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

  • The Advective Fisher-Rao Geometry of Deterministic Measure Transport

    A novel advective Fisher-Rao metric is introduced for optimization tasks on paths of probability measures governed by the continuity equation. This metric is shown to lead to optimal descent directions. It is then shown that this metric arises naturally from three different…

  • Introduce thermal noise before discretizing, not afterwards!

    Workshop report · Oberwolfach Reports 23(1), 803–806 (2026). Incorporating thermal noise before discretising a continuum model.

  • 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…

  • 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…

  • 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…

  • 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,…

  • 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…

  • 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…