopters demonstrate the computational efficiency of the proposed approach.
97 MATHEMATICS AND COMPUTING↗
Multilevel Spectral Coarsening for Graph Laplacian Problems with Application to Reservoir Simulation
We extend previously developed two-level coarsening procedures for graph Laplacian problems written in a mixed saddle point form to the fully recursive multilevel case. The resulting hierarchy of discretizations gives rise to a hierarchy of upscaled models, in the sense that they provide approximation in the natural norms (in the mixed setting). This property enables us to utilize them in three applications: (i) as an accurate reduced model, (ii) as a tool in multilevel Monte Carlo simulations (in application to finite volume discretizations), and (iii) for providing a sequence of nonlinear operators in a full approximation scheme for solving nonlinear pressure equations discretized by the conservative two-point flux approximation. Finally, we illustrate the potential of the proposed multilevel technique in all three applications on a number of popular benchmark problems used in reservoir simulation.
multilevel Monte Carlo↗
Thresholded quantum sensing with a frustrated Kitaev trimer
We investigate the response of a Ramsey interferometric quantum sensor based on a frustrated, three-spin system (a Kitaev trimer) to a classical time-dependent field (signal). The system eigenspectrum is symmetric about a critical point, |𝒃| = 0, with four of the spectral components varying approximately linearly with the magnetic field and four exhibiting a nonlinear dependence. Under the adiabatic approximation and for appropriate initial states, we show that the sensor's response to a zero-mean signal is such that below a threshold, |𝒃| < 𝑏 th , the sensor does not respond to the signal, whereas above the threshold, the sensor acts as a detector that the signal has occurred. This thresholded response is approximately omnidirectional. Moreover, when deployed in an entangled multisensor configuration, the sensor achieves sensitivity at the Heisenberg limit. Such detectors could be useful both as stand-alone units for signal detection above a noise threshold and in two- or three-dimensional arrays, analogous to a quantum bubble chamber, for applications such as particle track detection and long-baseline telescopy.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
TNet: A Model-Constrained Tikhonov Network Approach for Inverse Problems
Deep learning (DL), in particular deep neural networks, by default is purely data-driven and in general does not require physics. This is the strength of DL but also one of its key limitations when applied to science and engineering problems in which underlying physical properties—such as stability, conservation, and positivity—and accuracy are required. DL methods in their original forms are often not capable of respecting the underlying mathematical models or achieving desired accuracy even in big-data regimes. On the other hand, many data-driven science and engineering problems, such as inverse problems, typically have limited experimental or observational data, and DL would overfit the data in this case. Leveraging information encoded in the underlying mathematical models, we argue, not only compensates for missing information in low data regimes but also provides opportunities to equip DL methods with the underlying physics, hence promoting better generalization. This paper develops a model-constrained DL approach and its variant TNet—a Tikhonov neural network—which are capable of learning not only information hidden in the training data but also in the underlying mathematical models to solve inverse problems governed by partial differential equations in low data regimes. We provide the constructions and some theoretical results for the proposed approaches for both linear and nonlinear inverse problems. Since TNet is designed to learn inverse solutions with Tikhonov regularization, it is interpretable: in fact it recovers Tikhonov solutions for linear cases while potentially approximating Tikhonov solutions for nonlinear inverse problems. We also prove that data randomization can enhance not only the smoothness of the networks but also their generalizations. Comprehensive numerical results confirm the theoretical findings and show that with even as little as 1 training data sample for one-dimensional (1D) deconvolution, 5 for an inverse 2D heat conductivity problem, 100 for inverse initial conditions for a time-dependent 2D Burgers’s equation, and 50 for inverse initial conditions for 2D Navier–Stokes equations, TNet solutions can be as accurate as Tikhonov solutions while being several orders of magnitude faster. Furthermore, this is possible owing to the model-constrained term, replications, and randomization.
97 MATHEMATICS AND COMPUTING↗
Learning Optimal Power Flow Solutions using Linearized Models in Power Distribution Systems
Solving nonlinear optimal power flow (OPF) problem is computationally expensive, and poses scalability challenges for power distribution networks. An alternative to solving the original nonlinear OPF is the linear approximated OPF models. Although, these linear approximated OPF models are fast, the resulting solutions may result in significant optimality gap. Lately, the application of machine learning (ML) methods in successfully solving the nonlinear OPF has been reported. These methods learn and estimate the nonlinear control policies using a purely data-driven approach. In this paper, we propose an approach to complements the ML based approach to solving OPF using solutions from known linearized OPF model. Specifically, we use supervised learning to map the solutions of linear OPF to nonlinear control variables. Unlike, the traditional ML based methods for OPF that approximate the full distribution feeder model using function approximation, our approach uses a two-node approximation of radial networks. The proposed approach is validated using IEEE 123 bus test system for OPF solutions obtained using the nonlinear OPF models.
optimal power flow, power distribution systems, su↗
Deep Neural Networks are Adaptive to Function Regularity and Data Distribution in Approximation and Estimation
Deep learning has exhibited remarkable results across diverse areas. To understand its success, substantial research has been directed towards its theoretical foundations. Nev- ertheless, the majority of these studies examine how well deep neural networks can model functions with uniform regularities. In this paper, we explore a different angle: how deep neural networks can adapt to varying degrees of smoothness in functions and nonuni- form data distributions across different locations and scales. More precisely, we focus on a broad class of functions defined by nonlinear tree-based approximation methods. This class encompasses a range of function types, such as functions with uniform regularities and discontinuous functions. We develop nonparametric approximation and estimation theories for this class using deep ReLU networks. Our results show that deep neural networks are adaptive to the nonuniform smoothness of functions and nonuniform data distributions at different locations and scales. We apply our results to several function classes, and derive the corresponding approximation and generalization errors. The validity of our results is demonstrated through numerical experiments.
97 MATHEMATICS AND COMPUTING↗
Optimal Network Topology for Node-Breaker Representations With AC Power Flow Constraints
It has been demonstrated that network topology optimization (NTO) may change the topology of power system networks, and consequently, provide additional flexibility to reduce network congestion and violations. Most NTO problems are formulated based on the bus-branch model in which it is challenging to represent a realistic picture of all substation configurations. In this paper, we explore advantages of substation reconfiguration modeling based on node-breaker representations for NTO problem with full nonlinear alternating current power flow. It also proposes a tailored solution algorithm to solve this nonconvex mixed-integer nonlinear programming through the outer approximation method. The proposed solution approach iterates between a mixed-integer linear programming and a nonlinear subproblem. Additional enhancements to further accelerate the iteration process are illustrated. Numerical case studies demonstrate the relative economic and operational impact of optimal network topology with node-breaker representations.
42 ENGINEERING↗
Generalized analytical and numerical modeling of optical second harmonic generation in anisotropic crystals and complex heterostructures using #SHAARP package
Optical second harmonic generation (SHG) is a nonlinear optical effect widely used for nonlinear optical microscopy and laser frequency conversion. The closed-form analytical solution of the nonlinear optical responses is essential for evaluating the optical responses of new materials whose optical properties are unknown a priori. Many approximations have therefore been employed in the existing analytical approaches, such as slowly varying approximation, weak reflection of the nonlinear polarization, transparent medium, high crystallographic symmetry, Kleinman symmetry, easy crystal orientation along a high-symmetry direction, phase matching conditions and negligible interference among nonlinear waves, which may lead to large errors in the reported material properties. To avoid these approximations, here we have developed an open-source package named Second Harmonic Analysis of Anisotropic Rotational Polarimetry (#SHAARP) for single interface (si) and in multilayers (ml) for homogeneous crystals. The reliability and accuracy are established by experimentally benchmarking with both the SHG polarimetry and Maker fringes predicted from the package using standard materials. SHAARP.si and SHAARP.ml are available through GitHub https://github.com/Rui-Zu/SHAARP and https://github.com/bzw133/SHAARP.ml, respectively.
complex systems↗
ETG turbulent transport in the Mega Ampere Spherical Tokamak (MAST) pedestal
The results of a gyrokinetic analysis of turbulent transport driven by the electron temperature gradient (ETG) in the MAST pedestal are presented. Local nonlinear simulations from the gyrokinetic code, GENE, show that heat flux produced by ETG turbulence is 10%–30% of the total applied heating power in the upper pedestal and pedestal top during both the pre-edge-localized mode (ELM) (80%–99% inter-ELM period) and post-ELM (0%–20%) periods. Increasing strongly with the ETG, the ETG transport appears to be stiff. Considering radiation losses, ion thermal transport, and the strong sensitivity of the transport to the ETG, we propose that ETG transport is a plausible mechanism mediating the inter-ELM temperature profile on MAST. Cognizant of the possibility that sharp variations in the pedestal profiles may violate the local approximation, we conducted global nonlinear simulations; the results are in good agreement with local simulations except near the pedestal top, where extended radial structures and high transport levels (far beyond experimental) develop in the local simulations. We quantify and parameterize the discrepancy between local and global simulations by calculating the ratio of the radial correlation length to a length scale representative of the profile curvature. When this ratio is sufficiently small, local and global simulations agree as expected.
ETG↗
Probing topology in nonlinear topological materials using numerical K -theory
Nonlinear topological insulators have garnered substantial recent attention as they have both enabled the discovery of new physics due to interparticle interactions, and may have applications in photonic devices such as topological lasers and frequency combs. However, due to the local nature of nonlinearities, previous attempts to classify the topology of nonlinear systems have required significant approximations that must be tailored to individual systems. Here, we develop a general framework for classifying the topology of nonlinear materials in any discrete symmetry class and any physical dimension. Our approach is rooted in a numerical $K$ -theoretic method called the spectral localizer, which leverages a real-space perspective of a system to define local topological markers and a local measure of topological protection. Here, our nonlinear spectral localizer framework yields a quantitative definition of topologically nontrivial nonlinear modes that are distinguished by the appearance of a topological interface surrounding the mode. Moreover, we show how the nonlinear spectral localizer can be used to understand a system's topological dynamics, i.e., the time evolution of nonlinearly induced topological domains within a system. We anticipate that this framework will enable the discovery and development of novel topological systems across a broad range of nonlinear materials.
75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
Data-Efficient Methods for Determining Flory–Huggins χ Parameters in Multicomponent Polymer Formulations
Polymer formulations are essential in diverse applications including personal care products, coatings, paints, adhesives, and plastic materials. Designing these formulations requires navigating large, complex design spaces, where phase and self-assembly behavior critically impact performance. The Flory–Huggins χ parameter, which quantifies segmental miscibility, is widely used to parametrize the excess free energy of mixing in formulation models. In this work, we introduce two data-efficient, top-down methods for estimating χ parameters using the Random Phase Approximation (RPA): (i) Boundary Nonlinear Regression (Boundary-NLR), which fits theoretical spinodal boundaries to experimental phase boundaries, and (ii) Surrogate Model Inverse Parameter Estimation (SMIPE), which uses a Gaussian Process Classifier to fit sparse phase maps via a surrogate model. Both methods allow rapid parametrization of polymer field-theoretic models without the need for additional experiments. We evaluate these approaches on data sets involving polymer–solvent–nonsolvent ternary mixtures and block copolymer–solvent systems, demonstrating their robustness to experimental noise and their relevance for real-world formulation design.
copolymers↗
Stochastic Microgrid Scheduling With Chance‐Constrained Resilience Consideration
Traditionally, it is assumed that microgrids transition seamlessly from grid‐connected operation to islanded mode in the event of sudden main grid outages. In reality, the islanding process, especially unintentional islanding, is rarely seamless. Instead, it is subject to voltage and frequency fluctuations caused by the instantaneous disconnection of the point of common coupling (PCC) switch, variations in loads and renewable generation output and even the protection tripping of distributed energy resources (DERs). To mitigate these fluctuations and facilitate a smooth islanding process, we propose a stochastic microgrid scheduling model that incorporates chance‐constrained resilience measures. Specifically, the resilience measure is defined as the probability of successful islanding (PSI), that is, the probability that a microgrid can mitigate the generation‐demand imbalance caused by the disconnection of the PCC switch, variations in load and renewable generation and DER tripping. This measure is modelled using chance constraints. Unlike existing reliability and resilience indices, which typically neglect the possibility of microgrid/DER failure under extreme events and assume their survival while primarily focussing on reducing impact duration or magnitude, the proposed PSI‐based framework explicitly addresses microgrid and DER survival during the islanding transition. The formulated nonlinear chance constraints are approximated using a multiinterval approach and equivalently represented as a mixed‐integer linear programming (MILP) formulation. Case study results validate the proposed method, showing that the PSI estimation error is reduced to less than 8%, compared to approximately 28% with existing methods. Various sensitivity analyses on the DER tripping rate and PSI settings were performed to validate the robustness of the proposed method. In particular, the necessity of accounting for DER tripping in the PSI calculation was demonstrated.
chance constrained optimization↗
Theoretical methods to design and test quantum simulators for the compact Abelian Higgs model
The lattice compact Abelian Higgs model is a nonperturbative regularized formulation of low-energy scalar quantum electrodynamics. In 1+1 dimensions, this model can be quantum simulated using a ladder-shaped optical lattice with Rydberg-dressed atoms. In this setup, one spatial dimension is used to carry the angular momentum of the quantum rotors. One can use truncations corresponding to spin-2 and spin-1 to build local Hilbert spaces associated with the links of the lattice. We argue that ladder-shaped configurable arrays of Rydberg atoms can be used for the same purpose. We make concrete proposals involving two and three Rydberg atoms to build one local spin-1 space (a qutrit). We show that the building blocks of the Hamiltonian calculations are models with one and two spins. We compare target and simulators using perturbative and numerical methods. The two-atom setup provides an easily controllable simulator of the one-spin model while the three-atom setup involves solving nonlinear equations. We discuss approximate methods to couple two spin-1 spaces. The article provides analytical and numerical tools necessary to design and build the proposed simulators with current technology.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
JET EROSION OF PARTICLE BEDS: PROJECTING CRITICAL SUSPENSION VELOCITIES FROM EFFECTIVE CLEARING / CLEANING RADII
Here we explore the relationship between effective cleaning/clearing radii, ECR, and critical suspension velocities, Ucs. Although one set of physics controls both the radial extent of erosion (i.e., the ECR) and the velocity needed to erode from the center of jet impingement to any given location (i.e., Ucs, typically defined from nozzle center to the vessel center), the relationship between these two remains unexplored quantitatively. Here we advance the model of Kuhn, et al., (PNNL-22816, 2013) as described by Pease, et al. (FEDSM2017-69444, 2017) to evaluate the relationship between the effective clearing/cleaning radius and the nozzle velocity on flat surfaces including flat bottomed vessels. Two governing dimensionless groups are identified. We present both a closed form analytical but transcendental solution and a non-iterative approximation modeled after the Serghides approximation of Colebrook’s nonlinear equation for both ECR versus nozzle velocity and Ucs. Comparison of the model to data from flume testing on a flat surface finds reasonable agreement.
jet, erosion, critical suspension velocity, effect↗
Confronting Domain Shift in Trained Neural Networks
Neural networks (NNs) are known as universal function approximators and can interpolate nonlinear functions between observed data points. However, when the target domain for deployment shifts from the training domain and NNs must extrapolate, the results are notoriously poor. Prior work Martinez et al. (2019) has shown that NN uncertainty estimates can be used to correct binary predictions in shifted domains without retraining the model. We hypothesize that this approach can be extended to correct real-valued time series predictions. As an exemplar, we consider two mechanical systems with nonlinear dynamics. The first system consists of a spring-mass system where the stiffness changes abruptly, and the second is a real experimental system with a frictional joint that is an open challenge for structural dynamicists to model efficiently. Our experiments will test whether 1) NN uncertainty estimates can identify when the input domain has shifted from the training domain and 2) whether the information used to calculate uncertainty estimates can be used to correct the NN’s time series predictions. While the method as proposed did not significantly improve predictions, our results did show potential for modifications that could improve models’ predictions and play a role in structural health monitoring systems that directly impact public safety.
97 MATHEMATICS AND COMPUTING↗
Optical second harmonic generation in anisotropic multilayers with complete multireflection of linear and nonlinear waves using ♯SHAARP.ml package
Abstract Optical second harmonic generation (SHG) is a nonlinear optical effect widely used for nonlinear optical microscopy and laser frequency conversion. Closed-form analytical solution of the nonlinear optical responses is essential for evaluating materials whose optical properties are unknown a priori. A recent open-source code, ♯SHAARP. si , can provide such closed form solutions for crystals with arbitrary symmetries, orientations, and anisotropic properties at a single interface. However, optical components are often in the form of slabs, thin films on substrates, and multilayer heterostructures with multiple reflections of both the fundamental and up to ten different SHG waves at each interface, adding significant complexity. Many approximations have therefore been employed in the existing analytical approaches, such as slowly varying approximation, weak reflection of the nonlinear polarization, transparent medium, high crystallographic symmetry, Kleinman symmetry, easy crystal orientation along a high-symmetry direction, phase matching conditions and negligible interference among nonlinear waves, which may lead to large errors in the reported material properties. To avoid these approximations, we have developed an open-source package named Second Harmonic Analysis of Anisotropic Rotational Polarimetry in Multilayers (♯SHAARP. ml ). The reliability and accuracy are established by experimentally benchmarking with both the SHG polarimetry and Maker fringes using standard and commonly used nonlinear optical materials as well as twisted 2-dimensional heterostructures.
36 MATERIALS SCIENCE↗