Neural field based sequential networks for parametric spatial-temporal PDEs

Real-world engineering applications often involve complex Partial Differential Equations (PDEs), leading to high-dimensional spatial-temporal discrete systems, which can be computationally intensive. To address this challenge, dimension-reduction technologies are necessary to mitigate the computational burden. Previous methods, including principal component analysis (PCA), generalized discriminant analysis (GDA), and neural networks based autoencoders, are susceptible to issues arising from non-uniform mesh resolutions, parameter dependencies, nonlinearity, etc. In this work, we propose a novel approach that combines Neural Fields (NF) for spatial learning and Sequence Networks for temporal learning to handle the complexities of parametric spatial-temporal PDEs. The Neural Field is employed for nonlinear embedding of physics fields, capturing spatial distributions, while the PDE-informed sequence network captures the temporal dynamics within the latent space. Notably, the learnable parameters within the sequence model are partially conditioned on the PDE parameters, utilizing another NF. This dual NF approach enhances the expressiveness and adaptability of our neural field-based sequence networks for parametric spatial-temporal PDEs.