4 research outputs found
Diffusion is All You Need for Learning on Surfaces
We introduce a new approach to deep learning on 3D surfaces such as meshes or
point clouds. Our key insight is that a simple learned diffusion layer can
spatially share data in a principled manner, replacing operations like
convolution and pooling which are complicated and expensive on surfaces. The
only other ingredients in our network are a spatial gradient operation, which
uses dot-products of derivatives to encode tangent-invariant filters, and a
multi-layer perceptron applied independently at each point. The resulting
architecture, which we call DiffusionNet, is remarkably simple, efficient, and
scalable. Continuously optimizing for spatial support avoids the need to pick
neighborhood sizes or filter widths a priori, or worry about their impact on
network size/training time. Furthermore, the principled, geometric nature of
these networks makes them agnostic to the underlying representation and
insensitive to discretization. In practice, this means significant robustness
to mesh sampling, and even the ability to train on a mesh and evaluate on a
point cloud. Our experiments demonstrate that these networks achieve
state-of-the-art results for a variety of tasks on both meshes and point
clouds, including surface classification, segmentation, and non-rigid
correspondence
Generalised Latent Assimilation in Heterogeneous Reduced Spaces with Machine Learning Surrogate Models
Reduced-order modelling and low-dimensional surrogate models generated using
machine learning algorithms have been widely applied in high-dimensional
dynamical systems to improve the algorithmic efficiency. In this paper, we
develop a system which combines reduced-order surrogate models with a novel
data assimilation (DA) technique used to incorporate real-time observations
from different physical spaces. We make use of local smooth surrogate functions
which link the space of encoded system variables and the one of current
observations to perform variational DA with a low computational cost. The new
system, named Generalised Latent Assimilation can benefit both the efficiency
provided by the reduced-order modelling and the accuracy of data assimilation.
A theoretical analysis of the difference between surrogate and original
assimilation cost function is also provided in this paper where an upper bound,
depending on the size of the local training set, is given. The new approach is
tested on a high-dimensional CFD application of a two-phase liquid flow with
non-linear observation operators that current Latent Assimilation methods can
not handle. Numerical results demonstrate that the proposed assimilation
approach can significantly improve the reconstruction and prediction accuracy
of the deep learning surrogate model which is nearly 1000 times faster than the
CFD simulation
Neural function approximation on graphs: shape modelling, graph discrimination & compression
Graphs serve as a versatile mathematical abstraction of real-world phenomena in numerous scientific disciplines. This thesis is part of the Geometric Deep Learning subject area, a family of learning paradigms, that capitalise on the increasing volume of non-Euclidean data so as to solve real-world tasks in a data-driven manner. In particular, we focus on the topic of graph function approximation using neural networks, which lies at the heart of many relevant methods. In the first part of the thesis, we contribute to the understanding and design of Graph Neural Networks (GNNs). Initially, we investigate the problem of learning on signals supported on a fixed graph. We show that treating graph signals as general graph spaces is restrictive and conventional GNNs have limited expressivity. Instead, we expose a more enlightening perspective by drawing parallels between graph signals and signals on Euclidean grids, such as images and audio. Accordingly, we propose a permutation-sensitive GNN based on an operator analogous to shifts in grids and instantiate it on 3D meshes for shape modelling (Spiral Convolutions). Following, we focus on learning on general graph spaces and in particular on functions that are invariant to graph isomorphism. We identify a fundamental trade-off between invariance, expressivity and computational complexity, which we address with a symmetry-breaking mechanism based on substructure encodings (Graph Substructure Networks). Substructures are shown to be a powerful tool that provably improves expressivity while controlling computational complexity, and a useful inductive bias in network science and chemistry. In the second part of the thesis, we discuss the problem of graph compression, where we analyse the information-theoretic principles and the connections with graph generative models. We show that another inevitable trade-off surfaces, now between computational complexity and compression quality, due to graph isomorphism. We propose a substructure-based dictionary coder - Partition and Code (PnC) - with theoretical guarantees that can be adapted to different graph distributions by estimating its parameters from observations. Additionally, contrary to the majority of neural compressors, PnC is parameter and sample efficient and is therefore of wide practical relevance. Finally, within this framework, substructures are further illustrated as a decisive archetype for learning problems on graph spaces.Open Acces