Publications
The catalogue keeps one record per scientific work. A preprint record is updated with journal information rather than duplicated when a journal version appears.
Display scientific works by current members that first became publicly available while the author was a member of SAiS.
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- Benjamin Gess
- Max Sauerbrey
- Adrian Martini
- Paul Nikolaev
- Johannes Müller
- Sarah Geiss
- Dennis Chemnitz
- Javier Castro
- Shanshan Hu
- Thomas Müller
- Tom Hapke
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Mathematical challenges
Publication catalogue
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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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The Incompressible Navier–Stokes–Fourier System with Thermal Noise
We establish a solution theory for the incompressible Navier–Stokes–Fourier system with thermal noise, posed on the three-dimensional torus. While in the incompressible deterministic setting the equation for the velocity can be solved independently of the temperature, the inclusion of the effects of…
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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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The Porous Medium Equation: Multiscale Integrability in Large Deviations
We consider a zero-range process $η^N_t(x)$ with superlinear local jump rate, which in a hydrodynamic-small particle rescaling converges to the porous medium equation $\partial_t u=\frac12Δu^α, α>1$. As a main result we obtain a large deviation principle in any scaling regime of vanishing…
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Probabilistically Strong Solutions to Stochastic Euler Equations
In this paper, we establish the existence of probabilistically strong, measure-valued solutions for the stochastic incompressible Navier–Stokes equations and prove their convergence, in the vanishing viscosity limit, to probabilistically strong solutions for the stochastic incompressible Euler equations. In particular, this solves the…
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Ergodicity for SPDEs driven by divergence-free transport noise
We study the ergodic behaviour of the McKean-Vlasov equations driven by common, divergence-free transport noise. In particular, we show that in dimension $d\geq 2$, if the noise is mixing and sufficiently strong it can enforce the uniqueness of invariant probability measures, even…
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Mixing at the Batchelor Scale for White-In-Time Flows
We consider the mixing properties of solutions to the advection-diffusion equation of a white-in-time velocity field on the 2-dimensional torus with four forced modes. As the diffusivity parameter goes to zero, we show that the almost-sure exponential dissipation rate stays bounded from…
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A stochastic flow approach to De Giorgi-Nash-Moser estimates for SPDEs with smooth transport noise
The celebrated De Giorgi-Nash-Moser theory ensures that solutions to uniformly elliptic or parabolic PDEs are bounded and Hölder continuous, even with merely bounded measurable coefficients. For parabolic SPDEs with transport noise, boundedness has recently been established, but Hölder continuity remains a key…
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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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Convergence rate for Fluctuations of mean field interacting diffusion and application to 2D viscous Vortex model and Coulomb potential
For a system of mean field interacting diffusion on $\mathbb{T}^d$, the empirical measure $μ^N$ converges to the solution $μ$ of the Fokker-Planck equation. Refining this mean field limit as a Central Limit Theorem, the fluctuation process $ρ^N_t= \sqrt{N}( μ^N_t -μ_t)$ convergences to…