In this paper, we propose a general framework for projection-based model order reduction assisted by deep neural networks. The proposed methodology, called ROM-net, consists in using deep learning techniques to adapt the reduced-order model to a stochastic input tensor whose nonparametrized variabilities strongly influence the quantities of interest for a given physics problem. In particular, we introduce the concept of dictionary-based ROM-nets, where deep neural networks recommend a suitable local reduced-order model from a dictionary. The dictionary of local reduced-order models is constructed from a clustering of simplified simulations enabling the identification of the subspaces in which the solutions evolve for different input tensors. The training examples are represented by points on a Grassmann manifold, on which distances are computed for clustering. This methodology is applied to an anisothermal elastoplastic problem in structural mechanics, where the damage field depends on a random temperature field. When using deep neural networks, the selection of the best reduced-order model for a given thermal loading is 60 times faster than when following the clustering procedure used in the training phase.

The reduced basis method is a model reduction technique yielding substantial savings of computational time when a solution to a parametrized equation has to be computed for many values of the parameter. Certification of the approximation is possible by means of an a posteriori error bound. Under appropriate assumptions, this error bound is computed with an algorithm of complexity independent of the size of the full problem. In practice, the evaluation of the error bound can become very sensitive to round-off errors. We propose herein an explanation of this fact. A first remedy has been proposed in [F. Casenave, Accurate a posteriori error evaluation in the reduced basis method. C. R. Math. Acad. Sci. Paris 350 (2012) 539-542.]. Herein, we improve this remedy by proposing a new approximation of the error bound using the empirical interpolation method (EIM). This method achieves higher levels of accuracy and requires potentially less precomputations than the usual formula. A version of the EIM stabilized with respect to round-off errors is also derived. The method is illustrated on a simple one-dimensional diffusion problem and a three-dimensional acoustic scattering problem solved by a boundary element method.Mathematics Subject Classification. 65N15, 65D05, 68W25, 76Q05.Article published by EDP Sciences c EDP Sciences, SMAI 2014 208 F. CASENAVE ET AL.As described in Section 2, the RB method consists in replacing the sequence P μHere, P denotes the parameter set, E μ : μ → u μ the model problem,Ê μ : μ →û μ its lower-dimensional approximation, Q(u μ ) the quantity of interest, andQ(û μ ) its RB approximation. More specifically, the RB method consists in two steps: (i) A so-called offline stage, where solutions to E μ for well-chosen values of the parameter μ are computed. During this stage,N problems of size N are solved (withN N ), and some quantities related to theN solutions are stored, and (ii) a so-called online stage, where the precomputed quantities are used to solveÊ μ for many values of μ. In this stage, a certification of the approximation is possible by means of an a posteriori error bound. An important feature in the RB method is the use of an online-efficient error bound. The notion of online-efficiency is defined in Section 2.4. Moreover, the error bound must be as sharp as possible to faithfully represent the error. However, as noticed for example in ([34], pp. 148-149), the error bound is subject to round-off errors, especially for the computation of accurate solutions. This difficulty can be encountered in complex industrial applications in the following two cases. First and most importantly, when the stability constant of the underlying bilinear (or sesquilinear) form is very small, the classical formula for the error bound fails to certify, even at a relatively crude error level, as illustrated in Section 5 where the stability constant is about 10 −6 and the classical error bound stagnates at about 10 −4 . Second, in some industrial codes, the single-precision format is used to speed up computat...

The Reduced Basis Method can be exploited in an efficient way only if the so-called affine dependence assumption on the operator and right-hand side of the considered problem with respect to the parameters is satisfied. When it is not, the Empirical Interpolation Method is usually used to recover this assumption approximately. In both cases, the Reduced Basis Method requires to access and modify the assembly routines of the corresponding computational code, leading to an intrusive procedure. In this work, we derive variants of the EIM algorithm and explain how they can be used to turn the Reduced Basis Method into a nonintrusive procedure. We present examples of aeroacoustic problems solved by integral equations and show how our algorithms can benefit from the linear algebra tools available in the considered code.

Summary In this work, we propose a framework that constructs reduced‐order models for nonlinear structural mechanics in a nonintrusive fashion and can handle large‐scale simulations. Three steps are carried out: (i) the production of high‐fidelity solutions by commercial software, (ii) the offline stage of the model reduction, and (iii) the online stage where the reduced‐order model is exploited. The nonintrusivity assumes that only the displacement field solution is known, and the proposed framework carries out operations on these simulation data during the offline phase. The compatibility with a new commercial code only needs the implementation of a routine converting the discretized solution into our in‐house data format. The nonintrusive capabilities of the framework are demonstrated on numerical experiments using commercial versions of Z‐set and Ansys Mechanical. The nonlinear constitutive equations are evaluated by using an external plugin. The large‐scale simulations are handled using domain decomposition and parallel computing with distributed memory. The features and performances of the framework are evaluated on two numerical applications involving elastoviscoplastic materials: the second one involves a model of high‐pressure blade, where the framework is used to extrapolate cyclic loadings in 6.5 hours, whereas the reference high‐fidelity computation would take 9.5 days.

Evapotranspiration (ET) controls the flux between the land surface and the atmosphere. Assessing the ET ecosystems remains a key challenge in hydrology. We have found that the ET water mass loss can be directly inferred from continuous gravity measurements: as water evaporates and transpires from terrestrial ecosystems, the mass distribution of water decreases, changing the gravity field. Using continuous superconducting gravity measurements, we were able to identify daily gravity changes at the level of, or smaller than, 10−9 nm s−2 (or 10−10 g) per day. This corresponds to 1.7 mm of water over an area of 50 ha. The strength of this method is its ability to enable a direct, traceable and continuous monitoring of actual ET for years at the mesoscale with a high accuracy.