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This paper deals with the following important research question.Traditionally, the neural network employs non-linear activation functionsconcatenated with linear operators to approximate a given physical phenomenon.They "fill the space" with the concatenations of the activation functions andlinear operators and adjust their coefficients to approximate the physicalphenomena. We claim that it is better to "fill the space" with linearcombinations of smooth higher-order B-splines base functions as employed byisogeometric analysis and utilize the neural networks to adjust thecoefficients of linear combinations. In other words, the possibilities of usingneural networks for approximating the B-spline base functions' coefficients andby approximating the solution directly are evaluated. Solving differentialequations with neural networks has been proposed by Maziar Raissi et al. in2017 by introducing Physics-informed Neural Networks (PINN), which naturallyencode underlying physical laws as prior information. Approximation ofcoefficients using a function as an input leverages the well-known capabilityof neural networks being universal function approximators. In essence, in thePINN approach the network approximates the value of the given field at a givenpoint. We present an alternative approach, where the physcial quantity isapproximated as a linear combination of smooth B-spline basis functions, andthe neural network approximates the coefficients of B-splines. This researchcompares results from the DNN approximating the coefficients of the linearcombination of B-spline basis functions, with the DNN approximating thesolution directly. We show that our approach is cheaper and more accurate whenapproximating smooth physical fields.

Deep neural networks for smooth approximation of physics with higher order and continuity B-spline base functions