标题： 基于物理信息的条件神经场降阶模型 显示英文标题

标题： Physics-informed reduced order model with conditional neural fields

摘要： 本研究提出了用于降阶建模（CNF-ROM）框架的条件神经场，以近似参数化偏微分方程（PDEs）的解。该方法结合了用于随时间建模潜在动力学的参数化神经ODE（PNODE）和一个解码器，该解码器从相应的潜在状态重建PDE解。我们为CNF-ROM引入了一个物理信息学习目标，该目标包含两个关键组成部分。首先，该框架使用基于坐标的神经网络通过自动微分计算空间导数并应用链式法则处理时间导数来计算和最小化PDE残差。其次，使用近似距离函数（ADFs）[Sukumar和Srivastava, CMAME, 2022]来施加精确的初始和边界条件（IC/BC）。然而，ADFs在边界连接点处的二阶或更高阶导数会变得不稳定。为了解决这个问题，我们引入了一个受[Gladstone等人, NeurIPS ML4PS研讨会, 2022]启发的辅助网络。我们的方法通过参数外推和内插、时间外推以及与解析解的比较进行了验证。 显示英文摘要

摘要： This study presents the conditional neural fields for reduced-order modeling (CNF-ROM) framework to approximate solutions of parametrized partial differential equations (PDEs). The approach combines a parametric neural ODE (PNODE) for modeling latent dynamics over time with a decoder that reconstructs PDE solutions from the corresponding latent states. We introduce a physics-informed learning objective for CNF-ROM, which includes two key components. First, the framework uses coordinate-based neural networks to calculate and minimize PDE residuals by computing spatial derivatives via automatic differentiation and applying the chain rule for time derivatives. Second, exact initial and boundary conditions (IC/BC) are imposed using approximate distance functions (ADFs) [Sukumar and Srivastava, CMAME, 2022]. However, ADFs introduce a trade-off as their second- or higher-order derivatives become unstable at the joining points of boundaries. To address this, we introduce an auxiliary network inspired by [Gladstone et al., NeurIPS ML4PS workshop, 2022]. Our method is validated through parameter extrapolation and interpolation, temporal extrapolation, and comparisons with analytical solutions.