In this paper we use deep feedforward artificial neural networks to approximate solutions to partial differential equations in complex geometries. We show how to modify the backpropagation algorithm to compute the partial derivatives of the network output with respect to the space variables which is needed to approximate the differential operator. The method is based on an ansatz for the solution which requires nothing but feedforward neural networks and an unconstrained gradient based optimization method such as gradient descent or a quasi-Newton method.We show an example where classical mesh based methods cannot be used and neural networks can be seen as an attractive alternative. Finally, we highlight the benefits of deep compared to shallow neural networks and device some other convergence enhancing techniques.

In this paper we prove new results for p harmonic functions, p = 2, 1 < p < ∞, in Lipschitz and starlike Lipschitz ring domains. In particular we prove the boundary Harnack inequality, Theorem 1, for the ratio of two positive p harmonic functions vanishing on a portion of the boundary of a Lipschitz domain, with constants only depending on p, n and the Lipschitz constant of the domain. For p capacitary functions, in starlike Lipschitz ring domains, we prove an even stronger result, Theorem 2, showing that the ratio is Hölder continuous up to the boundary. Moreover, for p capacitary functions in starlike Lipschitz ring domains we prove, Theorems 3 and 4, appropriate extensions to p = 2, 1 < p < ∞, of famous results of Dahlberg [12] and Jerison and Kenig [25] on the Poisson kernel associated to the Laplace operator (i.e. p = 2). © 2007 Elsevier Masson SAS RÉSUMÉ.-Dans cet article, nous présentons de nouveaux résultats pour des fonctions p-harmoniques, p = 2, 1 < p < ∞, dans des domaines annulaires lipschitziens et lipschitziens étoilés. En particulier, nous démontrons l'inégalité de Harnack au bord (Théorème 1) pour le rapport de deux fonctions p-harmoniques strictement positives quand les deux fonctions s'annulent sur une partie du bord d'un domaine lipschitzien, avec constantes ne dépendant que de p, de n et de la constante de Lipschitz. Pour les fonctions p-harmoniques de capacité, dans des domaines annulaires lipschitziens étoilés, nous prouvons un résultat encore plus fort (Théorème 2) démontrant que le rapport est Hölder continu jusqu'au bord. De plus, pour les fonctions p-harmoniques de capacité dans des domaines annulaires lipschitziens étoilés, nous montrons (Théorèmes 3 et 4) des extensions appropriées pour p = 2, 1 < p < ∞, de résultats très connus de Dahlberg [12] et de Jerison et Kenig [25] sur le noyau de Poisson associé à l'opérateur de Laplace (pour p = 2).

Many processes in science and engineering can be described by partial differential equations (PDEs). Traditionally, PDEs are derived by considering first principles of physics to derive the relations between the involved physical quantities of interest. A different approach is to measure the quantities of interest and use deep learning to reverse engineer the PDEs which are describing the physical process.In this paper we use machine learning, and deep learning in particular, to discover PDEs hidden in complex data sets from measurement data. We include examples of data from a known model problem, and real data from weather station measurements. We show how necessary transformations of the input data amounts to coordinate transformations in the discovered PDE, and we elaborate on feature and model selection. It is shown that the dynamics of a non-linear, second order PDE can be accurately described by an ordinary differential equation which is automatically discovered by our deep learning algorithm. Even more interestingly, we show that similar results apply in the context of more complex simulations of the Swedish temperature distribution.

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In this paper we prove the boundary Harnack inequality and Hölder continuity for ratios of p-harmonic functions vanishing on a portion of certain Reifenberg flat and Ahlfors regular NTA-domains. Applications are given to the p-Martin boundary problem for these domains.

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