Moreno Pintore

Moreno Pintore, Bruno Després

Deriving sharp and computable upper bounds of the Lipschitz constant of deep neural networks is crucial to formally guarantee the robustness of neural-network based models. We analyse three existing upper bounds written for the $l^2$ norm. We highlight the importance of working with the $l^1$ and $l^\infty$ norms and we propose two novel bounds for both feed-forward fully-connected neural networks and convolutional neural networks. We treat the technical difficulties related to convolutional neural networks with two different methods, called explicit and implicit. Several numerical tests empirically confirm the theoretical results, help to quantify the relationship between the presented bounds and establish the better accuracy of the new bounds. Four numerical tests are studied: two where the output is derived from an analytical closed form are proposed; another one with random matrices; and the last one for convolutional neural networks trained on the MNIST dataset. We observe that one of our bound is optimal in the sense that it is exact for the first test with the simplest analytical form and it is better than other bounds for the other tests.

Moreno Pintore, Bruno Després

We present a machine-learning based Volume Of Fluid method to simulate multi-material flows on three-dimensional domains. One of the novelties of the method is that the flux fraction is computed by evaluating a previously trained neural network and without explicitly reconstructing any local interface approximating the exact one. The network is trained on a purely synthetic dataset generated by randomly sampling numerous local interfaces and which can be adapted to improve the scheme on less regular interfaces when needed. Several strategies to ensure the efficiency of the method and the satisfaction of physical constraints and properties are suggested and formalized. Numerical results on the advection equation are provided to show the performance of the method. We observe numerical convergence as the size of the mesh tends to zero $h=1/N_h\searrow 0$, with a better rate than two reference schemes.

Stefano Berrone, Claudio Canuto, Moreno Pintore

In this work we analyze how quadrature rules of different precisions and piecewise polynomial test functions of different degrees affect the convergence rate of Variational Physics Informed Neural Networks (VPINN) with respect to mesh refinement, while solving elliptic boundary-value problems. Using a Petrov-Galerkin framework relying on an inf-sup condition, we derive an a priori error estimate in the energy norm between the exact solution and a suitable high-order piecewise interpolant of a computed neural network. Numerical experiments confirm the theoretical predictions and highlight the importance of the inf-sup condition. Our results suggest, somehow counterintuitively, that for smooth solutions the best strategy to achieve a high decay rate of the error consists in choosing test functions of the lowest polynomial degree, while using quadrature formulas of suitably high precision.