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Physics-informed neural networks (PINNs) have emerged as powerful tools for solving a wide range of partial differential equations (PDEs). However, despite their user-friendly interface and broad applicability, PINNs encounter challenges in accurately resolving PDEs, especially when dealing with singular cases that may lead to unsatisfactory local minima. To address these challenges and improve solution accuracy, we propose an innovative approach called Annealed Adaptive Importance Sampling (AAIS) for computing the discretized PDE residuals of the cost functions, inspired by the Expectation Maximization algorithm used in finite mixtures to mimic target density. Our objective is to approximate discretized PDE residuals by strategically sampling additional points in regions with elevated residuals, thus enhancing the effectiveness and accuracy of PINNs. Implemented together with a straightforward resampling strategy within PINNs, our AAIS algorithm demonstrates significant improvements in efficiency across a range of tested PDEs, even with limited training datasets. Moreover, our proposed AAIS-PINN method shows promising capabilities in solving high-dimensional singular PDEs. The adaptive sampling framework introduced here can be integrated into various PINN frameworks.

We propose a novel method, VectorPainter, for the task of stylized vector graphics synthesis. Given a text prompt and a reference style image, VectorPainter generates a vector graphic that aligns in content with the text prompt and remains faithful in style to the reference image. We recognize that the key to this task lies in fully leveraging the intrinsic properties of vector graphics. Innovatively, we conceptualize the stylization process as the rearrangement of vectorized strokes extracted from the reference image. VectorPainter employs an optimization-based pipeline. It begins by extracting vectorized strokes from the reference image, which are then used to initialize the synthesis process. To ensure fidelity to the reference style, a novel style preservation loss is introduced. Extensive experiments have been conducted to demonstrate that our method is capable of aligning with the text description while remaining faithful to the reference image.

Our aim is to study the backward problem, i.e. recover the initial data from the terminal observation, of the subdiffusion with time dependent coefficients. First of all, by using the smoothing property of solution operators and a perturbation argument of freezing the diffusion coefficients, we show a stability estimate in Sobolev spaces, under some smallness/largeness condition on the terminal time. Moreover, in case of noisy observation, we apply a quasi-boundary value method to regularize the problem and then show the convergence of the regularization scheme. Finally, to numerically reconstruct the initial data, we propose a completely discrete scheme by applying the finite element method in space and backward Euler convolution quadrature in time. An \textsl{a priori} error estimate is then established. The proof is heavily built on a perturbation argument dealing with time dependent coefficients and some nonstandard error estimates for the direct problem. The error estimate gives a useful guide for balancing discretization parameters, regularization parameter and noise level. Some numerical experiments are presented to illustrate our theoretical results., Comment: 25 pages

The aim of this paper is to study the recovery of a spatially dependent potential in a (sub)diffusion equation from overposed final time data. We construct a monotone operator one of whose fixed points is the unknown potential. The uniqueness of the identification is theoretically verified by using the monotonicity of the operator and a fixed point argument. Moreover, we show a conditional stability in Hilbert spaces under some suitable conditions on the problem data. Next, a completely discrete scheme is developed, by using Galerkin finite element method in space and finite difference method in time, and then a fixed point iteration is applied to reconstruct the potential. We prove the linear convergence of the iterative algorithm by the contraction mapping theorem, and present a thorough error analysis for the reconstructed potential. Our derived \textsl{a priori} error estimate provides a guideline to choose discretization parameters according to the noise level. The analysis relies heavily on some suitable nonstandard error estimates for the direct problem as well as the aforementioned conditional stability. Numerical experiments are provided to illustrate and complement our theoretical analysis., Comment: 23 pages, 21 Figures

We aim at the development and analysis of the numerical schemes for approximately solving the backward diffusion-wave problem, which involves a fractional derivative in time with order $\alpha\in(1,2)$. From terminal observations at two time levels, i.e., $u(T_1)$ and $u(T_2)$, we simultaneously recover two initial data $u(0)$ and $u_t(0)$ and hence the solution $u(t)$ for all $t > 0$. First of all, existence, uniqueness and Lipschitz stability of the backward diffusion-wave problem were established under some conditions about $T_1$ and $T_2$. Moreover, for noisy data, we propose a quasi-boundary value scheme to regularize the "mildly" ill-posed problem, and show the convergence of the regularized solution. Next, to numerically solve the regularized problem, a fully discrete scheme is proposed by applying finite element method in space and convolution quadrature in time. We establish error bounds of the discrete solution in both cases of smooth and nonsmooth data. The error estimate is very useful in practice since it indicates the way to choose discretization parameters and regularization parameter, according to the noise level. The theoretical results are supported by numerical experiments., Comment: 29 pages

The aim of this paper is to develop and analyze numerical schemes for approximately solving the backward problem of subdiffusion equation involving a fractional derivative in time with order $\alpha\in(0,1)$. After using quasi-boundary value method to regularize the "mildly" ill-posed problem, we propose a fully discrete scheme by applying finite element method (FEM) in space and convolution quadrature (CQ) in time. We provide a thorough error analysis of the resulting discrete system in both cases of smooth and nonsmooth data. The analysis relies heavily on smoothing properties of (discrete) solution operators, and nonstandard error estimate for the direct problem in terms of problem data regularity. The theoretical results are useful to balance discretization parameters, regularization parameter and noise level. Numerical examples are presented to illustrate the theoretical results., Comment: 26 pages, 2 figures