Operator Learning Using Wavelet Representations

Abstract

Operator learning has recently emerged as an effective approach for approximating mappings between function spaces. In this work, we study operator learning using wavelet representations (OLWavelet). It represents functions via discrete wavelet transforms (DWT), uses a neural network to learn mappings between the resulting wavelet coefficients, and reconstructs the output functions from the predicted coefficients to learn operators between functions. Several representative operators are considered, including 1D translation operators, operators mapping different source terms of 1D Poisson equations to the corresponding solutions, operators mapping different initial
conditions of an 1D Diffusion euqation and an 1D Burgers’ equation to their terminal solutions, and operators approximating functions with piecewise functions on dyadic partitions. We analyze the sources of error in the proposed framework by decomposing the total error into the neural network generalization error and the reconstruction error. This perspective provides insight into how different components of the model contribute to the final prediction accuracy. In a series of numerical experiments, we compare models trained with different numbers of DWT coefficients for function representation, motivated by the fact that functions can be well approximated using only a subset of DWT coefficients, which also reduces computational cost. The experimental results show that our model achieves higher prediction accuracy than comparable models based on Fourier transforms on certain tasks. Moreover, we observe a non-monotonic relationship between the model’s prediction accuracy and the number of DWT coefficients used. In addition, experiments show that using a single neural network to learn the mappings among all wavelet coefficients is often more accurate than using multiple neural networks to separately learn the mappings of wavelet coefficients at different decomposition levels. Overall, the study demonstrates the effectiveness of the proposed model and analyzes the sources of its errors, thereby revealing its strengths and limitations.