Related papers: Geometric Operator Learning with Optimal Transport
Neural operator learning accelerates PDE solution by approximating operators as mappings between continuous function spaces. Yet in many engineering settings, varying geometry induces discrete structural changes, including topological…
The optimal transport (OT) map offers the most economical way to transfer one probability measure distribution to another. Classical OT theory does not involve a discussion of preserving topological connections and orientations in…
Computing optimal transport (OT) for general high-dimensional data has been a long-standing challenge. Despite much progress, most of the efforts including neural network methods have been focused on the static formulation of the OT…
Industrial design evaluation often relies on high-fidelity simulations of governing partial differential equations (PDEs). While accurate, these simulations are computationally expensive, making dense exploration of design spaces…
The challenge of applying learned knowledge from one domain to solve problems in another related but distinct domain, known as transfer learning, is fundamental in operator learning models that solve Partial Differential Equations (PDEs).…
The ability to compare two degenerate probability distributions (i.e. two probability distributions supported on two distinct low-dimensional manifolds living in a much higher-dimensional space) is a crucial problem arising in the…
Optimal transport (OT) has recently found widespread interest in machine learning. It allows to define novel distances between probability measures, which have shown promise in several applications. In this work, we discuss how to…
Solving large scale Optimal Transport (OT) in machine learning typically relies on sampling measures to obtain a tractable discrete problem. While the discrete solver's accuracy is controllable, the rate of convergence of the discretization…
Learning partial differential equations' (PDEs) solution operators is an essential problem in machine learning. However, there are several challenges for learning operators in practical applications like the irregular mesh, multiple input…
Neural operators have emerged as promising surrogate models for solving partial differential equations (PDEs), but struggle to generalise beyond training distributions and are often constrained to a fixed temporal discretisation. This work…
Optimal transport (OT) aims to find a map $T$ that transports mass from one probability measure to another while minimizing a cost function. Recently, neural OT solvers have gained popularity in high dimensional biological applications such…
Optimal Transport (OT) distances such as Wasserstein have been used in several areas such as GANs and domain adaptation. OT, however, is very sensitive to outliers (samples with large noise) in the data since in its objective function,…
We introduce a method that combines neural operators, physics-informed machine learning, and standard numerical methods for solving PDEs. The proposed approach extends each of the aforementioned methods and unifies them within a single…
This thesis examines self-attention training through the lens of Optimal Transport (OT) and develops an OT-based alternative for tabular classification. The study tracks intermediate projections of the self-attention layer during training…
Reduced order models (ROMs) are widely used in scientific computing to tackle high-dimensional systems. However, traditional ROM methods may only partially capture the intrinsic geometric characteristics of the data. These characteristics…
Optimal transport (OT) has gained popularity due to its various applications in fields such as machine learning, statistics, and signal processing. However, the balanced mass requirement limits its performance in practical problems. To…
Dataset distillation seeks to synthesize a compact distilled dataset, enabling models trained on it to achieve performance comparable to models trained on the full dataset. Recent methods for large-scale datasets focus on matching global…
Neural ordinary differential equations (ODEs) provide expressive representations of invertible transport maps that can be used to approximate complex probability distributions, e.g., for generative modeling, density estimation, and Bayesian…
The task of automatically segmenting 3-D surfaces representing boundaries of objects is important for quantitative analysis of volumetric images, and plays a vital role in biomedical image analysis. Recently, graph-based methods with a…
Neural network-based optimal transport (OT) is a recent and fruitful direction in the generative modeling community. It finds its applications in various fields such as domain translation, image super-resolution, computational biology and…