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We present a lightweighted neural PDE representation to discover the hiddenstructure and predict the solution of different nonlinear PDEs. Our key idea isto leverage the prior of ``translational similarity'' of numerical PDEdifferential operators to drastically reduce the scale of learning model andtraining data. We implemented three central network components, including aneural functional convolution operator, a Picard forward iterative procedure,and an adjoint backward gradient calculator. Our novel paradigm fully leveragesthe multifaceted priors that stem from the sparse and smooth nature of thephysical PDE solution manifold and the various mature numerical techniques suchas adjoint solver, linearization, and iterative procedure to accelerate thecomputation. We demonstrate the efficacy of our method by robustly discoveringthe model and accurately predicting the solutions of various types of PDEs withsmall-scale networks and training sets. We highlight that all the PDE exampleswe showed were trained with up to 8 data samples and within 325 networkparameters.

Neural Partial Differential Equations with Functional Convolution