Maxwell's equations are a collection of coupled partial differential equations (PDEs) that, together with the Lorentz force law, constitute the basis of classical electromagnetism and electric circuits. Effectively solving Maxwell's equations is crucial in various fields, like electromagnetic scattering and antenna design optimization. Physics-informed neural networks (PINNs) have shown powerful ability in solving PDEs. However, PINNs still struggle to solve Maxwell's equations in heterogeneous media. To this end, we propose a domain-adaptive PINN (da-PINN) to solve inverse problems of Maxwell's equations in heterogeneous media. First, we propose a location parameter of media interface to decompose the whole domain into several sub-domains. Furthermore, the electromagnetic interface conditions are incorporated into a loss function to improve the prediction performance near the interface. Then, we propose a domain-adaptive training strategy for da-PINN. Finally, the effectiveness of da-PINN is verified with two case studies.
相關內容
Networking：IFIP International Conferences on Networking。
Explanation：國際網絡會議(yi)。
Publisher：IFIP。
SIT：
We propose efficient algorithms for enumerating the notorious combinatorial structures of maximal planar graphs, called canonical orderings and Schnyder woods, and the related classical graph drawings by de Fraysseix, Pach, and Pollack [Combinatorica, 1990] and by Schnyder [SODA, 1990], called canonical drawings and Schnyder drawings, respectively. To this aim (i) we devise an algorithm for enumerating special $e$-bipolar orientations of maximal planar graphs, called canonical orientations; (ii) we establish bijections between canonical orientations and canonical drawings, and between canonical orientations and Schnyder drawings; and (iii) we exploit the known correspondence between canonical orientations and canonical orderings, and the known bijection between canonical orientations and Schnyder woods. All our enumeration algorithms have $O(n)$ setup time, space usage, and delay between any two consecutively listed outputs, for an $n$-vertex maximal planar graph.
Local fields, and fields complete with respect to a discrete valuation, are essential objects in commutative algebra, with applications to number theory and algebraic geometry. We formalize in Lean the basic theory of discretely valued fields. In particular, we prove that the unit ball with respect to a discrete valuation on a field is a discrete valuation ring and, conversely, that the adic valuation on the field of fractions of a discrete valuation ring is discrete. We define finite extensions of valuations and of discrete valuation rings, and prove some global-to-local results. Building on this general theory, we formalize the abstract definition and some fundamental properties of local fields. As an application, we show that finite extensions of the field $\mathbb{Q}_p$ of $p$-adic numbers and of the field $\mathbb{F}_p(\!(X)\!)$ of Laurent series over $\mathbb{F}_p$ are local fields.
Current AI-based methods do not provide comprehensible physical interpretations of the utilized data, extracted features, and predictions/inference operations. As a result, deep learning models trained using high-resolution satellite imagery lack transparency and explainability and can be merely seen as a black box, which limits their wide-level adoption. Experts need help understanding the complex behavior of AI models and the underlying decision-making process. The explainable artificial intelligence (XAI) field is an emerging field providing means for robust, practical, and trustworthy deployment of AI models. Several XAI techniques have been proposed for image classification tasks, whereas the interpretation of image segmentation remains largely unexplored. This paper offers to bridge this gap by adapting the recent XAI classification algorithms and making them usable for muti-class image segmentation, where we mainly focus on buildings' segmentation from high-resolution satellite images. To benchmark and compare the performance of the proposed approaches, we introduce a new XAI evaluation methodology and metric based on "Entropy" to measure the model uncertainty. Conventional XAI evaluation methods rely mainly on feeding area-of-interest regions from the image back to the pre-trained (utility) model and then calculating the average change in the probability of the target class. Those evaluation metrics lack the needed robustness, and we show that using Entropy to monitor the model uncertainty in segmenting the pixels within the target class is more suitable. We hope this work will pave the way for additional XAI research for image segmentation and applications in the remote sensing discipline.
Probabilistic numerical solvers for ordinary differential equations (ODEs) treat the numerical simulation of dynamical systems as problems of Bayesian state estimation. Aside from producing posterior distributions over ODE solutions and thereby quantifying the numerical approximation error of the method itself, one less-often noted advantage of this formalism is the algorithmic flexibility gained by formulating numerical simulation in the framework of Bayesian filtering and smoothing. In this paper, we leverage this flexibility and build on the time-parallel formulation of iterated extended Kalman smoothers to formulate a parallel-in-time probabilistic numerical ODE solver. Instead of simulating the dynamical system sequentially in time, as done by current probabilistic solvers, the proposed method processes all time steps in parallel and thereby reduces the span cost from linear to logarithmic in the number of time steps. We demonstrate the effectiveness of our approach on a variety of ODEs and compare it to a range of both classic and probabilistic numerical ODE solvers.
We show that it is possible to learn an open-loop policy in simulation for the dynamic manipulation of a deformable linear object (DLO) -- e.g., a rope, wire, or cable -- that can be executed by a real robot without additional training. Our method is enabled by integrating an existing state-of-the-art DLO model (Discrete Elastic Rods) with MuJoCo, a robot simulator. We describe how this integration was done, check that validation results produced in simulation match what we expect from analysis of the physics, and apply policy optimization to train an open-loop policy from data collected only in simulation that uses a robot arm to fling a wire precisely between two obstacles. This policy achieves a success rate of 76.7% when executed by a real robot in hardware experiments without additional training on the real task.
We derive a family of efficient constrained dynamics algorithms by formulating an equivalent linear quadratic regulator (LQR) problem using Gauss principle of least constraint and solving it using dynamic programming. Our approach builds upon the pioneering (but largely unknown) O(n + m^2d + m^3) solver by Popov and Vereshchagin (PV), where n, m and d are the number of joints, number of constraints and the kinematic tree depth respectively. We provide an expository derivation for the original PV solver and extend it to floating-base kinematic trees with constraints allowed on any link. We make new connections between the LQR's dual Hessian and the inverse operational space inertia matrix (OSIM), permitting efficient OSIM computation, which we further accelerate using matrix inversion lemma. By generalizing the elimination ordering and accounting for MUJOCO-type soft constraints, we derive two original O(n + m) complexity solvers. Our numerical results indicate that significant simulation speed-up can be achieved for high dimensional robots like quadrupeds and humanoids using our algorithms as they scale better than the widely used O(nd^2 + m^2d + d^2m) LTL algorithm of Featherstone. The derivation through the LQR-constrained dynamics connection can make our algorithm accessible to a wider audience and enable cross-fertilization of software and research results between the fields
Performance analysis is carried out in a near-field multiple-input multiple-output (MIMO) system for both discrete and continuous aperture antennas. The effective degrees of freedom (EDoF) is first derived. It is shown that near-field MIMO systems have a higher EDoF than free-space far-field ones. Additionally, the near-field EDoF further depends on the communication distance. Based on the derived EDoF, closed-form expressions of channel capacity with a fixed distance are obtained. As a further advance, with randomly deployed receivers, ergodic capacity is derived. Simulation results reveal that near-field MIMO has an enhanced multiplexing gain even under line-of-sight transmissions. In addition, the performance of discrete MIMO converges to that of continuous aperture MIMO.
We introduce a neural-preconditioned iterative solver for Poisson equations with mixed boundary conditions. The Poisson equation is ubiquitous in scientific computing: it governs a wide array of physical phenomena, arises as a subproblem in many numerical algorithms, and serves as a model problem for the broader class of elliptic PDEs. The most popular Poisson discretizations yield large sparse linear systems. At high resolution, and for performance-critical applications, iterative solvers can be advantageous for these -- but only when paired with powerful preconditioners. The core of our solver is a neural network trained to approximate the inverse of a discrete structured-grid Laplace operator for a domain of arbitrary shape and with mixed boundary conditions. The structure of this problem motivates a novel network architecture that we demonstrate is highly effective as a preconditioner even for boundary conditions outside the training set. We show that on challenging test cases arising from an incompressible fluid simulation, our method outperforms state-of-the-art solvers like algebraic multigrid as well as some recent neural preconditioners.
Due to the large state space of the two-qubit system, and the adoption of ladder reward function in the existing quantum state preparation methods, the convergence speed is slow and it is difficult to prepare the desired target quantum state with high fidelity under limited conditions. To solve the above problems, a difference-driven reinforcement learning (RL) algorithm for quantum state preparation of two-qubit system is proposed by improving the reward function and action selection strategy. Firstly, a model is constructed for the problem of preparing quantum states of a two-qubit system, with restrictions on the type of quantum gates and the time for quantum state evolution. In the preparation process, a weighted differential dynamic reward function is designed to assist the algorithm quickly obtain the maximum expected cumulative reward. Then, an adaptive e-greedy action selection strategy is adopted to achieve a balance between exploration and utilization to a certain extent, thereby improving the fidelity of the final quantum state. The simulation results show that the proposed algorithm can prepare quantum state with high fidelity under limited conditions. Compared with other algorithms, it has different degrees of improvement in convergence speed and fidelity of the final quantum state.
Business optimisation is the process of finding and implementing efficient and cost-effective means of operation to bring a competitive advantage for businesses. Synthesizing problem formulations is an integral part of business optimisation which is centred around human expertise, thus with a high potential of becoming a bottleneck. With the recent advancements in Large Language Models (LLMs), human expertise needed in problem formulation can potentially be minimized using Artificial Intelligence (AI). However, developing a LLM for problem formulation is challenging, due to training data requirements, token limitations, and the lack of appropriate performance metrics in LLMs. To minimize the requirement of large training data, considerable attention has recently been directed towards fine-tuning pre-trained LLMs for downstream tasks, rather than training a LLM from scratch for a specific task. In this paper, we adopt this approach and propose an AI-Copilot for business optimisation by fine-tuning a pre-trained LLM for problem formulation. To address token limitations, we introduce modularization and prompt engineering techniques to synthesize complex problem formulations as modules that fit into the token limits of LLMs. In addition, we design performance evaluation metrics that are more suitable for assessing the accuracy and quality of problem formulations compared to existing evaluation metrics. Experiment results demonstrate that our AI-Copilot can synthesize complex and large problem formulations for a typical business optimisation problem in production scheduling.
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