提出GEPS方法，提升神经微分方程求解器对未见参数的泛化能力
GEPS: Boosting Generalization in Parametric PDE Neural Solvers through Adaptive Conditioning
- 通过一阶优化与低秩适应，动态调整少量上下文参数实现自适应条件建模
- 在多种时空预测任务中，对未见初始条件、系数、外力项等均表现优异
- 适用于纯数据驱动和物理感知两类神经求解器，通用性强
求解参数化偏微分方程（PDE）对数据驱动方法构成重大挑战，因其时空动态对PDE参数高度敏感。现有机器学习方法常难以捕捉这种变化性。为此，数据驱动方法通常通过采样大量不同参数下的轨迹进行训练。我们首先证明，引入条件机制对学习参数化PDE至关重要，其中自适应条件机制可实现更强泛化。然而，现有自适应方法在需适应的参数数量增加时扩展性差。为此，我们提出GEPS：一种基于一阶优化与低秩快速适应的小规模上下文参数调整机制，显著提升神经求解器的泛化能力。该方法在多种时空预测任务中验证有效，能良好泛化至未见的初始条件、PDE系数、外力项及解域。适用于全数据驱动与物理感知型神经求解器。
原文摘要 · Abstract (English)
Solving parametric partial differential equations (PDEs) presents significant challenges for data-driven methods due to the sensitivity of spatio-temporal dynamics to variations in PDE parameters. Machine learning approaches often struggle to capture this variability. To address this, data-driven approaches learn parametric PDEs by sampling a very large variety of trajectories with varying PDE parameters. We first show that incorporating conditioning mechanisms for learning parametric PDEs is essential and that among them, $\textit{adaptive conditioning}$, allows stronger generalization. As existing adaptive conditioning methods do not scale well with respect to the number of parameters to adapt in the neural solver, we propose GEPS, a simple adaptation mechanism to boost GEneralization in Pde Solvers via a first-order optimization and low-rank rapid adaptation of a small set of context parameters. We demonstrate the versatility of our approach for both fully data-driven and for physics-aware neural solvers. Validation performed on a whole range of spatio-temporal forecasting problems demonstrates excellent performance for generalizing to unseen conditions including initial conditions, PDE coefficients, forcing terms and solution domain. $\textit{Project page}$: https://geps-project.github.io
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