Physics-informed neural networks (PINNs) combine the expressiveness of neural networks with the interpretability of physical modeling. Their good practical performance has been demonstrated both in the context of solving partial differential equations and in the context of hybrid modeling, which consists of combining an imperfect physical model with noisy observations. However, most of their...

Neural networks are increasingly used in scientific computing. Indeed, once trained, they can approximate highly complex, non-linear, and high dimensional functions with significantly reduced computational overhead compared to traditional simulation codes based on finite-differences methods. However, unlike conventional simulation whose error can be controlled, neural networks are statistical,...

Gaussian process regression (GPR) is the Bayesian formulation of kernel regression methods used in machine learning. This method may be used to treat regression problems stemming from physical models, the latter typically taking the form of partial differential equations (PDEs).

In this presentation, we study the question of the design of GPR methods, in relation with a target PDE model....

Surrogate modeling of costly mathematical models representing physical systems is challenging since it is necessary to fulfill physical constraints in the whole design domain together with specific boundary conditions of investigated systems. Moreover, it is typically not possible to create a large experimental design covering whole input space due to computational burden of original models....

Many parametric PDEs have solutions that possess a high degree of regularity with respect to their parameters. Low-rank tensor formats can leverage this regularity to overcome the curse of dimensionality and achieve optimal convergence rates in a wide range of approximation spaces. A particular advantage of these formats is their highly structured nature, which enables us to control the...