Multiwavelet-based Operator Learning for Differential Equations (2109.13459v2)

Abstract: The solution of a partial differential equation can be obtained by computing the inverse operator map between the input and the solution space. Towards this end, we introduce a \textit{multiwavelet-based neural operator learning scheme} that compresses the associated operator's kernel using fine-grained wavelets. By explicitly embedding the inverse multiwavelet filters, we learn the projection of the kernel onto fixed multiwavelet polynomial bases. The projected kernel is trained at multiple scales derived from using repeated computation of multiwavelet transform. This allows learning the complex dependencies at various scales and results in a resolution-independent scheme. Compare to the prior works, we exploit the fundamental properties of the operator's kernel which enable numerically efficient representation. We perform experiments on the Korteweg-de Vries (KdV) equation, Burgers' equation, Darcy Flow, and Navier-Stokes equation. Compared with the existing neural operator approaches, our model shows significantly higher accuracy and achieves state-of-the-art in a range of datasets. For the time-varying equations, the proposed method exhibits a ($2X-10X$) improvement ($0.0018$ ($0.0033$) relative $L2$ error for Burgers' (KdV) equation). By learning the mappings between function spaces, the proposed method has the ability to find the solution of a high-resolution input after learning from lower-resolution data.

• The paper presents a novel multiwavelet-based neural operator that compresses and represents PDE kernels for efficient, resolution-independent learning.
• It utilizes inverse multiwavelet filters to capture multi-scale dependencies, achieving significant numerical efficiency and accuracy.
• Experimental results on various PDEs demonstrate up to 10X improvement in relative L2 errors, highlighting the model's robustness for time-varying dynamics.

Multiwavelet-based Operator Learning for Differential Equations

The paper "Multiwavelet-based Operator Learning for Differential Equations" presents a novel approach for solving partial differential equations (PDEs) by introducing a multiwavelet-based neural operator learning framework. The proposed methodology is noteworthy for its utilization of the multiwavelet transform to efficiently compress and represent the kernel of an operator associated with a PDE. This representation leverages fine-grained wavelets that allow for a detailed and resolution-independent learning scheme.

Key Contributions and Methodology

• Multiwavelet Compression: The core innovation lies in embedding inverse multiwavelet filters to project the operator’s kernel onto multiwavelet polynomial bases. These multiwavelets sparsify the representation by capturing dependencies across various scales, which makes the model numerically efficient. The approach exploits smoothness and vanishing moments properties inherent to the wavelet domain, facilitating sparse and accurate mappings.
• Resolution-Independent Scheme: The proposed operator learning is resolution-independent, meaning that once the model is trained on data sampled at a particular resolution, it can predict accurate solutions even at different resolutions. This feature is achieved through the recursive application of the multiwavelet decomposition across multiple scales.
• Experimental Evaluation: The methodology is evaluated across several PDE datasets, namely the Korteweg-de Vries (KdV) equation, Burgers' equation, Darcy Flow, and Navier-Stokes equation. In these experiments, the proposed model consistently exhibits superior accuracy compared to existing neural operator techniques. Particularly, the model shows a relative $L2$ error improvement of $2X-10X$ when tackling time-varying equations, underscoring its robustness and efficiency in comparison to traditional approaches.

Strong Numerical Results

For specific equations like Burgers' and KdV, the model demonstrates orders of magnitude improvements in low relative $L2$ errors. Notably, when resolving dynamics for datasets with high fluctuation strengths, the model maintains accuracy, demonstrating robustness to varying input conditions.

Implications and Future Directions

The implications of this research are manifold. Practically, the ability to efficiently solve PDEs at arbitrary resolutions without retraining positions this model as a significant tool in engineering and scientific applications that involve complex dynamical systems, such as fluid dynamics and material sciences. Theoretically, the paper opens the door to further exploration of wavelet-based deep learning frameworks and their applicability in learning operators in infinite-dimensional function spaces.

Looking forward, exploration into a more diverse class of wavelet transformations and their potential impact on operator learning could be insightful. Additionally, addressing the challenges of non-compact support in wavelet transformations might expand the applicability of this approach to a wider array of PDEs, further integrating the power of deep learning with classical mathematical techniques in solving real-world problems.

In conclusion, the proposed multiwavelet-based operator learning framework stands as a promising advancement in the domain of neural operator methods for differential equations, with significant potential for future developments in both research and application fields.