Data-driven methods for partial differential equations

Session

Poster blitz, poster session, and welcome reception

Mar 2, 2026, 6:15 PM

Karlsruhe Institute of Technology

2. Neural semi-Lagrangian method for high-dimensional advection-diffusion problems

Laurent Navoret (University of Strasbourg, Inria)

3/2/26, 6:15 PM

Contributed poster

We are interested in numerically solving high-dimensional advection-diffusion equations, such as kinetic equations or parametric problems. Traditional numerical methods suffer from the curse of dimensionality, as the number of degrees of freedom grows exponentially with dimension. Recently, methods based on neural networks have proven effective in reducing the number of degrees of freedom by...

6. Prediction error certification for PINNs: Theory, computation and application

Birgit Hillebrecht (KIT)

3/2/26, 6:18 PM

Contributed poster

We present two residual‑based a posteriori error estimators for physics‑informed neural networks (PINNs) that are applicable to the approximation of solutions of partial differential equations (PDEs) on complex geometries. Building on the semigroup‑based framework introduced previously, we incorporate the concept of input‑to‑state stability (ISS), or suitable modifications thereof, to quantify...

5. Posterior contraction under misspecification and heteroskedasticity in non-linear inverse problems

Maximilian Siebel (Heidelberg University)

3/2/26, 6:21 PM

Contributed poster

In many practical and numerical inverse problems, the exact data log-likelihood is not fully accessible, motivating the use of surrogate likelihoods. We study heteroscedastic statistical nonparametric nonlinear inverse problems and establish posterior contraction results when inference is based on a surrogate log-likelihood constructed from proxy error variances and an approximate forward map....

7. Geometric Optimization in Scientific Machine Learning

Johannes Müller

3/2/26, 6:24 PM

Contributed poster

We discusses an “optimize-then-project” approach for applications in scientific machine learning. The key idea is to design algorithms at the infinite-dimensional level and subsequently discretize them in the tangent space of the neural network ansatz, similar to a natural gradient style ansatz. We illustrate this approach in the context of the variational Monte Carlo method for quantum...

10. Hyperbolic Traffic Models with Uncertain Accident Dynamics

Thomas Schillinger (University of Mannheim)

3/2/26, 6:27 PM

Contributed poster

We consider hyperbolic partial differential equations (PDEs) with a space-dependent flux function to describe traffic flow dynamics. The PDE is coupled with a stochastic process modeling traffic accidents, thereby capturing the interplay between traffic dynamics and accident occurrence. This framework enables the analysis of accident risk and its consequences in road networks.
A key aspect of...

8. Exploratory computation of statistical Navier–Stokes solutions

Stephan Simonis (ETH Zurich)

3/2/26, 6:30 PM

Contributed poster

Global well-posedness for three-dimensional fluid flow equations remains a profound open problem. Recent efforts have shifted toward statistical solutions as a robust framework for describing turbulence, yet efficient computational tools to explore these solutions in three dimensions are scarce.
We develop novel stochastic numerical schemes to compute and analyze statistical solutions for...