Machine Learning
Main fields of application
We investigate the mathematical foundations of machine learning through the dynamics, geometry and thermodynamics of learning algorithms. Randomness is both a source of fluctuations and a mechanism that can improve stability and exploration.
Current directions include stochastic-gradient dynamics, reinforcement learning, neural PDE solvers, diffusion and generative models, and thermodynamically informed learning. We analyse effective equations and large-deviation behaviour, connect optimization to gradient-flow and information-geometric structures, and develop algorithms that exploit these structures. This creates a two-way exchange: stochastic analysis explains learning at large scale, while questions from machine learning motivate new problems in conservative SPDEs, sampling and numerical analysis.
Current directions
- Stochastic-gradient dynamics
- Generative and diffusion models
- Reinforcement learning
- Scientific and thermodynamically informed learning
- Information geometry and optimization
Related people
Related projects
Selected publications
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Effective fluctuating continuum models for stochastic gradient descent
Workshop report · Oberwolfach Reports 23(1), 821–822 (2026). Fluctuating continuum descriptions of stochastic gradient descent.
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Large spikes in SGD: a large-deviations view of catapults
Workshop report · Oberwolfach Reports 23(1), 698–700 (2026). A large-deviations perspective on spikes and catapult behaviour in stochastic gradient descent.
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Projected Inverse Iteration: An Eigenvalue Approach to Ground-State Computation with Neural Quantum States
Deep learning offers a powerful approach to quantum many-body problems via neural network wavefunctions, but their optimization remains a severe bottleneck. Existing optimization methods, including natural gradient descent and stochastic reconfiguration, suffer from spectral gap-dependent convergence that limits their effectiveness on systems…
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Asymptotics of Multi-Scale McKean–Vlasov Diffusions with Super-Linear Kernels: a Lifted Semigroup Approach
In this work, we establish the small-noise asymptotic behaviour (namely, the functional law of large numbers and the large deviation principle) for multi-scale McKean–Vlasov diffusions with super-linear kernels. In this setting, the interaction depends on the laws of both the slow component…
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Curvature-Aware Optimization for High-Accuracy Physics-Informed Neural Networks
Published · Computer Methods in Applied Mechanics and Engineering 462, 119289 (2026). Efficient and robust optimization is essential for neural networks, enabling scientific machine learning models to converge rapidly to very high accuracy — faithfully capturing complex physical behavior governed by differential equations. In…
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Large Spikes in Stochastic Gradient Descent: A Large-Deviations View
Large loss spikes in stochastic gradient descent are studied through a rigorous large-deviations analysis for a shallow, fully connected network in the NTK scaling. In contrast to full-batch gradient descent, the catapult phase is shown to split into inflationary and deflationary regimes,…
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THINNs: Thermodynamically Informed Neural Networks
Physics-Informed Neural Networks (PINNs) are a class of deep learning models aiming to approximate solutions of PDEs by training neural networks to minimize the residual of the equation. Focusing on non-equilibrium fluctuating systems, we propose a physically informed choice of penalization that…
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Functional Neural Wavefunction Optimization
We propose a framework for the design and analysis of optimization algorithms in variational quantum Monte Carlo, drawing on geometric insights into the corresponding function space. The framework translates infinite-dimensional optimization dynamics into tractable parameter-space algorithms through a Galerkin projection onto the…