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The inverse medium problem, inherently ill-posed and nonlinear, presents significant computational challenges. This study introduces a novel approach by integrating a Neumann series structure within a neural network framework to effectively handle multiparameter inputs. Experiments demonstrate that our methodology not only accelerates computations but also significantly enhances generalization performance, even with varying scattering properties and noisy data. The robustness and adaptability of our framework provide crucial insights and methodologies, extending its applicability to a broad spectrum of scattering problems. These advancements mark a significant step forward in the field, offering a scalable solution to traditionally complex inverse problems.

In this paper, we propose Neumann Series Neural Operator (NSNO) to learn the solution operator of Helmholtz equation from inhomogeneity coefficients and source terms to solutions. Helmholtz equation is a crucial partial differential equation (PDE) with applications in various scientific and engineering fields. However, efficient solver of Helmholtz equation is still a big challenge especially in the case of high wavenumber. Recently, deep learning has shown great potential in solving PDEs especially in learning solution operators. Inspired by Neumann series in Helmholtz equation, we design a novel network architecture in which U-Net is embedded inside to capture the multiscale feature. Extensive experiments show that the proposed NSNO significantly outperforms the state-of-the-art FNO with at least 60\% lower relative $L^2$-error, especially in the large wavenumber case, and has 50\% lower computational cost and less data requirement. Moreover, NSNO can be used as the surrogate model in inverse scattering problems. Numerical tests show that NSNO is able to give comparable results with traditional finite difference forward solver while the computational cost is reduced tremendously.

We propose a novel iterative numerical method to solve the three-dimensional inverse obstacle scattering problem of recovering the shape of the obstacle from far-field measurements. To address the inherent ill-posed nature of the inverse problem, we advocate the use of a trained latent representation of surfaces as the generative prior. This prior enjoys excellent expressivity within the given class of shapes, and meanwhile, the latent dimensionality is low, which greatly facilitates the computation. Thus, the admissible manifold of surfaces is realistic and the resulting optimization problem is less ill-posed. We employ the shape derivative to evolve the latent surface representation, by minimizing the loss, and we provide a local convergence analysis of a gradient descent type algorithm to a stationary point of the loss. We present several numerical examples, including also backscattered and phaseless data, to showcase the effectiveness of the proposed algorithm.

Multimodal emotion recognition (MER) aims to detect the emotional status of a given expression by combining the speech and text information. Intuitively, label information should be capable of helping the model locate the salient tokens/frames relevant to the specific emotion, which finally facilitates the MER task. Inspired by this, we propose a novel approach for MER by leveraging label information. Specifically, we first obtain the representative label embeddings for both text and speech modalities, then learn the label-enhanced text/speech representations for each utterance via label-token and label-frame interactions. Finally, we devise a novel label-guided attentive fusion module to fuse the label-aware text and speech representations for emotion classification. Extensive experiments were conducted on the public IEMOCAP dataset, and experimental results demonstrate that our proposed approach outperforms existing baselines and achieves new state-of-the-art performance., Comment: Accepted by Interspeech 2023

Exploring the origin and properties of magnetic fields is crucial to the development of many fields such as physics, astronomy and meteorology. We focus on the edge element approximation and theoretical analysis of celestial dynamo system with quasi-vacuum boundary conditions. The system not only ensures that the magnetic field on the spherical shell is generated from the dynamo model, but also provides convenience for the application of the edge element method. We demonstrate the existence, uniqueness and stability of the solution to the system by the fixed point theorem. Then, we approximate the system using the edge element method, which is more efficient in dealing with electromagnetic field problems. Moreover, we also discuss the stability of the corresponding discrete scheme. And the convergence is demonstrated by later numerical tests. Finally, we simulate the three-dimensional time evolution of the spherical interface dynamo model, and the characteristics of the simulated magnetic field are consistent with existing work.

We propose reverse time migration (RTM) methods for the imaging of periodic obstacles using only measurements from lower or upper side of the obstacle arrays at a fixed frequency. We analyze the resolution of the lower side and upper side RTM methods in terms of the propagating part of the Rayleigh expansion, Helmholtz-Kirchhoff equation and the distance of the measurement surface to the obstacle arrays. We give some numerical experiments to justify the competitive efficiency of our imaging functionals and the robustness against noises.

Conductivity reconstruction in an inverse eddy current problem is considered in the present paper. With the electric field measurement on part of domain boundary, we formulate the reconstruction problem to a constrained optimization problem with total variation regularization. Existence and stability are proved for the solution to the optimization problem. The finite element method is employed to discretize the optimization problem. The gradient Lipschitz properties of the objective functional are established for the the discrete optimization problems. We propose the alternating direction method of multipliers to solve the discrete problem. Based on the the gradient Lipschitz property, we prove the convergence by extending the admissible set to the whole finite element space. Finally, we show some numerical experiments to illustrate the efficiency of the proposed methods.

We propose an alternating direction method of multipliers (ADMM) to solve an optimization problem stemming from inverse lithography. The objective functional of the optimization problem includes three terms: the misfit between the imaging on wafer and the target pattern, the penalty term which ensures the mask is binary and the total variation regularization term. By variable splitting, we introduce an augmented Lagrangian for the original objective functional. In the framework of ADMM method, the optimization problem is divided into several subproblems. Each of the subproblems can be solved efficiently. We give the convergence analysis of the proposed method. Specially, instead of solving the subproblem concerning sigmoid, we solve directly the threshold truncation imaging function which can be solved analytically. We also provide many numerical examples to illustrate the effectiveness of the method.

Green's function plays a significant role in both theoretical analysis and numerical computing of partial differential equations (PDEs). However, in most cases, Green's function is difficult to compute. The troubles arise in the following three folds. Firstly, compared with the original PDE, the dimension of Green's function is doubled, making it impossible to be handled by traditional mesh-based methods. Secondly, Green's function usually contains singularities which increase the difficulty to get a good approximation. Lastly, the computational domain may be very complex or even unbounded. To override these problems, we leverage the fundamental solution, boundary integral method and neural networks to develop a new method for computing Green's function with high accuracy in this paper. We focus on Green's function of Poisson and Helmholtz equations in bounded domains, unbounded domains. We also consider Poisson equation and Helmholtz domains with interfaces. Extensive numerical experiments illustrate the efficiency and the accuracy of our method for solving Green's function. In addition, we also use the Green's function calculated by our method to solve a class of PDE, and also obtain high-precision solutions, which shows the good generalization ability of our method on solving PDEs.

Mainstream numerical Partial Differential Equation (PDE) solvers require discretizing the physical domain using a mesh. Mesh movement methods aim to improve the accuracy of the numerical solution by increasing mesh resolution where the solution is not well-resolved, whilst reducing unnecessary resolution elsewhere. However, mesh movement methods, such as the Monge-Ampere method, require the solution of auxiliary equations, which can be extremely expensive especially when the mesh is adapted frequently. In this paper, we propose to our best knowledge the first learning-based end-to-end mesh movement framework for PDE solvers. Key requirements of learning-based mesh movement methods are alleviating mesh tangling, boundary consistency, and generalization to mesh with different resolutions. To achieve these goals, we introduce the neural spline model and the graph attention network (GAT) into our models respectively. While the Neural-Spline based model provides more flexibility for large deformation, the GAT based model can handle domains with more complicated shapes and is better at performing delicate local deformation. We validate our methods on stationary and time-dependent, linear and non-linear equations, as well as regularly and irregularly shaped domains. Compared to the traditional Monge-Ampere method, our approach can greatly accelerate the mesh adaptation process, whilst achieving comparable numerical error reduction.

We propose a method combining boundary integral equations and neural networks (BINet) to solve partial differential equations (PDEs) in both bounded and unbounded domains. Unlike existing solutions that directly operate over original PDEs, BINet learns to solve, as a proxy, associated boundary integral equations using neural networks. The benefits are three-fold. Firstly, only the boundary conditions need to be fitted since the PDE can be automatically satisfied with single or double layer representations according to the potential theory. Secondly, the dimension of the boundary integral equations is typically smaller, and as such, the sample complexity can be reduced significantly. Lastly, in the proposed method, all differential operators of the original PDEs have been removed, hence the numerical efficiency and stability are improved. Adopting neural tangent kernel (NTK) techniques, we provide proof of the convergence of BINets in the limit that the width of the neural network goes to infinity. Extensive numerical experiments show that, without calculating high-order derivatives, BINet is much easier to train and usually gives more accurate solutions, especially in the cases that the boundary conditions are not smooth enough. Further, BINet outperforms strong baselines for both one single PDE and parameterized PDEs in the bounded and unbounded domains., Comment: 27 pages, 15 figures

This paper presents our pioneering effort for emotion recognition in conversation (ERC) with pre-trained language models. Unlike regular documents, conversational utterances appear alternately from different parties and are usually organized as hierarchical structures in previous work. Such structures are not conducive to the application of pre-trained language models such as XLNet. To address this issue, we propose an all-in-one XLNet model, namely DialogXL, with enhanced memory to store longer historical context and dialog-aware self-attention to deal with the multi-party structures. Specifically, we first modify the recurrence mechanism of XLNet from segment-level to utterance-level in order to better model the conversational data. Second, we introduce dialog-aware self-attention in replacement of the vanilla self-attention in XLNet to capture useful intra- and inter-speaker dependencies. Extensive experiments are conducted on four ERC benchmarks with mainstream models presented for comparison. The experimental results show that the proposed model outperforms the baselines on all the datasets. Several other experiments such as ablation study and error analysis are also conducted and the results confirm the role of the critical modules of DialogXL., Comment: Accepted by AAAI 2021 main conference

We study an inverse problem associated with an eddy current model. We first address the ill-posedness of the inverse problem by proving the compactness of the forward map with respect to the conductivity and the non-uniqueness of the recovery process. Then by virtue of non-radiating source conceptions, we establish a regularity result for the tangential trace of the true solution on the boundary, which is necessary to justify our subsequent mathematical formulation. After that, we formulate the inverse problem as a constrained optimization problem with an appropriate regularization and prove the existence and stability of the regularized minimizers. To facilitate the numerical solution of the nonlinear non-convex constrained optimization, we introduce a feasible Lagrangian and its discrete variant. Then the gradient of the objective functional is derived using the adjoint technique. By means of the gradient, a nonlinear conjugate gradient method is formulated for solving the optimization system, and a Sobolev gradient is incorporated to accelerate the iterative process. Numerical examples are provided to demonstrate the feasibility of the proposed algorithm.

We propose a new single frequency reverse time migration (RTM) algorithm for imaging extended targets using electromagnetic waves. The imaging functional is defined as the imaginary part of the cross-correlation of the Green function for Helmholtz equation and the back-propagated electromagnetic field. The resolution of our RTM method for both penetrable and non-penetrable extended targets is studied by virtue of Helmholtz-Kirchhoff identity for the time-harmonic Maxwell equation. The analysis implies that our imaging functional is always positive and thus may have better stability properties. Numerical examples are provided to demonstrate the powerful imaging quality and confirm our theoretical results., Comment: 19 pages, 14 figures