一种用于带有复杂先验的 PDE 约束逆问题的基于分数的生成求解器
A Score-based Generative Solver for PDE-constrained Inverse Problems with Complex Priors
AI总结：
针对带有复杂先验的 PDE 约束逆问题，提出一种结合随机微分方程的基于分数的扩散模型与物理信息 CNN 代理，高效实现后验采样并显著提升逆估计精度。
摘要：
在由偏微分方程（PDEs）建模的系统的逆估计领域中，估计高维（甚至无限维）参数时出现了挑战。通常，此类问题的不适定性使得必须利用先验信息来实现适定性。在大多数现有的逆求解器中，先验分布被假设为高斯或拉普拉斯形式，这在许多实际场景中是一种过度简化。在先验复杂且似然模型计算代价高昂（例如由于昂贵的正向模型）的情况下，从此类后验中抽取样本在计算上可能是不可行的，特别是当未知参数为高维时。在本工作中，为了高效采样，我们提出了一种基于分数的扩散模型，该模型将基于分数的生成采样工具与由随机微分方程驱动的加噪和去噪过程相结合。该工具用于根据后验分布进行迭代样本生成，同时学习并利用给定复杂先验中固有的底层信息和约束。提出了一种时变时间表以使该方法适应后验采样。为了加速非参数化 PDE 的模拟并增强泛化能力，我们为正向模型引入了物理信息卷积神经网络（CNN）代理。最后，数值实验（包括超弹性问题和多尺度力学问题）证明了所提方法的有效性。具体而言，基于分数的扩散模型与物理信息 CNN 代理相结合，有效地从提供的先验样本中学习几何特征，与最先进的技术相比，产生了更好的逆估计结果。
英文摘要：
In the field of inverse estimation for systems modeled by partial differential equations (PDEs), challenges arise when estimating high- (or even infinite-) dimensional parameters. Typically, the ill-posed nature of such problems necessitates leveraging prior information to achieve well-posedness. In most existing inverse solvers, the prior distribution is assumed to be of either Gaussian or Laplace form which, in many practical scenarios, is an oversimplification. In case the prior is complex and the likelihood model is computationally expensive (e.g., due to expensive forward models), drawing the sample from such posteriors can be computationally intractable, especially when the unknown parameter is high-dimensional. In this work, to sample efficiently, we propose a score-based diffusion model, which combines a score-based generative sampling tool with a noising and denoising process driven by stochastic differential equations. This tool is used for iterative sample generation in accordance with the posterior distribution, while simultaneously learning and leveraging the underlying information and constraints inherent in the given complex prior. A time-varying time schedule is proposed to adapt the method for posterior sampling. To expedite the simulation of non-parameterized PDEs and enhance the generalization capacity, we introduce a physics-informed convolutional neural network (CNN) surrogate for the forward model. Finally, numerical experiments, including a hyper-elastic problem and a multi-scale mechanics problem, demonstrate the efficacy of the proposed approach. In particular, the score-based diffusion model, coupled with the physics-informed CNN surrogate, effectively learns geometrical features from provided prior samples, yielding better inverse estimation results compared to the state-of-the-art techniques.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
