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Ensemble and Mixture-of-Experts DeepONets for Enhancing Operator Learning Capabilities


Core Concepts
Ensemble DeepONet architectures, including a novel Partition-of-Unity Mixture-of-Experts (PoU-MoE) trunk, can significantly improve the accuracy of operator learning compared to standard DeepONets, especially for problems involving output functions with steep spatial gradients.
Abstract

The paper presents a novel deep operator network (DeepONet) architecture called the ensemble DeepONet, which allows for enriching the trunk network of a single DeepONet with multiple distinct trunk networks. This trunk enrichment enables greater expressivity and generalization capabilities over a range of operator learning problems.

The authors also introduce a spatial Mixture-of-Experts (MoE) DeepONet trunk network architecture, called the PoU-MoE trunk, that utilizes a Partition-of-Unity (PoU) approximation to promote spatial locality and model sparsity in the operator learning problem.

The authors first prove that both the ensemble and PoU-MoE DeepONets are universal approximators. They then demonstrate that ensemble DeepONets containing a trunk ensemble of a standard trunk, the PoU-MoE trunk, and/or a Proper Orthogonal Decomposition (POD) trunk can achieve 2-4x lower relative ℓ2 errors than standard DeepONets and POD-DeepONets on both standard and challenging new operator learning problems involving partial differential equations (PDEs) in two and three dimensions.

The key highlights are:

  • The ensemble DeepONet provides a powerful and general framework for incorporating basis enrichment in scientific machine learning architectures for operator learning.
  • The PoU-MoE formulation offers a natural way to incorporate spatial locality and model sparsity into any neural network architecture.
  • Ensemble DeepONets with a combination of global (POD) and local (PoU-MoE) basis functions outperform standalone DeepONets and overparametrized DeepONets, especially on problems with output functions exhibiting steep spatial gradients.
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Stats
The relative ℓ2 error on the 2D Darcy flow problem was reduced from 0.857% for the vanilla DeepONet to 0.187% for the ensemble Vanilla-POD-PoU DeepONet. The relative ℓ2 error on the 2D reaction-diffusion problem was reduced from 0.144% for the vanilla DeepONet to 0.0539% for the ensemble POD-PoU DeepONet. The relative ℓ2 error on the 3D reaction-diffusion problem was reduced from 0.127% for the vanilla DeepONet to 0.0576% for the ensemble POD-PoU DeepONet.
Quotes
"Ensemble DeepONets containing a trunk ensemble of a standard trunk, the PoU-MoE trunk, and/or a Proper Orthogonal Decomposition (POD) trunk can achieve 2-4x lower relative ℓ2 errors than standard DeepONets and POD-DeepONets on both standard and challenging new operator learning problems involving partial differential equations (PDEs) in two and three dimensions." "The PoU-MoE formulation provides a natural way to incorporate spatial locality and model sparsity into any neural network architecture."

Key Insights Distilled From

by Ramansh Shar... at arxiv.org 10-01-2024

https://arxiv.org/pdf/2405.11907.pdf
Ensemble and Mixture-of-Experts DeepONets For Operator Learning

Deeper Inquiries

How can the training and inference time of the ensemble DeepONet architectures, especially those with the PoU-MoE trunk, be further improved?

To enhance the training and inference time of ensemble DeepONet architectures, particularly those utilizing the PoU-MoE trunk, several strategies can be implemented. First, a parallelization strategy for the PoU-MoE trunk can significantly reduce the computational overhead. By allowing simultaneous processing of the overlapping patches, the forward pass and back-propagation can be executed concurrently, leading to faster training times. Additionally, optimizing the implementation of the weight functions ( w_k(y) ) can improve efficiency. Utilizing more efficient data structures or algorithms for computing the blending of trunk networks can reduce the computational complexity. For instance, employing sparse matrix representations can minimize memory usage and speed up calculations. Furthermore, leveraging hardware accelerators such as GPUs or TPUs can enhance performance. Fine-tuning the hyperparameters, such as batch size and learning rate, can also lead to more efficient training. Lastly, reducing the number of trainable parameters in the ensemble by employing techniques like pruning or quantization can help maintain accuracy while improving speed.

What other types of basis functions or expert networks could be incorporated into the ensemble DeepONet framework to address specific operator learning challenges?

Incorporating various types of basis functions or expert networks into the ensemble DeepONet framework can enhance its adaptability to specific operator learning challenges. For instance, wavelet basis functions can be integrated to capture localized features in the data, which is particularly useful for problems with sharp gradients or discontinuities. Additionally, radial basis functions (RBFs) can be employed to create localized expert networks that focus on specific regions of the input space, enhancing the model's ability to learn complex mappings. Another promising approach is to use polynomial chaos expansions, which can effectively represent uncertainty in operator learning problems, especially in stochastic settings. Moreover, incorporating domain-specific knowledge through physics-informed neural networks (PINNs) can improve the model's performance on problems governed by partial differential equations (PDEs). This can be achieved by embedding physical constraints directly into the loss function, guiding the learning process. Lastly, integrating generative models, such as variational autoencoders or generative adversarial networks, can help in learning complex distributions of the output functions, thereby enriching the ensemble with diverse expert networks that can handle a wider range of operator learning tasks.

Can the ensemble DeepONet approach be extended to other neural operator architectures beyond the DeepONet, such as the Fourier Neural Operator or Graph Neural Operators, to achieve similar performance gains?

Yes, the ensemble DeepONet approach can be effectively extended to other neural operator architectures, including Fourier Neural Operators (FNOs) and Graph Neural Operators (GNOs). The fundamental principle of ensemble learning—combining multiple models to improve performance—remains applicable across different architectures. For FNOs, the ensemble approach can involve integrating multiple Fourier basis functions or different configurations of the Fourier transform to capture various frequency components of the input functions. This can enhance the model's ability to learn complex periodic behaviors and improve generalization to unseen data. In the case of GNOs, the ensemble strategy can be implemented by utilizing different graph structures or node features to create diverse expert networks. Each expert can focus on specific aspects of the graph, such as local connectivity or global structure, thereby enriching the overall representation and improving the model's performance on graph-based operator learning tasks. Furthermore, the incorporation of the PoU-MoE trunk into these architectures can provide spatial locality and model sparsity, which are beneficial for learning operators in high-dimensional spaces. By leveraging the strengths of each architecture while applying the ensemble framework, significant performance gains can be achieved across a variety of operator learning challenges.
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