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

Abstract

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: \textbf{1) inadequate representation learning}, resulting from separate embeddings of observed and target points in popular encoder-decoder frameworks and \textbf{2) limited generalization power}, caused by overlooking prior interpolation knowledge shared across different domains. To overcome these limitations, we present a \textbf{N}umerical \textbf{I}nterpolation approach using \textbf{E}ncoder \textbf{R}epresentation of \textbf{T}ransformers (called \textbf{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$\times$ lower MSE on Mathit/PhysioNet/PTV datasets. The source code of NIERT is available at https://github.com/DingShizhe/NIERT.