• Abstract

This paper presents an a priori error analysis of the Deep Mixed Residual method (MIM) for solving high-order elliptic equations with non-homogeneous boundary conditions, including Dirichlet, Neumann, and Robin conditions. We examine MIM with two types of loss functions, referred to as first-order and second-order least squares systems. By providing boundedness and coercivity analysis, we leverage Céa’s Lemma to decompose the total error into the approximation, generalization, and optimization errors. Utilizing the Barron space theory and Rademacher complexity, an a priori error is derived regarding the training samples and network size that are exempt from the curse of dimensionality. Our results reveal that MIM significantly reduces the regularity requirements for activation functions compared to the deep Ritz method, implying the effectiveness of MIM in solving high-order equations.

• General Summary

• Abstract

In this paper, we consider the algorithms and convergence for a general optimization problem, which has a wide range of applications in image segmentation, topology optimization, flow network formulation, and surface reconstruction. In particular, the problem focuses on interface related optimization problems where the interface is implicitly described by characteristic functions of the corresponding domains. Under such representation and discretization, the problem is then formulated into a discretized optimization problem where the objective function is concave with respect to characteristic functions and convex with respect to state variables. We show that under such structure, the iterative scheme based on alternative minimization can converge to a local minimizer. Extensive numerical examples are performed to support the theory.

• General Summary

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Deep generative models aim to learn the underlying distribution of data and generate new ones. Despite the diversity of generative models and their high-quality generation performance in practice, most of them lack rigorous theoretical convergence proofs. In this work, we aim to establish some convergence results for OT-Flow, one of the deep generative models. First, by reformulating the framework of OT-Flow model, we establish the $Γ$-convergence of the formulation of OT-Flow to the corresponding optimal transport (OT) problem as the regularization term parameter $α$ goes to infinity. Second, since the loss function will be approximated by Monte Carlo method in training, we established the convergence between the discrete loss function and the continuous one when the sample number $N$ goes to infinity as well. Meanwhile, the approximation capability of the neural network provides an upper bound for the discrete loss function of the minimizers. The proofs in both aspects provide convincing assurances for the stability of OT-Flow.

• General Summary

• Abstract

We consider a general convex-concave saddle point problem that frequently arises in large-scale image processing. First-order primal-dual algorithms have garnered significant attention due to their promising results in solving saddle point problems. Notably, these algorithms exhibit improved performance with larger step sizes. In a recent series of articles, the upper bound on step sizes has been increased, thereby relaxing the convergence-guaranteeing condition. This paper analyzes the generalized primal-dual method proposed in [B. He, F. Ma, S. Xu, X. Yuan, SIAM J. Imaging Sci. 15 (2022)] and introduces a better condition to ensure its convergence. This enhanced condition also encompasses the optimal upper bound of step sizes in the primal-dual hybrid gradient method. We establish both the global convergence of the iterates and the ergodic $\mathcal{O}(1/N)$ convergence rate for the objective function value in the generalized primal-dual algorithm under the enhanced condition.

• General Summary

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Inspired by the row and column action methods for solving large-scale linear systems, in this work, we explore the use of frontal slices for solving tensor linear systems. In particular, this paper presents a novel approach for using frontal slices of a tensor $\mathcal{A}$ to solve tensor linear systems $\mathcal{A}∗\mathcal{X} =\mathcal{B}$ where ∗ denotes the $t$-product. In addition, we consider variations of this method, including cyclic, block, and randomized approaches, each designed to optimize performance in different operational contexts. Our primary contribution lies in the development and convergence analysis of these methods. Experimental results on synthetically generated and real-world data, including applications such as image and video deblurring, demonstrate the efficacy of our proposed approaches and validate our theoretical findings.

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In this paper, we solve the inverse source problem of fractional evolution PDEs by MC-fPINNs. We construct the loss function in terms of the governing equation residual, boundary residual, initial residual and measurement data with noise. Meanwhile, we present a rigorous error analysis of this method. In the experimental section, we present the reconstruction outcomes of the source term for three evolutionary fractional partial differential equations (fPDEs): the evolutionary fractional Laplacian equation, the time-space fractional diffusion equation, and the fractional advection-diffusion equation. These experiments illustrate robust performance of MC-fPINNs in both low-dimensional and high-dimensional scenarios. Our results confirm the effectiveness of MC-fPINNs in solving such inverse source problem, and also provide a theoretical foundation to choose neural networks parameters in this algorithm.

• General Summary

• Abstract

This paper proposes a class of physics-informed neural networks called noise robust physics-informed generative adversarial networks (NR-PIGANs) to solve stochastic differential equations in the presence of noisy measurements. In these scenarios, while the governing equations are known, only a limited number of sensor measurements of the system parameters are available, and some may contain significant measurement errors. To address this, NR-PIGAN incorporates an additional noise generator with specific distribution constraints into a physics-informed generative adversarial network framework. The noise generator is trained alongside the clean data generators in an end-to-end manner, enabling the model to effectively capture both clean and noisy data distributions under the given physical constraints. Numerical experiments demonstrate that NR-PIGAN excels in handling forward and inverse problems under diverse noise perturbations, and its advantage becomes more pronounced as the noise level increases.

• General Summary