Recently, Conditional Neural Fields (NeFs) have emerged as a powerful modelling paradigm for PDEs, by learning solutions as flows in the latent space of the Conditional NeF. Although benefiting from favourable properties of NeFs such as grid-agnosticity and space-time-continuous dynamics modelling, this approach limits the ability to impose known constraints of the PDE on the solutions -- e.g. symmetries or boundary conditions -- in favour of modelling flexibility. Instead, we propose a space-time continuous NeF-based solving framework that - by preserving geometric information in the latent space - respects known symmetries of the PDE. We show that modelling solutions as flows of pointclouds over the group of interest $G$ improves generalization and data-efficiency. We validated that our framework readily generalizes to unseen spatial and temporal locations, as well as geometric transformations of the initial conditions - where other NeF-based PDE forecasting methods fail - and improve over baselines in a number of challenging geometries.

翻译：近期，条件神经场（Conditional NeFs）作为一种强大的偏微分方程建模范式崭露头角，通过在条件神经场的潜在空间中将解建模为流（flows）来实现。尽管该方法受益于神经场在网格无关性和时空连续动力学建模等方面的有利特性，但为了建模灵活性，其限制了将偏微分方程的已知约束（例如对称性或边界条件）施加到解上的能力。相反，我们提出了一种基于时空连续神经场的求解框架——通过在潜在空间中保持几何信息——来尊重偏微分方程的已知对称性。我们证明，将解建模为感兴趣群 $G$ 上点云流（flows of pointclouds）能够提高泛化能力和数据效率。我们验证了我们的框架能够轻易泛化到未见过的时空位置以及初始条件的几何变换（在此类情况下其他基于神经场的偏微分方程预测方法会失败），并在多个具有挑战性的几何结构中优于基线方法。