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Paul Bogdan (University of Southern California)


Wavelet-inspired Neural Operators: NeuralWave Architectures Enabling Discovering and Multiresolution Learning of Partial Differential Equations


Human perception relies on detecting and processing waves. While our eyes detect waves of electromagnetic radiation, our ears detect waves of compression in the surrounding air. Going beyond waves, from complex dynamics of blood flow to sustain tissue growth and life, to navigating underwater, ground and aerial vehicles at high speeds requires discovering, learning and controlling the partial differential equations (PDEs) governing individual or webs of biological, physical and chemical phenomena. Within this context, neural operators (NOs) proved successful to learn and solve various PDEs. In this talk, we will discuss a multiwavelet-based neural operator learning architecture that compresses the associated operator’s kernel using fine-grained multiwavelets. 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. For initial value problems, we will discuss a Pade´ approximation based exponential neural operator scheme for efficiently learning the map between a given initial condition and the activities at a later time. By embedding the exponential operators in the model, this reduces the training parameters and makes it more data-efficient which is essential in dealing with scarce and noisy real-world datasets. The Pade´exponential operator uses a recurrent structure with shared parameters to model the non-linearity compared to recent neural operators that rely on using multiple linear operator layers in succession. To solve coupled partial differential equations, we propose the coupled multiwavelets neural operator (CMWNO) learning scheme by decoupling the coupled integral kernels during the multiwavelet decomposition and reconstruction procedures in the wavelet space. The proposed model can effectively solve coupled PDEs, including non-local mean field game (MFG) problems.