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.

翻译：本研究提出了一种用于参数化偏微分方程（PDE）求解的降阶建模框架——条件神经场降阶模型（CNF-ROM）。该方法将用于建模潜在时间动力学的参数化神经常微分方程（PNODE）与一个从相应潜在状态重构PDE解的解码器相结合。我们为CNF-ROM引入了一个物理信息驱动的学习目标，该目标包含两个关键组成部分。首先，该框架利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，从而计算并最小化PDE残差。其次，使用近似距离函数（ADF）[Sukumar and Srivastava, CMAME, 2022] 来施加精确的初始条件和边界条件（IC/BC）。然而，ADF在边界连接点处的二阶或更高阶导数会变得不稳定，这引入了一种权衡。为了解决这个问题，我们引入了一个受 [Gladstone et al., NeurIPS ML4PS workshop, 2022] 启发的辅助网络。我们的方法通过参数外推与内插、时间外推以及与解析解的比较得到了验证。