NIERT: Accurate Numerical Interpolation through Unifying Scattered Data Representations using Transformer Encoder

Numerical interpolation for scattered data aims to estimate values for target points based on those of some observed points. Traditional approaches produce estimations through constructing an interpolation function that combines multiple basis functions. These approaches require the basis functions to be pre-defined explicitly, thus greatly limiting their applications in practical scenarios. Recent advances exhibit an alternative strategy that learns interpolation functions directly from observed points using machine learning techniques, say deep neural networks. This strategy, although promising, cannot effectively exploit the correlations between observed points and target points as it treats these types of points separately. Here, we present a learning-based approach to numerical interpolation using encoder representations of Transformers (thus called NIERT). NIERT treats the value of each target point as a masked token, which enables processing target points and observed points in a unified fashion. By calculating the partial self-attention between target points and observed points at each layer, NIERT gains advantages of exploiting the correlations among these points and, more importantly, avoiding the unexpected interference of target points on observed points. NIERT also uses the pre-training technique to further improve its accuracy. On three representative datasets, including two synthetic datasets and a real-world dataset, NIERT outperforms the existing approaches, e.g., on the TFRD-ADlet dataset for temperature field reconstruction, NIERT achieves an MAE of $1.897\times 10^{-3}$, substantially better than the transformer-based approach (MAE: $27.074\times 10^{-3}$). These results clearly demonstrate the accuracy of NIERT and its potential to apply in multiple practical fields.