Numerics
Mathematical challenges
Our numerical programme develops reliable approximations for nonlinear and stochastic systems whose conservation laws, geometry or multiscale structure matter to the observed behaviour.
Rather than treating discretization as a separate final step, we use analysis to design and assess methods that retain essential properties of the continuum model. Topics include structure-preserving schemes for conservative SPDEs, efficient algorithms for generative and diffusion models, numerical approaches to stochastic fluid dynamics, and optimization methods adapted to the geometry of learning problems. We study stability, convergence and computational efficiency, seeking methods that remain informative in singular, high-dimensional or small-noise regimes.
Current directions
- Structure-preserving discretization
- Stability and convergence
- Multiscale and small-noise computation
- Scientific machine learning
- Efficient generative-model algorithms
Related people
Related projects
Selected publications
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A Dynamical Systems Perspective on the Analysis of Neural Networks
In this chapter, we utilize dynamical systems to analyze several aspects of machine learning algorithms. As an expository contribution we demonstrate how to re-formulate a wide variety of challenges from deep neural networks, (stochastic) gradient descent, and related topics into dynamical statements.…
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Central Path Proximal Policy Optimization
Published · Exploration in AI Today Workshop at ICML 2025. In constrained Markov decision processes, enforcing constraints during training is often thought of as decreasing the final return. Recently, it was shown that constraints can be incorporated directly into the policy geometry,…
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Non-Asymptotic Analysis of Projected Gradient Descent for Physics-Informed Neural Networks
Published · Scientific Machine Learning: Emerging Topics, SEMA SIMAI Springer Series (2026). In this work, we provide a non-asymptotic convergence analysis of projected gradient descent for physics-informed neural networks for the Poisson equation. Under suitable assumptions, we show that the optimization error can be…
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Characterizing Dynamical Stability of Stochastic Gradient Descent in Overparameterized Learning
For overparameterized optimization tasks, such as those found in modern machine learning, global minima are generally not unique. In order to understand generalization in these settings, it is vital to study to which minimum an optimization algorithm converges. The possibility of having…
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Stochastic Modified Flows for Riemannian Stochastic Gradient Descent
Published · SIAM journal on control and optimization, 62 (2024) 6, pp. 3288-3314. We give quantitative estimates for the rate of convergence of Riemannian stochastic gradient descent (RSGD) to Riemannian gradient flow and to a diffusion process, the so-called Riemannian stochastic modified flow (RSMF). Using…
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Stochastic Modified Flows, Mean-Field Limits and Dynamics of Stochastic Gradient Descent
Published · Journal of machine learning research, 25 (2024) 30, pp. 1-27. We propose new limiting dynamics for stochastic gradient descent in the small learning rate regime called stochastic modified flows. These SDEs are driven by a cylindrical Brownian motion and improve the so-called…
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Exponential convergence rates for momentum stochastic gradient descent in the overparametrized setting
Published · Mathematical programming, (2026). We prove explicit bounds on the exponential rate of convergence for the momentum stochastic gradient descent scheme (MSGD) for arbitrary, fixed hyperparameters (learning rate, friction parameter) and its continuous-in-time counterpart in the…
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Conservative SPDEs as fluctuating mean field limits of stochastic gradient descent
Published · Probability theory and related fields, 192 (2025) 3/4, pp. 1447-1515. The convergence of stochastic interacting particle systems in the mean-field limit to solutions of conservative stochastic partial differential equations is established, with optimal rate of convergence. As a second main result, a…