Operator learning is a promising paradigm for surrogate modeling by learning mappings between function spaces rather than discretized data points. Unlike CNNs and GNNs, Neural Operators learn and treat physical variables as continuous fields, enabling evaluation at any arbitrary point, removing any constraints on discretization. The Fourier Neural Operator (FNO) [29] does this by approximating the Greens Function: the integral of kernel of the unknown operator in a learned latent space using spectral methods. They are limited to uniform grids for efficient Fast Fourier Transform operations, limiting their applicability to unstructured meshes. FNOs are extended by Graph based counterparts - GINOs to handle non-uniform discretizations and geometric variability.
Neural Fields, also known as Implicit Neural Representations (INRs), offer an alternative approach. The core idea of the representation lies in directly mapping spatial coordinates to function values - continuous representations of physical fields through neural network parameterizations.
Methodology
First, both geometric variations (represented by signed distance function fields) and output physical fields are encoded into compact latent representations. Second, a mapping between the input and output latent encodings is approximated by a separate neural network: the processor model. Despite proving good performance, such approaches face overfitting challenges on small datasets. The separate latent processing stage introduces an optimization that is decoupled from the final output field reconstruction: small regression errors in the latent space can induce larger errors in the decoded reconstructions.