结构保持变分神经场：非线性守恒律的不确定性量化降阶建模
Structure-preserving variational neural fields: Uncertainty-quantified reduced-order modeling of nonlinear conservation laws
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中文总结 AI 辅助
研究针对非线性守恒律控制的物理系统模拟，开发变分潜在神经场框架，集成高斯过程代理模型，有三种变体，能估计预测置信度，通过时空无散度表示嵌入守恒律流形实现守恒结构保持，经实验验证其有效性和鲁棒性。
中文摘要 AI 辅助
降阶模型，如潜在动力学模型，正成为加速由非线性守恒律控制的参数化物理系统模拟的主流方法。然而，大多数现有潜在动力学框架存在两个重要局限性：它们不提供模型预测的不确定性估计，也不保证遵循潜在的守恒律。虽然这些挑战在先前工作中已分别得到解决，但一个同时提供不确定性量化和精确守恒律保持的统一框架在很大程度上仍未被探索。在这项工作中，我们开发了一个变分潜在神经场框架，该框架集成了受高斯过程启发的代理模型，能够估计分布内和分布外参数区域的预测置信度。考虑了该框架的三种变体：IRS-UQ、PI-IRS-UQ和ECLEIRS-UQ，分别对应无约束、物理 informed 和守恒结构保持公式。通过将解通量场的时空无散度表示嵌入守恒律流形中的解动力学来实现精确的守恒结构保持。我们通过三个数值实验证明了该框架的适用性：1-D平流、2-D欧拉和2-D浅水方程在参数化设置下。数值实验表明，所提出的方法在提供准确预测和不确定性估计的同时，对稀疏和嘈杂的训练数据具有鲁棒性。所提出的三种方法之间进行比较表明，守恒结构保持潜在表示在保持有竞争力的预测精度和不确定性量化能力的同时，提高了对退化训练数据的鲁棒性。
英文摘要
Reduced-order models, such as latent dynamics models, are becoming mainstream for accelerating simulations for parameterized physical systems governed by nonlinear conservation laws. However, most existing latent dynamics frameworks suffer from two important limitations: they do not provide uncertainty estimates for model predictions, and they do not guarantee adherence to the underlying conservation laws. While these challenges have been addressed separately in prior work, a unified framework that simultaneously provides uncertainty quantification and exact conservation-law preservation remains largely unexplored. In this work, we develop a variational latent neural field framework that integrates Gaussian process-inspired surrogates, enabling estimation of predictive confidence for both in-distribution and out-of-distribution parameter regimes. Three variants of the framework are considered: IRS-UQ, PI-IRS-UQ, and ECLEIRS-UQ, corresponding to unconstrained, physics-informed, and conservation-structure-preserving formulations, respectively. Exact conservation-structure preservation is achieved by embedding the solution dynamics within a conservation-law manifold through a space-time divergence-free representation of the solution-flux field. We demonstrate the applicability of the framework through three numerical experiments: 1) 1-D advection, 2) 2-D Euler and 3) 2-D shallow water equations in parameterized settings. Numerical experiments demonstrate that the proposed approach provides accurate predictions together with uncertainty estimates, while remaining robust to sparse and noisy training data. Comparisons between the proposed three approaches show that conservation-structure preserving latent representations improve robustness to degraded training data while maintaining competitive predictive accuracy and uncertainty quantification capability.
