GridMix: Exploring Spatial Modulation for Neural Fields in PDE Modeling

Significant advancements have been achieved in PDE modeling using neural fields. Despite their effectiveness, existing methods rely on global modulation, limiting their ability to reconstruct local details. While spatial modulation with vanilla grid-based representations offers a promising alternative, it struggles with inadequate global information modeling and over-fitting to the training spatial domain. To address these challenges, we propose GridMix, a novel approach that models spatial modulation as a mixture of grid-based representations. GridMix effectively explores global structures while preserving locality for fine-grained modulation. Furthermore, we introduce spatial domain augmentation to enhance the robustness of the modulated neural fields against spatial domain variations. With all these innovations,our comprehensive approach culminates in MARBLE, a framework that significantly advancing the capabilities of neural fields in PDE modeling. The effectiveness of MARBLE is extensivelyvalidated on diverse benchmarks encompassing dynamics modeling and geometric prediction.