Pretraining Codomain Attention Neural Operators for Solving Multiphysics PDEs

Existing neural operator architectures face challenges when solving multiphysics problems with coupled partial differential equations (PDEs), due to complex geometries, interactions between physical variables, and the lack of large amounts of high-resolution training data. To address these issues, we propose Codomain Attention Neural Operator (CoDA-NO), which tokenizes functions along the codomain or channel space, enabling self-supervised learning or pretraining of multiple PDE systems. Specifically, we extend positional encoding, self-attention, and normalization layers to the function space. CoDA-NO can learn representations of different PDE systems with a single model. We evaluate CoDA-NO's potential as a backbone for learning multiphysics PDEs over multiple systems by considering few-shot learning settings. On complex downstream tasks with limited data, such as fluid flow simulations and fluid-structure interactions, we found CoDA-NO to outperform existing methods on the few-shot learning task by over $36\%$. The code is available at https://github.com/ashiq24/CoDA-NO.

Incremental Fourier Neural Operator

Recently, neural networks have proven their impressive ability to solve partial differential equations (PDEs). Among them, Fourier neural operator (FNO) has shown success in learning solution operators for highly non-linear problems such as turbulence flow. FNO is discretization-invariant, where it can be trained on low-resolution data and generalizes to problems with high-resolution. This property is related to the low-pass filters in FNO, where only a limited number of frequency modes are selected to propagate information. However, it is still a challenge to select an appropriate number of frequency modes and training resolution for different PDEs. Too few frequency modes and low-resolution data hurt generalization, while too many frequency modes and high-resolution data are computationally expensive and lead to over-fitting. To this end, we propose Incremental Fourier Neural Operator (IFNO), which augments both the frequency modes and data resolution incrementally during training. We show that IFNO achieves better generalization (around 15% reduction on testing L2 loss) while reducing the computational cost by 35%, compared to the standard FNO. In addition, we observe that IFNO follows the behavior of implicit regularization in FNO, which explains its excellent generalization ability.