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Variational Physics Informed Neural Networks: the role of quadratures and test functions

Moreno Pintore
Friday, December 10, 2021 - 14:00

Physics Informed Neural Networks (PINNs) are neural networks used to approximate the solution of differential equations exploiting the equation itself or some known solutions. Since it has been observed that they can efficiently approximate the solution of nonlinear, high-dimensional or parametric problems, their popularity is increasing and different types of networks have been developed; one of them is the Variational Physics Informed Neural Network (VPINN). The main difference between PINNs and VPINNs is the fact that the former are trained minimizing the strong form of the equation, while the latter focus on its variational formulation. In today's talk we analyze the relation between the VPINN accuracy and the way in which it is trained while solving second order elliptic boundary-value problems. In particular, the VPINN is trained minimizing the integral residuals against a predefined set of test functions on a fixed mesh. We prove that, under suitable assumptions, the convergence rate can be computed as a function of the order of the chosen quadrature rule and of the involved test functions. Our estimate suggests that, in order to increase the convergence rate, it is convenient to raise the quadrature rule order as much as possible and using piecewise linear test functions. To derive such an a priori error estimate, we stabilize the VPINN with a piecewise interpolant to satisfy the required inf-sup condition. Nevertheless, even if the a priori estimate has been proved only when the VPINN is interpolated, we observe almost identical convergence rates for non interpolated VPINNs. We also show that such a condition is necessary to avoid spurious modes. The error estimate is proved for classical feed-forward fully connected neural networks, the only required assumption is on its dimension, which has to be large enough to ensure good approximation properties. Several numerical experiments have been performed to confirm the theoretical prediction on different test cases.

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