NIERT: Accurate Numerical Interpolation through Unifying Scattered Data Representations using Transformer Encoder
Abstract
Numerical interpolation for scattered data aims to estimate values for target pointsbased on those of some observed points.
Traditional approaches produce estimations through constructing an interpolation function that combines multiple basisfunctions.
These approaches require the basis functions to be pre-defined explicitly,thus greatly limiting their applications in practical scenarios.
Recent advancesexhibit an alternative strategy that learns interpolation functions directly fromobserved points using machine learning techniques, say deep neural networks.
Thisstrategy, although promising, cannot effectively exploit the correlations betweenobserved points and target points as it treats these types of points separately.
Here,we present a self-supervised approach to numerical interpolation using encoderrepresentations of Transformers (thus called NIERT).
NIERT treats the value ofeach target point as a masked token, which enables processing target points andobserved points in a unified fashion.
By calculating the partial self-attention between target points and observed points at each layer, NIERT gains advantagesof exploiting the correlations among these points and, more importantly, avoidingthe unexpected interference of target points on observed points.
NIERT also usesthe pre-training technique to further improve its accuracy.
On three representativedatasets, including two synthetic datasets and a real-world dataset, NIERT outperforms the existing approaches, e.g., on the TFRD-ADlet dataset for temperaturefield reconstruction, NIERT achieves an MAE of 1 .
× 10 − 3 , substantially betterthan the transformer-based approach (MAE: 27 .
× 10 − 3 ).
These results clearlydemonstrate the accuracy of NIERT and its potential to apply in multiple practicalfields.
Introduction
Numerical interpolation plays important and fundamental roles in a wide range of practical scenarios,including solving partial differential equations [1], temperature field reconstruction [2], single imagesuper-resolution [3], time series interpolation [4, 5].
The interpolation error often leads to deviationsin subsequent calculations which seriously affects the performance of the entire task.
Thus, accurateapproaches to numerical interpolation are highly desirable.
Essentially, numerical interpolation for scattered data aims to approximate a target function accordingto the values of some observed scattered points.
To make accurate interpolation, an idealized approachshould reasonably model the following two key components: i ) the space of interpolation functionsand the distribution over it, which imply priori of the interpolation tasks, and ii ) the contributions ofobserved points to the target function, which imply the posteriori of the interpolation task [6].
Major progress has been made and a large number of approaches have been proposed for interpolationof scattered data, which can be divided into two categories, namely, traditional non-learning basedmethods and recent learning-based methods.
The typical traditional interpolation schemes constructthe target function by a linear combination of basis functions [7].
These schemes require explicitlydefined basis functions to model the contributions of observed points. At the same time, the useof different basis functions essentially implies different assumptions on the interpolation functiondistribution. Thus, the selection of appropriate basis functions to suit application scenarios requiresconsiderable expertise of the algorithm designers.
Recent progress has exhibited an alternative strategy that uses neural networks to learn interpolationfunctions directly from the given observed points.
For example, conditional neural processes (CNPs)[8] and their extensions [9–11] model the conditional distribution of regression functions giventhe observed points.
In addition, Chen et al.
[2] proposed to use vanilla Transformer [12] to solveinterpolation task in temperature field reconstruction.
All of these approaches use an “encoderdecoder” architecture, in which the encoder learns the representations of observed points while thedecoder estimates values for target points.
Intuitively, observed points and target points should beprocessed in a unified fashion because they are from the same domain.
However, this architecturetreats them separately and cannot effectively exploit the correlation between them.
Inspired by the recent advances of language/image models, especially BERT [13] and BE I T [14], wedesigned an approach to numerical interpolation that can effectively exploit the correlations betweenobserved points and target points.
Our approach is a self-supervised approach using the encoderrepresentations of Transformers (thus called NIERT).
The key elements of NIERT include: i ) theuse of the mask mechanism, which enables processing target points and observed points in a unifiedfashion, ii ) a novel partial self-attention model, which calculates attentions between target points andobserved points at each layer, thus gaining the advantages of exploiting the correlations between thesetwo types of points and, more importantly, avoiding the unexpected interference of target points onobserved points simultaneously, and iii ) the use of the pre-training technique, which further improvesthe interpolation accuracy of NIERT.
The main contributions of this study are summarized as follows.
We propose an accurate approach to numerical interpolation for scattered data.
On represen-
tative datasets, including both synthetic and real-world datasets, our approach outperforms existing approaches.
The experimental results demonstrate the potential of our approachin a wide range of application fields.
The source code of NIERT will be released for open source use.
We propose a novel partial self-attention mechanism to make Transformer incorporated with strong inductive bias for interpolation tasks; i e., it can effectively exploit the correlation
among two types of points but simultaneously avoid the interference of one type of points
onto the others.
We propose to use the pre-training technique to enhance interpolation approaches.
When facing an interpolation task in a newly-appearing application field, we can benefit from theexperience learned from low-cost synthesized interpolation tasks.
Related works2.1 Traditional interpolation approaches for scattered data
Traditional interpolation approaches for scattered data use explicit basis functions to constructinterpolation function, e.g., Lagrange interpolation, Newton interpolation [7], B-spline interpolation[15], Shepard’s method [16], Kriging [17], and radial basis function interpolation (RBF) [18, 19].
Among these approaches, the classical Lagrange interpolation, Newton interpolation and B-splinesinterpolation are usually used for univariate interpolation.
Wang et al.
[20] proposed a high ordermultivariate approximation scheme for scattered data sets, in which approximation error is representedwith Taylor expansions at data points, and basis functions are determined through minimizing thesquares of approximation error. [Figure]
The partial self-attention mechanism facilitates exploiting the correlations between observed points and target points and avoiding the unexpected interference of target points on observed points
2.2 Neural network-based interpolation approaches
Equipped with deep neural networks, data-driven interpolation and reconstruction methods show greatadvantages and potential.
For instance, convolutional neural networks (CNNs) have been applied inthe interpolation tasks of single image super-resolution [21, 3], and recurrent neural networks (RNNs)and Transformers have been used for interpolation of sequences like time series data [5, 22].
Recently, Garnelo et al.
[8] proposed to model the conditional distribution of regression functionsgiven observed points.
The proposed approach, conditional neural processes (CNPs), has shownincreased estimation accuracy and generalizing ability.
Kim et al.
[9] designed an enhanced model,attentive neural processes (ANPs), with improved accuracy.
Lee et al.
[11] leveraged Bayesian lastlayer (BLL) [23] for faster training and better prediction.
In addition, the bootstrap technique wasalso employed for further improvement [10].
To solve the interpolation task in 2D temperature fieldreconstruction, Chen et al.
[2] proposed an approach that uses vanilla Transformer. This approachcan also be applied to solve interpolation tasks for scattered data with high dimensions.
2.3 Masked language/image models and the pre-training technique
The design of NIERT is also inspired by the recent advances in masked language/image models[13, 14, 24, 25] and pre-trained models for symbolic regression [26, 27].
The masked pre-trainedmodels have been shown to be successful in learning representations of languages and imagesfrom large-scale data and improving the performance of downstream tasks.
In addition, the maskmechanism makes the model able to reconstruct missing data from their context.
Utilizing large-scalesynthetic symbolic functions and sampled scattered data, Biggio et al.
[26] and Valipour et al.
[27]pre-trained Transformers to learn the map from scattered data to corresponding symbolic formulas.
Different from these approaches, NIERT uses synthetic data to learn interpolating functions fromscattered data numerically.
Method
3.1 Overview of NIERT
In the study, we focus on the interpolation task that can be formally described as follows: We aregiven n observed points with known values O = { ( x i , y i ) } ni =1 , and m target points with values to bedetermined, denoted as T =
{ x i } n + m i = n +1 .
Here, x i ∈
X denotes position of a point, y i =
f ( x i ) ∈
Ydenotes the value of a point, and f : X → Y denotes a function mapping positions to values.
Thefunction f is from a function distribution F , which can be explicitly defined using a mathematicalformula or implicitly represented using a set of scattered data in the form ( x i , y i ) .
The goal of
Algorithm
Require: NIERT model M θ with parameters θ , epoch number N , batch size B , domain X , function distribution F and error metric function Error( · , · ) for k in { 1 ..N } do
J ←for b in { 1 ..B } do f, { x i } i ← sample a function and scatter points from F , X { y i } i ← calculate f on { x i } i
O, T ← split and mask scatter points with values { ( x i , y i ) } i { ˆ y i } i ← M θ ( O, T )
J ← J + (cid:80) i Error( ˆ y i , y i ) end for
Compute the gradient ∇ θ J and update θ end for interpolation task is to accurately estimate the values f ( x ) for each target point x ∈ T according tothe observed points in O .
Figure 1 depicts the schematic diagram of our NIERT approach.
Briefly speaking, our approachemploys a self-supervised approach to numerical interpolation using encoder representations of Transformers.
The main element of our approach is a neural interpolator that learns to estimate valuesfor target points.
The interpolator is featured by the characteristic that it treats the value of each targetpoint as a masked token, thus enabling the unifying fashion to process both target points and observedpoints in the subsequent encoding and estimation procedures.
To suit the interpolation task, we design a partial self-attention mechanism: on one side, we calculatethe attention between target points and observed points at each layer, which gains NIERT theadvantage to effectively exploit the correlations between these two types of points.
On the other side,the attention is a partial one as we do not consider the effects of a target point on all other points.
This way, the unexpected interference of target points onto the observed points, and the interferenceamong target points, are completely avoided.
The training process is depicted in Algorithm 1.
Specifically, we prepare the training interpolationtasks by first sampling functions from a distribution F and then sampling observed points O andtarget points T on each function.
When training NIERT, we set the loss function as the error betweenthe estimated values and the corresponding ground-truth.
It should be pointed out that errors acquiredon both observed points and target points are accounted into loss function.
3.2 Architecture of the NIERT interpolator
The neural interpolator in NIERT adopts the Transformer encoder framework; however, to suit theinterpolation task, significant modifications and extensions were made in embedding, Transformerand output layers, which are described in details below.
Embedding with masked tokens: NIERT embeds both observed points and target points into theunified high-dimensional embedding space.
As the position x of a data point and its value y are fromdifferent domains, we use two multilayer perceptrons (MLPs) : MLP x embeds the positions while MLP y embeds the values.
It should be noted that for target points, their values are absent when embedding as they are to bedetermined.
In this case, we use a masked token as substitutes, which is embedded as a trainableparameter MASK y as performed in BERT [13].
This way, the interpolator processes both targetpoints and observed points in an unifying fashion.
We concatenate the embeddings of position and value of a data point as the point’s embedding,denoted as h
i , i.e.,
[Equation]
Transformer layer with partial self-attention mechanism: NIERT feeds the embeddings of thepoints into a stack of L Transformer layers, producing encodings of these points as results.
Each
Transformer layer contains two subsequent sub-layers, namely, a multi-head self-attention module,and a point-wise fully-connected network.
These sub-layers are interlaced with residual connectionsand layer normalization between them.
To avoid the unexpected interference of target points on observed points and target points themselves,
NIERT replaces the original self-attention in Transformer layer with a partial self-attention , whichcalculates the feature of point i at the l + 1 -st layer as follows:
[Equation]
[Equation]
where v li and α lij represent the value embedding and ordinary attention weights at the l -th layer ascalculated in Transformer [12].
In this formula, we introduce a new term w ij that represents thepartial self-attention pattern, i.e.,
[Equation]
By forcing the weight w ij to be 0 for a target point i and any point j , we completely avoid theunexpected interference of target points on the other points.
Estimating values for target points: For each target point i , we estimate its value ˆ y i through feedingits features at the final Transformer layer into a fully connected feed-forward network, i.e.,ˆ y i =
MLP out ( h Li ) .
We calculate the error between the estimation and the corresponding ground-truth value, and composethe errors for all points into a loss function to be minimized.
3.3 Enhancing NIERT using pre-training technique
The interpolation functions from different applications usually differ greatly in their forms; however,the interpolation tasks might still share some common characteristics, say the correlation betweenobserved points and target points.
These common characteristics enable enhancing NIERT usingthe pre-training technique.
Here, we pre-train NIERT using a synthetic dataset (see 4.1 for furtherdetails) and fine-tune it on other datasets in application fields.
Experiments and results
We evaluated NIERT and compared it with ten representative scattered data interpolation approacheson both synthetic and real-world datasets.
We also examined the effects of the key elements of NIERT,including the partial self-attention, and the pre-training technique.
4.1 Experiment setting
The datasets, metrics and approaches for comparison are briefly described below.
Further details ofexperiment settings are provided in Supplementary text.
Datasets: Three representative datasets in various application fields, including two synthetic datasets
NeSymReS and TFRD-ADlet, and real-world dataset PhysioNet, are used for evaluation.
NeSymReS is a synthetic dataset for mathematical function interpolation, which is built using datagenerator proposed by Biggio et al.
[26] and Lample and Charton [28].
We construct a function setwith various dimensionality of data points, including 1D, 2D, 3D, and 4D. Scattered points in eachinstance are randomly sampled and divided into observed points and target points.
TFRD-ADlet [2] is a synthetic dataset for 2D temperature field reconstruction where each instancerepresents a simulated 2D temperature field containing several heat source components and a specific
Dirichlet conditioned boundary.
The goal of each task instance is to reconstruct the whole temperaturefield according to a limited number of observed points with measured temperature.
PhysioNet Challenge 2012 dataset [29] is a real world dataset collected from intensive care unit(ICU) records for time-series data interpolation.
Each point in an instance represents a measurementat a specific time, where each measurement contains up to 37 physiological indices.
Following thestudy [22], we randomly divided the points into observed points and target points by setting the ratioof observed points at five levels, i.e., 50%, 60%, 70%, 80%, and 90%.
Note that this dataset is arepresentative of hard interpolation tasks due to the sparsity and irregularity of the records.
Pre-training dataset: In this study, NeSymReS dataset was used for pre-training NIERT to furtherimprove its interpolation accuracy on TFRD-ADlet and PhysioNet dataset.
For TFRD-ADlet datasetwe directly use 2D TFRD-ADlet dataset for pre-training.
As the PhysioNet dataset has a dimensionality of 37, we construct the pre-training instances by stacking random 37 functions from 1D
TFRD-ADlet dataset and then sampling interpolation task instances.
Metrics: When evaluating NIERT and other interpolation approaches, the prediction error of targetpoints are calculated as interpolation accuracy.
For the NeSymReS and PhysioNet dataset, we adoptedmean squared error (MSE) as the error metric.
For TFRD-ADlet dataset, we use three error metrics:mean absolute error (MAE), MAE in component area (CMAE) and MAE at boundary (BMAE)following Chen et al.
Accordingly, we use L 2 -form loss function for NeSymReS and PhysioNetdataset and L 1 -form loss function for TFRD-ADlet dataset for training.
Approaches for comparison: For NeSymReS dataset, we compared NIERT with six representativeinterpolation approaches, including RBF[30], MIR [20], CNP[8], ANP[9], BANP[10] and TFRtransformer [2].
For TFRD-ADlet we compared NIERT with CNP, ANP, BANP and TFR-transformer.
For PhysioNet we compared NIERT with four approaches designed for time-series data interpolation,including RNN-VAE [31], L-ODE-RNN [32], L-ODE-ODE[33], and mTAND-Full[22].
4.2 Interpolation accuracy on synthetic and real-world datasets
For each instance of the test datasets [Table] we applied the trained NIERT to estimate values for the targetpoints.
We calculate the errors between the estimation and the ground-truth as interpolation accuracy.
[Table]
[Table] [Table]
transformer,which use neural networks to learn interpolation, show relatively lower errors (MSE: 14.913, 15.556).
Compared with these approaches, our NIERT approach achieves the best interpolation accuracy
(MSE: 8.964).
Table 1 also demonstrates the advantage of NIERT over the existing approach on the2D, 3D, and 4D instances.
[Figure]
f p r e d i c t i o n o f t a r g e t p o i n t s
RBFMIRCNPANPBANPTFR -
[Figure]
Here we
2D instances
Note that the number of observed points variesgreatly in test instances, making the averageinterpolation error calculated over all test instances insufficient to measure interpolation performance.
Therefore, we further divide test instances into subsets according to the number ofobserved points.
As shown in Figure 2, as thenumber of observed points increases, the interpolation error decreases as expected.
In addition,the relative advantages of these approaches varywith the number of observed points, e.g., CNP isbetter than RBF and MIR initially but finally becomes worse as the number of observed pointsincreases.
Among all approaches, NIERT stablyshows the best performance over all test subsets,regardless of the number of observed points.
Accuracy on the TFRD-ADlet dataset: As shown in Table 2, CNP, although employing the neuralnetwork technique, still performs poorly with MAE as high as of 96.674.
In contrast, NIERT achievesthe lowest interpolation error (MAE: 3.473), which is over one order of magnitude lower than CNP,
ANP, BANP and TFR-transformer.
Moreover, when enhanced with the pre-training technique, NIERTcan further decrease its interpolation MAE to be 1.897.
Besides MAE, other metrics, say CMAE and
BMAE, also show the superior of NIERT over the existing approaches (Table 2 ).
Accuracy on the PhysioNet dataset: Table 3 suggests that on the PhysioNet dataset, NIERT alsooutperforms the existing approaches, e.g., when controlling the ratio of observed points to be 50%,
NIERT achieves an average MSE ( × 10 − 3 ) of 2.868, significantly lower than RNN-VAE (13.418),
L-ODE-RNN (8.132), L-ODE-ODE (6.721) and mTAND-Full (4.139).
Again, NIERT with thepre-training technique shows better performance.
The advantages of NIERT hold across varioussettings of the ratio of the observed points.
Taken together, these results clearly demonstrate the power of NIERT for numerical interpolation inmultiple application fields, including interpolating the scattered data generated using mathematicalfunctions, reconstructing temperature fields, and interpolating time-series data.
4.3 Case studies of interpolation results
To further understand the advantages of NIERT, we carried out case studies through visualizing theobserved points, the reconstructed interpolation functions and the interpolation errors.
Figure 3 and 4 show 1D and 2D instance in the NeSymReS test set, respectively. As illustrated bythese two figures, RBF performs poorly in the application scenario with sparse observed data. Inaddition, RBF and MIR, especially ANP, cannot accurately predict values for the target points thatfall out of the range restricted by observed points. The CNP approach can only learn the rough trendstated by the observed points, thus leading to significant errors.
In contrast, BANP, TFR-transformerand NIERT can accurately estimate values for target points within considerably large range, andcompared with BANP and TFR-transformer, NIERT can produce more accurate results.
Figure5 shows an instance of temperature field reconstruction extracted from TFRD-ADlet. Fromthis figure, we can observe that when using pre-training technique, NIERT further improves itsinterpolation accuracy in the whole area. 4.4 Contribution analysis of observed points for interpolation An idealized interpolation approach is expected to effectively exploit all observed points withappropriate consideration of relative positions among observed points and target points as well.
Toexamine this issue, we visualized the attention weight of each observed point to all target points.
These attention weights provide an intuitive description of the contribution by observed points.
RBF 1.0 0.5 0.0 0.5 1.0 0.0 0.
0.4 MSE: 8.19e1.0 0.5 0.0 0.5 1.0 0.0 0.
MIR 1.0 0.5 0.0 0.5 1.0 0.0 0.
0.4 MSE: 2.17e1.0 0.5 0.0 0.5 1.0 0.0 0.
CNP 1.0 0.5 0.0 0.5 1.0 0.0 0.
0.4 MSE: 9.97e1.0 0.5 0.0 0.5 1.0 0.0 0.
ANP 1.0 0.5 0.0 0.5 1.0 0.0 0.
0.4 MSE: 2.59e1.0 0.5 0.0 0.5 1.0 0.0 0.
BANP 1.0 0.5 0.0 0.5 1.0 0.0 0.
0.4 MSE: 2.61e1.0 0.5 0.0 0.5 1.0 0.0 0.
1.0 TFR-transformer 1.0 0.5 0.0 0.5 1.0 0.0 0.
0.4 MSE: 2.02e1.0 0.5 0.0 0.5 1.0 0.0 0.
1.0 NIERT (Ours) 1.0 0.5 0.0 0.5 1.0 0.0 0. [Figure] [Figure]
[Figure] [Figure] [Figure] [Figure]
[Figure] [Figure] [Figure] [Figure]
[Figure] [Figure] [Figure] [Figure]
[Figure] [Figure] [Figure]
[Figure]
1.0 MSE: 1.77e1.0 0.5 0.0 0.5 1.0 1.0 0.
1.0 MSE: 1.18e1.0 0.5 0.0 0.5 1.0 1.0 0.
1.0 MSE: 7.29e1.0 0.5 0.0 0.5 1.0 1.0 0.
1.0 MSE: 1.61e1.0 0.5 0.0 0.5 1.0 1.0 0.
1.0 MSE: 2.00e1.0 0.5 0.0 0.5 1.0 1.0 0.
1.0 MSE: 3.40e0.00 0.02 0.
[Figure]
0.1 NIERT (Ours) 0.
0.1 NIERT w/ PT (Ours) 300.0 302.5 305.0 307.
0.1 Observed points 0.
0.1 MAE: 1.2e0.
0.1 MAE: 6.17e0.
0.1 MAE: 2.84e0. 0.1 MAE: 2.34e0.
0.1 MAE: 3.70e0.
0.1 MAE: 1.93e0.00 0.
The up-left figure shows the ground-truth temperature field while the bottom-left figure shows theobserved points. The reconstructed results reported by NIERT and the existing approaches are listed on the top panel with their differences with the ground-truth temperature field are list below
As shown in Figure 6, when using TFR-transformer, the contributions by observed points areconsiderably imbalanced: on one side, some observed points might affect their neighboring targetpoints in a large region; on the other side, the other observed points have little contributions tointerpolation.
In contrast, when using NIERT, contributions by an observed point are much morelocal and thus targeted.
More importantly, all observed points have contributions to interpolation.
These results demonstrate that NIERT can exploit the correlation between observed points and targetpoints more effectively.
[Figure]
4.5 Ablation study
The effects of partial self-attention: For a specific interpolation task, the interpolation function isdetermined by the observed points only.
Thus, an idealized encoding of observed points should notbe affected by target points.
To investigate the affects of target points, we evaluated NIERT on thetest sets with various number of target points.
Here, we compared two variants of NIERT, one withpartial self-attention, and the other with vanilla self-attention.
Both of these two variants were trainedusing the same training sets (the number of observed points varies within [206, 246]).
As illustrated by Figure 7, the variant with vanilla self-attention shows poor performance for the taskswith few target points, say less than 64 target points.
In contrast, the variant with partial self-attentionalways performs stably without significant changes of accuracy.
The results clearly demonstrate that the partial self-attention mechanism allows NIERT to be freefrom the unexpected affects by the target points.
The effects of pre-training technique: To investigate the effects of the pre-training technique, weshow in Figure 8 the training process of two versions of NIERT, one without pre-training technique,and the other enhanced with pre-training.
As depicted by the figure, even at the first epoch, thepre-trained NIERT shows a sufficiently high interpolation accuracy, which is comparable with thefully-trained BANP and TFR-transformer.
Moreover, the performance of the pre-trained NIERTimproves in roughly the same convergence speed to the original NIERT.
At the final epoch, thepre-trained NIERT decreases the interpolation error to be nearly half of that of the original NIERT.
These results clearly suggest that the experience learned by NIERT from the interpolation task in oneapplication field can be transferred to the interpolation tasks in other application fields.
[Figure]
points.
Here, two variants of NIERT are trained on 1D NeSymReS dataset
[Figure]
Here, models are trained on TFRD-ADlet dataset
[Figure]
Discussion and conclusion
We present in the study an accurate approach to numerical interpolation for scattered data.
Thespecific features of our NIERT approach are highlighted by the full exploitation of the correlationbetween observed points and target points through unifying scattered data representation.
At the sametime, the use of partial self-attention mechanism can effectively avoid the interference of target pointsonto the observed points.
The enhancement with pre-training technique is another special featureof NIERT.
The advantages of NIERT in interpolation accuracy have been clearly demonstrated byexperimental results on both synthetic and real-world datasets.
The current version of NIERT has a space complexity of O ( n ( m + n )) , thus cannot handle theinterpolation tasks with extremely large amounts of observed points due to the limitations of GPUmemory size.
How to reduce the memory requirement is one of the future works.
Additionally, it isinteresting to combine NIERT and the traditional approaches based on basis functions to yield anapproach with both high accuracy and interpretability.
We expect NIERT, with extensions and modifications, to greatly facilitate numerical interpolations ina wide range of engineering and science fields.
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