用神经场建模物理方程解，兼顾精度与边界稳定性。
Physics-informed reduced order model with conditional neural fields
- 用参数化神经微分方程+解码器构建低维模型，学习动态演化。
- 通过自动微分计算残差，结合精确初边值条件提升预测准确率。
- 改进边界处理机制，解决高阶导数不稳问题，适合物理仿真任务。
本文提出条件神经场降阶模型（CNF-ROM）框架，用于近似参数化偏微分方程（PDE）的解。该方法结合参数化神经常微分方程（PNODE）以建模时序隐状态动态，并通过解码器从隐状态重构PDE解。引入物理信息学习目标，包含两个关键部分：首先，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，从而最小化PDE残差；其次，采用近似距离函数（ADFs）[Sukumar and Srivastava, CMAME, 2022] 精确施加初值和边界条件（IC/BC）。然而，ADFs在边界连接点处的二阶及以上导数会变得不稳定。为此，我们引入受[Gladstone et al., NeurIPS ML4PS workshop, 2022]启发的辅助网络来缓解该问题。方法在参数内插与外推、时间外推以及与解析解对比中得到验证。
原文摘要 · Abstract (English)
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.
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