Convolutional Neural Operators for robust and accurate learning of PDEs (2302.01178v3)

Abstract: Although very successfully used in conventional machine learning, convolution based neural network architectures -- believed to be inconsistent in function space -- have been largely ignored in the context of learning solution operators of PDEs. Here, we present novel adaptations for convolutional neural networks to demonstrate that they are indeed able to process functions as inputs and outputs. The resulting architecture, termed as convolutional neural operators (CNOs), is designed specifically to preserve its underlying continuous nature, even when implemented in a discretized form on a computer. We prove a universality theorem to show that CNOs can approximate operators arising in PDEs to desired accuracy. CNOs are tested on a novel suite of benchmarks, encompassing a diverse set of PDEs with possibly multi-scale solutions and are observed to significantly outperform baselines, paving the way for an alternative framework for robust and accurate operator learning. Our code is publicly available at https://github.com/bogdanraonic3/ConvolutionalNeuralOperator

• The paper presents a novel convolutional neural operator architecture that preserves continuous-discrete equivalence to avoid aliasing errors in learning PDE operators.
• The paper establishes a universality theorem showing that CNOs can approximate a wide range of PDE operators with high accuracy and robust generalization.
• Experimental results on Representative PDE Benchmarks demonstrate that CNOs achieve superior performance and resolution invariance compared to models like DeepONets and Fourier Neural Operators.

Convolutional Neural Operators for Robust and Accurate Learning of PDEs: An Overview

The paper entitled "Convolutional Neural Operators for Robust and Accurate Learning of PDEs" presents an innovative approach to solving Partial Differential Equations (PDEs) by leveraging convolution-based neural network architectures. The authors challenge the prevailing belief that convolutional neural networks (CNNs) are inconsistent in function space, particularly when learning solution operators of PDEs. They introduce Convolutional Neural Operators (CNOs) as a novel architecture that extends CNNs to effectively process functions as inputs and outputs, maintaining the integrity of continuous function spaces even when implemented in a discretized format.

Central Contributions

1. Novel Architecture Design: The authors propose significant modifications to standard CNNs to enforce a structure-preserving continuous-discrete equivalence. This condition is crucial to avoid aliasing errors and ensures that the network learns the underlying operator rather than a mere discrete approximation. The resulting CNO architecture is adapted from the widely-used U-Net architecture, tailored specifically for operator learning in the context of PDEs.
2. Universality in Approximating PDE Operators: A key theoretical contribution of the paper is the establishment of a universality theorem. It demonstrates that CNOs can approximate operators arising in a wide array of PDEs to a desired level of accuracy, thus broadening the application potential of CNNs in scientific computing.
3. Benchmark Performance: The authors test CNOs on a comprehensive suite of benchmarks termed Representative PDE Benchmarks (RPB), which encompass an assortment of PDEs with potentially multiscale solutions. The results show that CNOs consistently outperform existing baselines, such as DeepONets, Fourier Neural Operators (FNO), and other neural operator-based architectures.

Experimental Findings

The empirical analysis provided in the paper underscores several salient points:

• Accuracy and Generalization: CNOs not only deliver superior accuracy in approximating PDE solutions compared to other models but also exhibit robust generalization capabilities across varied distributional characteristics of input data.
• Resolution Invariance: The CNO architecture demonstrates resolution invariance, a critical feature that allows it to perform consistently well across different spatial resolutions of input data, which is a noted limitation in many conventional neural operator approaches.

Implications and Future Directions

The introduction and successful application of CNOs have significant implications for the future of data-driven scientific computation:

• Broader Applicability: By overcoming the limitations of traditional CNNs in operator learning, CNOs pave the way for their application in high-dimensional, complex systems modeled by PDEs. This capability could lead to breakthroughs in various fields such as climate modeling, fluid dynamics, and material science.
• Foundation for Further Research: The rigorous theoretical foundation and practical advantages of CNOs suggest several avenues for further research. Future work could explore extensions of CNOs to three-dimensional domains, improve computational efficiency, and even integrate CNOs into broader multi-physics simulation frameworks.

In conclusion, the paper reaffirms the value of convolutional networks in scenarios traditionally dominated by other architectures and highlights the transformative potential of CNOs in advancing the field of scientific machine learning. The thorough theoretical grounding, combined with strong empirical results, makes a compelling case for the broader adoption of this approach in PDE-related problems.