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

翻译：求解参数化偏微分方程（PDE）对数据驱动方法提出了重大挑战，因为时空动力学对PDE参数的变化非常敏感。机器学习方法通常难以捕捉这种变异性。为解决此问题，数据驱动方法通过采样大量具有不同PDE参数的轨迹来学习参数化PDE。我们首先证明，在学习参数化PDE时引入条件化机制至关重要，并且其中$\textit{自适应条件化}$能够实现更强的泛化能力。由于现有的自适应条件化方法在神经求解器中需要调整的参数数量增加时扩展性不佳，我们提出了GEPS，这是一种简单的适应机制，通过一阶优化和对一小部分上下文参数进行低秩快速适应，来提升偏微分方程求解器的泛化能力。我们证明了该方法在纯数据驱动和物理感知神经求解器中的通用性。在一系列时空预测问题上进行的验证表明，该方法在泛化到未见过的条件（包括初始条件、PDE系数、强迫项和解域）方面表现出色。$\textit{项目页面}$：https://geps-project.github.io