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Wavelet neural operator: a neural operator for parametric partial differential equations
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With massive advancements in sensor technologies and Internet-of-things, we now have access to terabytes of historical data; however, there is a lack of clarity in how to best exploit the data to predict future events. One possible alternative in this context is to utilize operator learning algorithm that directly learn nonlinear mapping between two functional spaces; this facilitates real-time prediction of naturally arising complex evolutionary dynamics. In this work, we introduce a novel operator learning algorithm referred to as the Wavelet Neural Operator (WNO) that blends integral kernel with wavelet transformation. WNO harnesses the superiority of the wavelets in time-frequency localization of the functions and enables accurate tracking of patterns in spatial domain and effective learning of the functional mappings. Since the wavelets are localized in both time/space and frequency, WNO can provide high spatial and frequency resolution. This offers learning of the finer details of the parametric dependencies in the solution for complex problems. The efficacy and robustness of the proposed WNO are illustrated on a wide array of problems involving Burger's equation, Darcy flow, Navier-Stokes equation, Allen-Cahn equation, and Wave advection equation. Comparative study with respect to existing operator learning frameworks are presented. Finally, the proposed approach is used to build a digital twin capable of predicting Earth's air temperature based on available historical data.
Forward citations
Cited by 7 Pith papers
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PDEfuncta: Spectrally-Aware Neural Representation for PDE Solution Modeling
A Fourier-based weight modulation for shared INR networks improves reconstruction of high-frequency PDE fields and enables bidirectional inference between paired solution spaces.
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PMNO: A novel physics guided multi-step neural operator predictor for partial differential equations
A multi-step neural operator trained with an implicit BDF-based physics residual loss predicts PDE dynamics for longer horizons than data-only baselines in five benchmark systems.
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Principled Approaches for Extending Neural Architectures to Function Spaces for Operator Learning
A practical recipe to convert common neural architectures into discretization-agnostic neural operators, validated by Navier-Stokes experiments showing cross-resolution generalization of FNO-style models.
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A Neural Operator based on Dynamic Mode Decomposition
A DMD-enhanced branch-trunk neural operator is proposed and tested on three 2D PDEs, but the claimed comparative results and key theoretical bound are not supported.
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