Emmanuel de Bezenac

Levi Lingsch, Mike Y. Michelis, Emmanuel de Bezenac et al.

The computational efficiency of many neural operators, widely used for learning solutions of PDEs, relies on the fast Fourier transform (FFT) for performing spectral computations. As the FFT is limited to equispaced (rectangular) grids, this limits the efficiency of such neural operators when applied to problems where the input and output functions need to be processed on general non-equispaced point distributions. Leveraging the observation that a limited set of Fourier (Spectral) modes suffice to provide the required expressivity of a neural operator, we propose a simple method, based on the efficient direct evaluation of the underlying spectral transformation, to extend neural operators to arbitrary domains. An efficient implementation* of such direct spectral evaluations is coupled with existing neural operator models to allow the processing of data on arbitrary non-equispaced distributions of points. With extensive empirical evaluation, we demonstrate that the proposed method allows us to extend neural operators to arbitrary point distributions with significant gains in training speed over baselines while retaining or improving the accuracy of Fourier neural operators (FNOs) and related neural operators.

Matthieu Kirchmeyer, Alain Rakotomamonjy, Emmanuel de Bezenac et al.

We consider the problem of unsupervised domain adaptation (UDA) between a source and a target domain under conditional and label shift a.k.a Generalized Target Shift (GeTarS). Unlike simpler UDA settings, few works have addressed this challenging problem. Recent approaches learn domain-invariant representations, yet they have practical limitations and rely on strong assumptions that may not hold in practice. In this paper, we explore a novel and general approach to align pretrained representations, which circumvents existing drawbacks. Instead of constraining representation invariance, it learns an optimal transport map, implemented as a NN, which maps source representations onto target ones. Our approach is flexible and scalable, it preserves the problem's structure and it has strong theoretical guarantees under mild assumptions. In particular, our solution is unique, matches conditional distributions across domains, recovers target proportions and explicitly controls the target generalization risk. Through an exhaustive comparison on several datasets, we challenge the state-of-the-art in GeTarS.

Arthur Pajot, Emmanuel de Bezenac, Patrick Gallinari

We consider inpainting in an unsupervised setting where there is neither access to paired nor unpaired training data. The only available information is provided by the uncomplete observations and the inpainting process statistics. In this context, an observation should give rise to several plausible reconstructions which amounts at learning a distribution over the space of reconstructed images. We model the reconstruction process by using a conditional GAN with constraints on the stochastic component that introduce an explicit dependency between this component and the generated output. This allows us sampling from the latent component in order to generate a distribution of images associated to an observation. We demonstrate the capacity of our model on several image datasets: faces (CelebA), food images (Recipe-1M) and bedrooms (LSUN Bedrooms) with different types of imputation masks. The approach yields comparable performance to model variants trained with additional supervision.

Emmanuel de Bezenac, Arthur Pajot, Patrick Gallinari

We consider the use of Deep Learning methods for modeling complex phenomena like those occurring in natural physical processes. With the large amount of data gathered on these phenomena the data intensive paradigm could begin to challenge more traditional approaches elaborated over the years in fields like maths or physics. However, despite considerable successes in a variety of application domains, the machine learning field is not yet ready to handle the level of complexity required by such problems. Using an example application, namely Sea Surface Temperature Prediction, we show how general background knowledge gained from physics could be used as a guideline for designing efficient Deep Learning models. In order to motivate the approach and to assess its generality we demonstrate a formal link between the solution of a class of differential equations underlying a large family of physical phenomena and the proposed model. Experiments and comparison with series of baselines including a state of the art numerical approach is then provided.