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Neural operators, as a powerful approximation to the non-linear operatorsbetween infinite-dimensional function spaces, have proved to be promising inaccelerating the solution of partial differential equations (PDE). However, itrequires a large amount of simulated data which can be costly to collect,resulting in a chicken-egg dilemma and limiting its usage in solving PDEs. Tojump out of the dilemma, we propose a general data-free paradigm where theneural network directly learns physics from the mean squared residual (MSR)loss constructed by the discretized PDE. We investigate the physicalinformation in the MSR loss and identify the challenge that the neural networkmust have the capacity to model the long range entanglements in the spatialdomain of the PDE, whose patterns vary in different PDEs. Therefore, we proposethe low-rank decomposition network (LordNet) which is tunable and alsoefficient to model various entanglements. Specifically, LordNet learns alow-rank approximation to the global entanglements with simple fully connectedlayers, which extracts the dominant pattern with reduced computational cost.The experiments on solving Poisson's equation and Navier-Stokes equationdemonstrate that the physical constraints by the MSR loss can lead to betteraccuracy and generalization ability of the neural network. In addition, LordNetoutperforms other modern neural network architectures in both PDEs with thefewest parameters and the fastest inference speed. For Navier-Stokes equation,the learned operator is over 50 times faster than the finite differencesolution with the same computational resources.

LordNet: Learning to Solve Parametric Partial Differential Equations without Simulated Data