NIERT: Accurate Numerical Interpolation Through Unifying Scattered Data Representations Using Transformer Encoder

Interpolation for scattered data is a classical problem in numerical analysis, with a long history of theoretical and practical contributions. Recent advances have utilized deep neural networks to construct interpolators, exhibiting excellent and generalizable performance. However, they still fall short in two aspects: 1) inadequate representation learning, resulting from separate embeddings of observed and target points in popular encoder-decoder frameworks and 2) limited generalization power, caused by overlooking prior interpolation knowledge shared across different domains. To overcome these limitations, we present a Numerical Interpolation approach using Encoder Representation of Transformers (called NIERT). On one hand, NIERT utilizes an encoder-only framework rather than the encoder-decoder structure. This way, NIERT can embed observed and target points into a unified encoder representation space, thus effectively exploiting the correlations among them and obtaining more precise representations. On the other hand, we propose to pre-train NIERT on large-scale synthetic mathematical functions to acquire prior interpolation knowledge, and transfer it to multiple interpolation domains with consistent performance gain. On both synthetic and real-world datasets, NIERT outperforms the existing approaches by a large margin, i.e., 4.3$\sim 14.3\times$ lower MAE on TFRD subsets, and 1.7/1.8/8.7× lower MSE on Mathit/PhysioNet/PTV datasets.