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Phase-field models have been widely used to investigate the phase transformation phenomena. However, it is difficult to solve the problems numerically due to their strong nonlinearities and higher-order terms. This work is devoted to solving forward and inverse problems of the phase-field models by a novel deep learning framework named Phase-Field Weak-form Neural Networks (PFWNN), which is based on the weak forms of the phase-field equations. In this framework, the weak solutions are parameterized as deep neural networks with a periodic layer, while the test function space is constructed by functions compactly supported in small regions. The PFWNN can efficiently solve the phase-field equations characterizing the sharp transitions and identify the important parameters by employing the weak forms. It also allows local training in small regions, which significantly reduce the computational cost. Moreover, it can guarantee the residual descending along the time marching direction, enhancing the convergence of the method. Numerical examples are presented for several benchmark problems. The results validate the efficiency and accuracy of the PFWNN. This work also sheds light on solving the forward and inverse problems of general high-order time-dependent partial differential equations.

This paper focuses on integrating the networks and adversarial training into constrained optimization problems to develop a framework algorithm for constrained optimization problems. For such problems, we first transform them into minimax problems using the augmented Lagrangian method and then use two (or several) deep neural networks(DNNs) to represent the primal and dual variables respectively. The parameters in the neural networks are then trained by an adversarial process. The proposed architecture is relatively insensitive to the scale of values of different constraints when compared to penalty based deep learning methods. Through this type of training, the constraints are imposed better based on the augmented Lagrangian multipliers. Extensive examples for optimization problems with scalar constraints, nonlinear constraints, partial differential equation constraints, and inequality constraints are considered to show the capability and robustness of the proposed method, with applications ranging from Ginzburg--Landau energy minimization problems, partition problems, fluid-solid topology optimization, to obstacle problems., Comment: 24 pages, 18 figures

Recently, studies have shown the potential of integrating field-type iterative methods with deep learning (DL) techniques in solving inverse scattering problems (ISPs). In this article, we propose a novel Variational Born Iterative Network, namely, VBIM-Net, to solve the full-wave ISPs with significantly improved flexibility and inversion quality. The proposed VBIM-Net emulates the alternating updates of the total electric field and the contrast in the variational Born iterative method (VBIM) by multiple layers of subnetworks. We embed the calculation of the contrast variation into each of the subnetworks, converting the scattered field residual into an approximate contrast variation and then enhancing it by a U-Net, thus avoiding the requirement of matched measurement dimension and grid resolution as in existing approaches. The total field and contrast of each layer's output is supervised in the loss function of VBIM-Net, which guarantees the physical interpretability of variables of the subnetworks. In addition, we design a training scheme with extra noise to enhance the model's stability. Extensive numerical results on synthetic and experimental data both verify the inversion quality, generalization ability, and robustness of the proposed VBIM-Net. This work may provide some new inspiration for the design of efficient field-type DL schemes., Comment: 14 pages, 21 figures

This paper proposes a new boundary integral equation (BIE) methodology based on the perfectly matched layer (PML) truncation technique for solving the electromagnetic scattering problems in a multi-layered medium. Instead of using the original PML stretched fields, artificial fields which are also equivalent to the solutions in the physical region are introduced. This significantly simplifies the study of the proposed methodology to derive the PML problem. Then some PML transformed layer potentials and the associated boundary integral operators (BIOs) are defined and the corresponding jump relations are shown. Under the assumption that the fields vanish on the PML boundary, the solution representations, as well as the related BIEs and regularization of the hyper-singular operators, in terms of the current density functions on the truncated interface, are derived. Numerical experiments are presented to demonstrate the efficiency and accuracy of the method., Comment: 26 pages, 13 figures

This paper investigates the scattering of biharmonic waves by a one-dimensional periodic array of cavities embedded in an infinite elastic thin plate. The transparent boundary conditions are introduced to formulate the problem from an unbounded domain to a bounded one. The well-posedness of the associated variational problem is demonstrated utilizing the Fredholm alternative theorem. The perfectly matched layer (PML) method is employed to reformulate the original scattering problem, transforming it from an unbounded domain to a bounded one. The transparent boundary conditions for the PML problem are deduced, and the well-posedness of its variational problem is established. Moreover, exponential convergence is achieved between the solution of the PML problem and that of the original scattering problem.

469219 Kamo`oalewa is selected as one of the primary targets of Tianwen-2 mission, which is currently believed to be the most stable quasi-satellite of Earth. Here we derive a weak detection of the Yarkovsky effect for Kamo`oalewa, giving $A_2 = -1.075\pm0.447\times 10^{-13} \rm{au/d}^2$, with the available ground-based optical observations from Minor Planet Center and a relatively conservative weighting scheme. Due to the quasi-satellite resonance with Earth, we show that the detection of Yarkovsky effect by orbital fitting with astrometric observations becomes difficult as its orbital drift shows a slow oscillatory growth resulting from the Yarkovsky effect. In addition, we extensively explore the characteristics of orbital uncertainty propagation and find that the positional uncertainty mainly arises from the geocentric radial direction in 2010-2020, and then concentrates in the heliocentric transverse direction in 2020-2030. Furthermore, the heliocentric transverse uncertainty is clearly monthly dependent, which can arrive at a minimum around January and a maximum around July as the orbit moves towards the leading and trailing edges, respectively, in 2025-2027. Finally, we investigate a long-term uncertainty propagation in the quasi-satellite regime, implying that the quasi-satellite resonance with Earth may play a crucial role in constraining the increase of uncertainty over time. Such interesting feature further implies that the orbital precision of Kamo`oalewa is relatively stable at its quasi-satellite phase, which may also be true for other quasi-satellites of Earth., Comment: 15 pages, 11 figures, accepted for publication in AJ

Deep neural networks (DNNs) have been widely used to solve partial differential equations (PDEs) in recent years. In this work, a novel deep learning-based framework named Particle Weak-form based Neural Networks (ParticleWNN) is developed for solving PDEs in the weak form. In this framework, the trial space is defined as the space of DNNs, while the test space consists of functions compactly supported in extremely small regions, centered around particles. To facilitate the training of neural networks, an R-adaptive strategy is designed to adaptively modify the radius of regions during training. The ParticleWNN inherits the benefits of weak/variational formulation, requiring less regularity of the solution and a small number of quadrature points for computing integrals. Additionally, due to the special construction of the test functions, ParticleWNN enables parallel implementation and integral calculations only in extremely small regions. This framework is particularly desirable for solving problems with high-dimensional and complex domains. The efficiency and accuracy of ParticleWNN are demonstrated through several numerical examples, showcasing its superiority over state-of-the-art methods. The source code for the numerical examples presented in this paper is available at https://github.com/yaohua32/ParticleWNN.

Accurate evaluation of nearly singular integrals plays an important role in many boundary integral equation based numerical methods. In this paper, we propose a variant of singularity swapping method to accurately evaluate the layer potentials for arbitrarily close targets. Our method is based on the global trapezoidal rule and trigonometric interpolation, resulting in an explicit quadrature formula. The method achieves spectral accuracy for nearly singular integrals on closed analytic curves. In order to extract the singularity from the complexified distance function, an efficient root finding method is proposed based on contour integration. Through the change of variables, we also extend the quadrature method to integrals on the piecewise analytic curves. Numerical examples for Laplace's and Helmholtz equations show that high order accuracy can be achieved for arbitrarily close field evaluation.

This work establishes a framework for solving inverse boundary problems with the geodesic based quadratic Wasserstein distance ($W_{2}$). A general form of the Fr\'echet gradient is systematically derived by optimal transportation (OT) theory. In addition, a fast algorithm based on the new formulation of OT on $\mathbb{S}^{1}$ is developed to solve the corresponding optimal transport problem. The computational complexity of the algorithm is reduced to $O(N)$ from $O(N^{3})$ of the traditional method. Combining with the adjoint-state method, this framework provides a new computational approach for solving the challenging electrical impedance tomography (EIT) problem. Numerical examples are presented to illustrate the effectiveness of our method.

Stability results for the Helmholtz equations in both deterministic and random periodic structures are proved in this paper. Under the assumption of excluding resonances, by a variational method and Fourier analysis in the energy space, the stability estimate for the Helmholtz equation in a deterministic periodic structure is established. For the stochastic case, by introducing a variable transform, the variational formulation of the scattering problem in a random domain is reduced to that in a definite domain with random medium. Combining the stability result for the deteministic case with regularity and stochastic regularity of the scattering surface, Pettis measurability theorem and Bochner's Theorem further yield the stability result for the scattering problem by random periodic structures. Both stability estimates are explicit with respect to the wavenumber.

This paper is concerned with a numerical solution to the scattering of a time-harmonic electromagnetic wave by a bounded and impenetrable obstacle in three dimensions. The electromagnetic wave propagation is modeled by a boundary value problem of Maxwell's equations in the exterior domain of the obstacle. Based on the Dirichlet-to-Neumann (DtN) operator, which is defined by an infinite series, an exact transparent boundary condition is introduced and the scattering problem is reduced equivalently into a bounded domain. An a posteriori error estimate based adaptive finite element DtN method is developed to solve the discrete variational problem, where the DtN operator is truncated into a sum of finitely many terms. The a posteriori error estimate takes into account both the finite element approximation error and the truncation error of the DtN operator. The latter is shown to decay exponentially with respect to the truncation parameter. Numerical experiments are presented to illustrate the effectiveness of the proposed method.

The paper is concerned with an inverse point source problem for the Helmholtz equation. It consists of recovering the locations and amplitudes of a finite number of radiative point sources inside a given inhomogeneous medium from the knowledge of a single boundary measurement. The main result of the paper is a new H\"{o}lder type stability estimate for the inversion under the assumption that the point sources are well separated. The proof of the stability is based on a combination of Carleman estimates and a technique for proving uniqueness of the Cauchy problem.

Big progress has been achieved in domain adaptation in decades. Existing works are always based on an ideal assumption that testing target domain are i.i.d. with training target domains. However, due to unpredictable corruptions (e.g., noise and blur) in real data like web images, domain adaptation methods are increasingly required to be corruption robust on target domains. In this paper, we investigate a new task, Corruption-agnostic Robust Domain Adaptation (CRDA): to be accurate on original data and robust against unavailable-for-training corruptions on target domains. This task is non-trivial due to large domain discrepancy and unsupervised target domains. We observe that simple combinations of popular methods of domain adaptation and corruption robustness have sub-optimal CRDA results. We propose a new approach based on two technical insights into CRDA: 1) an easy-to-plug module called Domain Discrepancy Generator (DDG) that generates samples that enlarge domain discrepancy to mimic unpredictable corruptions; 2) a simple but effective teacher-student scheme with contrastive loss to enhance the constraints on target domains. Experiments verify that DDG keeps or even improves performance on original data and achieves better corruption robustness that baselines., Comment: The first literature to investigate the topic of corruption-agnostic robust domain adaptation, a new practical and challenging domain adaptation setting

Consider the elastic scattering of an incident wave by a rigid obstacle in three dimensions, which is formulated as an exterior problem for the Navier equation. By constructing a Dirichlet-to-Neumann (DtN) operator and introducing a transparent boundary condition, the scattering problem is reduced equivalently to a boundary value problem in a bounded domain. The discrete problem with the truncated DtN operator is solved by using the a posteriori error estimate based adaptive finite element method. The estimate takes account of both the finite element approximation error and the truncation error of the DtN operator, where the latter is shown to converge exponentially with respect to the truncation parameter. Moreover, the generalized Woodbury matrix identity is utilized to solve the resulting linear system efficiently. Numerical experiments are presented to demonstrate the superior performance of the proposed method.

This is a perspective paper inspired from the study of Turing Test proposed by A.M. Turing (23 June 1912 - 7 June 1954) in 1950. Following one important implication of Turing Test for enabling a machine with a human-like behavior or performance, we define human-like robustness (HLR) for AI machines. The objective of the new definition aims to enforce AI machines with HLR, including to evaluate them in terms of HLR. A specific task is discussed only on object identification, because it is the most common task for every person in daily life. Similar to the perspective, or design, position by Turing, we provide a solution of how to achieve HLR AI machines without constructing them and conducting real experiments. The solution should consists of three important features in the machines. The first feature of HLR machines is to utilize common sense from humans for realizing a causal inference. The second feature is to make a decision from a semantic space for having interpretations to the decision. The third feature is to include a "human-in-the-loop" setting for advancing HLR machines. We show an "identification game" using proposed design of HLR machines. The present paper shows an attempt to learn and explore further from Turing Test towards the design of human-like AI machines., Comment: 6 pages, 6 figures

In this comprehensive review, we describe a new mathematical problem in artificial intelligence (AI) from a mathematical modeling perspective, following the philosophy stated by Rudolf E. Kalman that "Once you get the physics right, the rest is mathematics". The new problem is called "Generalized Constraints (GCs)", and we adopt GCs as a general term to describe any type of prior information in modelings. To understand better about GCs to be a general problem, we compare them with the conventional constraints (CCs) and list their extra challenges over CCs. In the construction of AI machines, we basically encounter more often GCs for modeling, rather than CCs with well-defined forms. Furthermore, we discuss the ultimate goals of AI and redefine transparent, interpretable, and explainable AI in terms of comprehension levels about machines. We review the studies in relation to the GC problems although most of them do not take the notion of GCs. We demonstrate that if AI machines are simplified by a coupling with both knowledge-driven submodel and data-driven submodel, GCs will play a critical role in a knowledge-driven submodel as well as in the coupling form between the two submodels. Examples are given to show that the studies in view of a generalized constraint problem will help us perceive and explore novel subjects in AI, or even in mathematics, such as generalized constraint learning (GCL)., Comment: 20 pages, 16 figures, 3 tables

This paper proposes a new and efficient numerical algorithm for recovering the damping coefficient from the spectrum of a damped wave operator, which is a classical Borg-Levinson inverse spectral problem. The algorithm is based on inverting a sequence of trace formulas, which are deduced by a recursive formula, bridging geometrical and spectrum information explicitly in terms of Fredholm integral equations. Numerical examples are presented to illustrate the efficiency of the proposed algorithm.

Consider the scattering of a time-harmonic elastic plane wave by a bi-periodic rigid surface. The displacement of elastic wave motion is modeled by the three-dimensional Navier equation in an open domain above the surface. Based on the Dirichlet-to-Neumann (DtN) operator, which is given as an infinite series, an exact transparent boundary condition is introduced and the scattering problem is formulated equivalently into a boundary value problem in a bounded domain. An a posteriori error estimate based adaptive finite element DtN method is proposed to solve the discrete variational problem where the DtN operator is truncated into a finite number of terms. The a posteriori error estimate takes account of the finite element approximation error and the truncation error of the DtN operator which is shown to decay exponentially with respect to the truncation parameter. Numerical experiments are presented to illustrate the effectiveness of the proposed method.

Consider the scattering of a time-harmonic electromagnetic plane wave by an open cavity which is embedded in a perfectly electrically conducting infinite ground plane. This paper concerns the numerical solutions of the open cavity scattering problems in both transverse magnetic and transverse electric polarizations. Based on the Dirichlet-to-Neumann (DtN) map for each polarization, a transparent boundary condition is imposed to reduce the scattering problem equivalently into a boundary value problem in a bounded domain. An a posteriori error estimate based adaptive finite element DtN method is proposed. The estimate consists of the finite element approximation error and the truncation error of the DtN operator, which is shown to decay exponentially with respect to the truncation parameter. Numerical experiments are presented for both polarizations to illustrate the competitive behavior of the adaptive method.

This paper develops an efficient numerical method for the inverse scattering problem of a time-harmonic plane wave incident on a perfectly reflecting random periodic structure. The method is based on a novel combination of the Monte Carlo technique for sampling the probability space, a continuation method with respect to the wavenumber, and the Karhunen-Lo$\grave{e}$ve expansion of the random structure, which reconstructs key statistical properties of the profile for the unknown random periodic structure from boundary measurements of the scattered fields away from the structure. Numerical results are presented to demonstrate the reliability and efficiency of the proposed method.

We consider a weak adversarial network approach to numerically solve a class of inverse problems, including electrical impedance tomography and dynamic electrical impedance tomography problems. We leverage the weak formulation of PDE in the given inverse problem, and parameterize the solution and the test function as deep neural networks. The weak formulation and the boundary conditions induce a minimax problem of a saddle function of the network parameters. As the parameters are alternatively updated, the network gradually approximates the solution of the inverse problem. We provide theoretical justifications on the convergence of the proposed algorithm. Our method is completely mesh-free without any spatial discretization, and is particularly suitable for problems with high dimensionality and low regularity on solutions. Numerical experiments on a variety of test inverse problems demonstrate the promising accuracy and efficiency of our approach.