56 citations · 160 across the 20 of their papers we have counts for
24 papers
Conformal Uncertainty Quantification Guarantees for Neural Operators
Tom Stent, Nicolas Boullé
Neural operators provide fast surrogate models for approximating operators between function spaces, but their predictions often lack uncertainty quantification. We develop a split…
Physics-guided correction for operator learning under model misspecification
Lei Ma, Nicolas Boullé, Yu-Sen Yang +2
Physics-informed operator learning provides an efficient framework for approximating solution operators of partial differential equations by combining observational data with gover…
A zero-one law for one-shot system identification
Nicolas Boullé, Diana Halikias, Samuel E. Otto +1
Can a model be identified from one experiment? We study analytic systems that are linearly parameterized by a combination of prescribed dictionary terms, such as partial differenti…
Generalized Discrete Diffusion from Snapshots
Oussama Zekri, Théo Uscidda, Nicolas Boullé +1
We introduce Generalized Discrete Diffusion from Snapshots (GDDS), a unified framework for discrete diffusion modeling that supports arbitrary noising processes over large discrete…
Trustworthy Koopman Operator Learning: Invariance Diagnostics and Error Bounds
Gustav Conradie, Nicolas Boullé, Jean-Christophe Loiseau +2
Koopman operator theory provides a global linear representation of nonlinear dynamics and underpins many data-driven methods. In practice, however, finite-dimensional feature space…
Jacobian Scopes: token-level causal attributions in LLMs
Toni J. B. Liu, Baran Zadeoğlu, Nicolas Boullé +3
Large language models (LLMs) make next-token predictions based on clues present in their context, such as semantic descriptions and in-context examples. Yet, elucidating which prio…
Subsystems (in)dependence in GIE proposals
Nicolas Boulle, Guilherme Franzmann
Recent proposals suggest that detecting entanglement between two spatially superposed masses would establish the quantum nature of gravity. However, these gravitationally induced e…
Text-Trained LLMs Can Zero-Shot Extrapolate PDE Dynamics, Revealing a Three-Stage In-Context Learning Mechanism
Jiajun Bao, Nicolas Boullé, Toni J. B. Liu +2
Large language models (LLMs) have demonstrated emergent in-context learning (ICL) capabilities across a range of tasks, including zero-shot time-series forecasting. We show that te…
Convergent Methods for Koopman Operators on Reproducing Kernel Hilbert Spaces
Nicolas Boullé, Matthew J. Colbrook, Gustav Conradie
Data-driven spectral analysis of Koopman operators is a powerful tool for understanding numerous real-world dynamical systems, from neuronal activity to variations in sea surface t…
Multi-Level Monte Carlo Training of Neural Operators
James Rowbottom, Stefania Fresca, Pietro Lio +2
Operator learning is a rapidly growing field that aims to approximate nonlinear operators related to partial differential equations (PDEs) using neural operators. These rely on dis…
Elliptic PDE learning is provably data-efficient
Nicolas Boullé, Diana Halikias, Alex Townsend
PDE learning is an emerging field that combines physics and machine learning to recover unknown physical systems from experimental data. While deep learning models traditionally re…
Bifurcation analysis of a two-dimensional magnetic Rayleigh-Bénard problem
Fabian Laakmann, Nicolas Boullé
We perform a bifurcation analysis of a two-dimensional magnetic Rayleigh--Bénard problem using a numerical technique called deflated continuation. Our aim is to study the influence…
Data-driven discovery of Green's functions
Nicolas Boullé
Discovering hidden partial differential equations (PDEs) and operators from data is an important topic at the frontier between machine learning and numerical analysis. This doctora…
Learning Green's functions associated with time-dependent partial differential equations
Nicolas Boullé, Seick Kim, Tianyi Shi +1
Neural operators are a popular technique in scientific machine learning to learn a mathematical model of the behavior of unknown physical systems from data. Neural operators are es…
Optimization of Hopf bifurcation points
Nicolas Boullé, Patrick E. Farrell, Marie E. Rognes
We introduce a numerical technique for controlling the location and stability properties of Hopf bifurcations in dynamical systems. The algorithm consists of solving an optimizatio…
Control of bifurcation structures using shape optimization
Nicolas Boullé, Patrick E. Farrell, Alberto Paganini
Many problems in engineering can be understood as controlling the bifurcation structure of a given device. For example, one may wish to delay the onset of instability, or bring for…
A generalization of the randomized singular value decomposition
Nicolas Boullé, Alex Townsend
The randomized singular value decomposition (SVD) is a popular and effective algorithm for computing a near-best rank approximation of a matrix using matrix-vector products…
Data-driven discovery of Green's functions with human-understandable deep learning
Nicolas Boullé, Christopher J. Earls, Alex Townsend
There is an opportunity for deep learning to revolutionize science and technology by revealing its findings in a human interpretable manner. To do this, we develop a novel data-dri…
An optimal complexity spectral method for Navier--Stokes simulations in the ball
Nicolas Boullé, Jonasz Słomka, Alex Townsend
We develop a spectral method for solving the incompressible generalized Navier--Stokes equations in the ball with no-flux and prescribed slip boundary conditions. The algorithm ach…
Accurate numerical simulation of electrodiffusion and water movement in brain tissue
Ada J. Ellingsrud, Nicolas Boullé, Patrick E. Farrell +1
Mathematical modelling of ionic electrodiffusion and water movement is emerging as a powerful avenue of investigation to provide new physiological insight into brain homeostasis. H…
Bifurcation analysis of two-dimensional Rayleigh--Bénard convection using deflation
Nicolas Boullé, Vassilios Dallas, Patrick E. Farrell
We perform a bifurcation analysis of the steady states of Rayleigh--Bénard convection with no-slip boundary conditions in two dimensions using a numerical method called deflated co…
Learning elliptic partial differential equations with randomized linear algebra
Nicolas Boullé, Alex Townsend
Given input-output pairs of an elliptic partial differential equation (PDE) in three dimensions, we derive the first theoretically-rigorous scheme for learning the associated Green…
Rational neural networks
Nicolas Boullé, Yuji Nakatsukasa, Alex Townsend
We consider neural networks with rational activation functions. The choice of the nonlinear activation function in deep learning architectures is crucial and heavily impacts the pe…
Computing with functions in the ball
Nicolas Boullé, Alex Townsend
A collection of algorithms in object-oriented MATLAB is described for numerically computing with smooth functions defined on the unit ball in the Chebfun software. Functions are nu…