NIERT: Accurate Numerical Interpolation through Unifying Scattered Data Representations using Transformer Encoder
This work addresses interpolation challenges in numerical analysis, offering a novel method that enhances accuracy and generalization across domains, though it is incremental in building on deep learning techniques.
The paper tackles the problem of numerical interpolation for scattered data by proposing NIERT, a transformer encoder-based method that unifies representations and incorporates prior knowledge through pre-training, achieving significant improvements such as 4.3 to 14.3 times lower MAE on TFRD subsets and 1.7 to 8.7 times lower MSE on other datasets compared to existing approaches.
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.