Mesh-Informed Neural Operator : A Transformer Generative Approach
This work addresses a critical gap for researchers and practitioners in scientific computing by enabling generative modeling on irregular domains, though it is incremental as it builds on existing neural operator frameworks.
The paper tackles the limitation of existing functional generative models to regular grids and rectangular domains by introducing the Mesh-Informed Neural Operator (MINO), which uses graph neural operators and cross-attention to enable domain- and discretization-agnostic generative modeling in function spaces, expanding applicability to diverse scientific and engineering tasks.
Generative models in function spaces, situated at the intersection of generative modeling and operator learning, are attracting increasing attention due to their immense potential in diverse scientific and engineering applications. While functional generative models are theoretically domain- and discretization-agnostic, current implementations heavily rely on the Fourier Neural Operator (FNO), limiting their applicability to regular grids and rectangular domains. To overcome these critical limitations, we introduce the Mesh-Informed Neural Operator (MINO). By leveraging graph neural operators and cross-attention mechanisms, MINO offers a principled, domain- and discretization-agnostic backbone for generative modeling in function spaces. This advancement significantly expands the scope of such models to more diverse applications in generative, inverse, and regression tasks. Furthermore, MINO provides a unified perspective on integrating neural operators with general advanced deep learning architectures. Finally, we introduce a suite of standardized evaluation metrics that enable objective comparison of functional generative models, addressing another critical gap in the field.