Julie Digne

Steeven Janny, Aurélien Béneteau, Madiha Nadri et al.

Estimating fluid dynamics is classically done through the simulation and integration of numerical models solving the Navier-Stokes equations, which is computationally complex and time-consuming even on high-end hardware. This is a notoriously hard problem to solve, which has recently been addressed with machine learning, in particular graph neural networks (GNN) and variants trained and evaluated on datasets of static objects in static scenes with fixed geometry. We attempt to go beyond existing work in complexity and introduce a new model, method and benchmark. We propose EAGLE, a large-scale dataset of 1.1 million 2D meshes resulting from simulations of unsteady fluid dynamics caused by a moving flow source interacting with nonlinear scene structure, comprised of 600 different scenes of three different types. To perform future forecasting of pressure and velocity on the challenging EAGLE dataset, we introduce a new mesh transformer. It leverages node clustering, graph pooling and global attention to learn long-range dependencies between spatially distant data points without needing a large number of iterations, as existing GNN methods do. We show that our transformer outperforms state-of-the-art performance on, both, existing synthetic and real datasets and on EAGLE. Finally, we highlight that our approach learns to attend to airflow, integrating complex information in a single iteration.

Bastien Doignies, Nicolas Bonneel, David Coeurjolly et al.

Much effort has been put into developing samplers with specific properties, such as producing blue noise, low-discrepancy, lattice or Poisson disk samples. These samplers can be slow if they rely on optimization processes, may rely on a wide range of numerical methods, are not always differentiable. The success of recent diffusion models for image generation suggests that these models could be appropriate for learning how to generate point sets from examples. However, their convolutional nature makes these methods impractical for dealing with scattered data such as point sets. We propose a generic way to produce 2-d point sets imitating existing samplers from observed point sets using a diffusion model. We address the problem of convolutional layers by leveraging neighborhood information from an optimal transport matching to a uniform grid, that allows us to benefit from fast convolutions on grids, and to support the example-based learning of non-uniform sampling patterns. We demonstrate how the differentiability of our approach can be used to optimize point sets to enforce properties.

Olivier Pradelle, Raphaelle Chaine, David Wendland et al.

Scene understanding has made tremendous progress over the past few years, as data acquisition systems are now providing an increasing amount of data of various modalities (point cloud, depth, RGB...). However, this improvement comes at a large cost on computation resources and data annotation requirements. To analyze geometric information and images jointly, many approaches rely on both a 2D loss and 3D loss, requiring not only 2D per pixel-labels but also 3D per-point labels. However, obtaining a 3D groundtruth is challenging, time-consuming and error-prone. In this paper, we show that image segmentation can benefit from 3D geometric information without requiring a 3D groundtruth, by training the geometric feature extraction and the 2D segmentation network jointly, in an end-to-end fashion, using only the 2D segmentation loss. Our method starts by extracting a map of 3D features directly from a provided point cloud by using a lightweight 3D neural network. The 3D feature map, merged with the RGB image, is then used as an input to a classical image segmentation network. Our method can be applied to many 2D segmentation networks, improving significantly their performance with only a marginal network weight increase and light input dataset requirements, since no 3D groundtruth is required.

Abderrahim Bendahi, Adrien Fradin, Johan Peralez et al.

Neural operators have emerged as a powerful, discretization-invariant framework for solving partial differential equations (PDEs). Although established approaches like the Deep Operator Network (DeepONet) have successfully achieved universal approximation for operators, and architectures such as Fourier Neural Operators (FNOs) have shown algebraic convergence rates, a precise theoretical connection between the continuous theory and its discrete numerical implementation remains a challenge. Specifically, the relationship between the continuous formulation and the discrete numerical stability has yet to be fully explored. In this paper, we address this gap by establishing theoretical guarantees for the discretization error and stability of neural operator approximation schemes. We prove analytical bounds that link solution regularity to input discretization, providing a formal quantification of neural operator accuracy under real-world numerical constraints. We derive these bounds to the specific cases of State Space Model-based Neural Operators (SS-NOs) and FNOs, thus providing a new discretization error theorem for these models. Additionally, through an input-to-state stability (ISS) analysis, we formally assess the impact of discretization on the stability of SS-NOs results obtained in the continuous domain. Our empirical experiments on 1D and 2D benchmarks validate our theoretical bounds and show the robustness of SS-NOs under varying resolutions.

Mattéo Clémot, Julie Digne, Julien Tierny

This paper presents a novel topology-aware dimensionality reduction approach aiming at accurately visualizing the cyclic patterns present in high dimensional data. To that end, we build on the Topological Autoencoders (TopoAE) formulation. First, we provide a novel theoretical analysis of its associated loss and show that a zero loss indeed induces identical persistence pairs (in high and low dimensions) for the $0$-dimensional persistent homology (PH$^0$) of the Rips filtration. We also provide a counter example showing that this property no longer holds for a naive extension of TopoAE to PH$^d$ for $d\ge 1$. Based on this observation, we introduce a novel generalization of TopoAE to $1$-dimensional persistent homology (PH$^1$), called TopoAE++, for the accurate generation of cycle-aware planar embeddings, addressing the above failure case. This generalization is based on the notion of cascade distortion, a new penalty term favoring an isometric embedding of the $2$-chains filling persistent $1$-cycles, hence resulting in more faithful geometrical reconstructions of the $1$-cycles in the plane. We further introduce a novel, fast algorithm for the exact computation of PH for Rips filtrations in the plane, yielding improved runtimes over previously documented topology-aware methods. Our method also achieves a better balance between the topological accuracy, as measured by the Wasserstein distance, and the visual preservation of the cycles in low dimensions. Our C++ implementation is available at https://github.com/MClemot/TopologicalAutoencodersPlusPlus.

Emmanouil Nikolakakis, Amine Ouasfi, Julie Digne et al.

We present RibPull, a methodology that utilizes implicit occupancy fields to bridge computational geometry and medical imaging. Implicit 3D representations use continuous functions that handle sparse and noisy data more effectively than discrete methods. While voxel grids are standard for medical imaging, they suffer from resolution limitations, topological information loss, and inefficient handling of sparsity. Coordinate functions preserve complex geometrical information and represent a better solution for sparse data representation, while allowing for further morphological operations. Implicit scene representations enable neural networks to encode entire 3D scenes within their weights. The result is a continuous function that can implicitly compesate for sparse signals and infer further information about the 3D scene by passing any combination of 3D coordinates as input to the model. In this work, we use neural occupancy fields that predict whether a 3D point lies inside or outside an object to represent CT-scanned ribcages. We also apply a Laplacian-based contraction to extract the medial axis of the ribcage, thus demonstrating a geometrical operation that benefits greatly from continuous coordinate-based 3D scene representations versus voxel-based representations. We evaluate our methodology on 20 medical scans from the RibSeg dataset, which is itself an extension of the RibFrac dataset. We will release our code upon publication.

Beatrix-Emőke Fülöp-Balogh, Eleanor Tursman, James Tompkin et al.

Structure from motion (SfM) enables us to reconstruct a scene via casual capture from cameras at different viewpoints, and novel view synthesis (NVS) allows us to render a captured scene from a new viewpoint. Both are hard with casual capture and dynamic scenes: SfM produces noisy and spatio-temporally sparse reconstructed point clouds, resulting in NVS with spatio-temporally inconsistent effects. We consider SfM and NVS parts together to ease the challenge. First, for SfM, we recover stable camera poses, then we defer the requirement for temporally-consistent points across the scene and reconstruct only a sparse point cloud per timestep that is noisy in space-time. Second, for NVS, we present a variational diffusion formulation on depths and colors that lets us robustly cope with the noise by enforcing spatio-temporal consistency via per-pixel reprojection weights derived from the input views. Together, this deferred approach generates novel views for dynamic scenes without requiring challenging spatio-temporally consistent reconstructions nor training complex models on large datasets. We demonstrate our algorithm on real-world dynamic scenes against classic and more recent learning-based baseline approaches.

Julien Lacombe, Julie Digne, Nicolas Courty et al.

Optimal transport is a notoriously difficult problem to solve numerically, with current approaches often remaining intractable for very large scale applications such as those encountered in machine learning. Wasserstein barycenters -- the problem of finding measures in-between given input measures in the optimal transport sense -- is even more computationally demanding as it requires to solve an optimization problem involving optimal transport distances. By training a deep convolutional neural network, we improve by a factor of 60 the computational speed of Wasserstein barycenters over the fastest state-of-the-art approach on the GPU, resulting in milliseconds computational times on $512\times512$ regular grids. We show that our network, trained on Wasserstein barycenters of pairs of measures, generalizes well to the problem of finding Wasserstein barycenters of more than two measures. We demonstrate the efficiency of our approach for computing barycenters of sketches and transferring colors between multiple images.