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Andrew M. Stuart

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5 papers

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JMLR Journal 2025 Journal Article

Autoencoders in Function Space

Justin Bunker
Mark Girolami
Hefin Lambley
Andrew M. Stuart
T. J. Sullivan

Autoencoders have found widespread application in both their original deterministic form and in their variational formulation (VAEs). In scientific applications and in image processing it is often of interest to consider data that are viewed as functions; while discretisation (of differential equations arising in the sciences) or pixellation (of images) renders problems finite dimensional in practice, conceiving first of algorithms that operate on functions, and only then discretising or pixellating, leads to better algorithms that smoothly operate between resolutions. In this paper function-space versions of the autoencoder (FAE) and variational autoencoder (FVAE) are introduced, analysed, and deployed. Well-definedness of the objective governing VAEs is a subtle issue, particularly in function space, limiting applicability. For the FVAE objective to be well defined requires compatibility of the data distribution with the chosen generative model; this can be achieved, for example, when the data arise from a stochastic differential equation, but is generally restrictive. The FAE objective, on the other hand, is well defined in many situations where FVAE fails to be. Pairing the FVAE and FAE objectives with neural operator architectures that can be evaluated on any mesh enables new applications of autoencoders to inpainting, superresolution, and generative modelling of scientific data. [abs] [ pdf ][ bib ] [ code ] &copy JMLR 2025. ( edit, beta )

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JMLR Journal 2025 Journal Article

Continuum Attention for Neural Operators

Edoardo Calvello
Nikola B. Kovachki
Matthew E. Levine
Andrew M. Stuart

Transformers, and the attention mechanism in particular, have become ubiquitous in machine learning. Their success in modeling nonlocal, long-range correlations has led to their widespread adoption in natural language processing, computer vision, and time series problems. Neural operators, which map spaces of functions into spaces of functions, are necessarily both nonlinear and nonlocal if they are universal; it is thus natural to ask whether the attention mechanism can be used in the design of neural operators. Motivated by this, we study transformers in the function space setting. We formulate attention as a map between infinite dimensional function spaces and prove that the attention mechanism as implemented in practice is a Monte Carlo or finite difference approximation of this operator. The function space formulation allows for the design of transformer neural operators, a class of architectures designed to learn mappings between function spaces. In this paper, we state and prove the first universal approximation result for transformer neural operators, using only a slight modification of the architecture implemented in practice. The prohibitive cost of applying the attention operator to functions defined on multi-dimensional domains leads to the need for more efficient attention-based architectures. For this reason we also introduce a function space generalization of the patching strategy from computer vision, and introduce a class of associated neural operators. Numerical results, on an array of operator learning problems, demonstrate the promise of our approaches to function space formulations of attention and their use in neural operators. [abs] [ pdf ][ bib ] [ code ] &copy JMLR 2025. ( edit, beta )

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JMLR Journal 2021 Journal Article

Continuous Time Analysis of Momentum Methods

Nikola B. Kovachki
Andrew M. Stuart

Gradient descent-based optimization methods underpin the parameter training of neural networks, and hence comprise a significant component in the impressive test results found in a number of applications. Introducing stochasticity is key to their success in practical problems, and there is some understanding of the role of stochastic gradient descent in this context. Momentum modifications of gradient descent such as Polyak's Heavy Ball method (HB) and Nesterov's method of accelerated gradients (NAG), are also widely adopted. In this work our focus is on understanding the role of momentum in the training of neural networks, concentrating on the common situation in which the momentum contribution is fixed at each step of the algorithm. To expose the ideas simply we work in the deterministic setting. Our approach is to derive continuous time approximations of the discrete algorithms; these continuous time approximations provide insights into the mechanisms at play within the discrete algorithms. We prove three such approximations. Firstly we show that standard implementations of fixed momentum methods approximate a time-rescaled gradient descent flow, asymptotically as the learning rate shrinks to zero; this result does not distinguish momentum methods from pure gradient descent, in the limit of vanishing learning rate. We then proceed to prove two results aimed at understanding the observed practical advantages of fixed momentum methods over gradient descent, when implemented in the non-asymptotic regime with fixed small, but non-zero, learning rate. We achieve this by proving approximations to continuous time limits in which the small but fixed learning rate appears as a parameter; this is known as the method of modified equations in the numerical analysis literature, recently rediscovered as the high resolution ODE approximation in the machine learning context. In our second result we show that the momentum method is approximated by a continuous time gradient flow, with an additional momentum-dependent second order time-derivative correction, proportional to the learning rate; this may be used to explain the stabilizing effect of momentum algorithms in their transient phase. Furthermore in a third result we show that the momentum methods admit an exponentially attractive invariant manifold on which the dynamics reduces, approximately, to a gradient flow with respect to a modified loss function, equal to the original loss function plus a small perturbation proportional to the learning rate; this small correction provides convexification of the loss function and encodes additional robustness present in momentum methods, beyond the transient phase. [abs] [ pdf ][ bib ] [ supplementary ] &copy JMLR 2021. ( edit, beta )

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ICLR Conference 2021 Conference Paper

Fourier Neural Operator for Parametric Partial Differential Equations

Zongyi Li
Nikola Borislavov Kovachki
Kamyar Azizzadenesheli
Burigede Liu
Kaushik Bhattacharya
Andrew M. Stuart
Anima Anandkumar

The classical development of neural networks has primarily focused on learning mappings between finite-dimensional Euclidean spaces. Recently, this has been generalized to neural operators that learn mappings between function spaces. For partial differential equations (PDEs), neural operators directly learn the mapping from any functional parametric dependence to the solution. Thus, they learn an entire family of PDEs, in contrast to classical methods which solve one instance of the equation. In this work, we formulate a new neural operator by parameterizing the integral kernel directly in Fourier space, allowing for an expressive and efficient architecture. We perform experiments on Burgers' equation, Darcy flow, and Navier-Stokes equation. The Fourier neural operator is the first ML-based method to successfully model turbulent flows with zero-shot super-resolution. It is up to three orders of magnitude faster compared to traditional PDE solvers. Additionally, it achieves superior accuracy compared to previous learning-based solvers under fixed resolution.

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JMLR Journal 2018 Journal Article

How Deep Are Deep Gaussian Processes?

Matthew M. Dunlop
Mark A. Girolami
Andrew M. Stuart
Aretha L. Teckentrup

Recent research has shown the potential utility of deep Gaussian processes. These deep structures are probability distributions, designed through hierarchical construction, which are conditionally Gaussian. In this paper, the current published body of work is placed in a common framework and, through recursion, several classes of deep Gaussian processes are defined. The resulting samples generated from a deep Gaussian process have a Markovian structure with respect to the depth parameter, and the effective depth of the resulting process is interpreted in terms of the ergodicity, or non-ergodicity, of the resulting Markov chain. For the classes of deep Gaussian processes introduced, we provide results concerning their ergodicity and hence their effective depth. We also demonstrate how these processes may be used for inference; in particular we show how a Metropolis-within-Gibbs construction across the levels of the hierarchy can be used to derive sampling tools which are robust to the level of resolution used to represent the functions on a computer. For illustration, we consider the effect of ergodicity in some simple numerical examples. [abs] [ pdf ][ bib ] &copy JMLR 2018. ( edit, beta )

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