arXiv subjects

Tobias Blickhan

Publications and source records attributed to Tobias Blickhan.

9 recordsLinked to original sources

arXiv2026

MRX: A differentiable 3D MHD equilibrium solver without nested flux surfaces

This article introduces the 3D magnetohydrodynamic (MHD) equilibrium solver MRX, based on relaxation of a magnetic field to a lower energy equilibrium state via admissible variations. We describe the mathematical theory behind this method and discuss what can be transferred to the fully discrete setting of a relaxation code. Our code is designed to address a number of traditional challenges to 3D MHD equilibrium solvers: enforcing physical constraints such as divergence-free magnetic field, fast convergence, handling strongly shaped polar geometries, and controlling topology changes of the magnetic field. Building on the JAX framework, we address questions of computational efficiency on modern computing architectures, user accessibility, and differentiability at each step. The solver is verified on manufactured vacuum solutions and against the vacuum field of a VMEC equilibrium of a quasi-axisymmetric stellarator. Automatic differentiation with respect to the boundary shape is demonstrated by optimizing stellarator shape for quasi-axisymmetry in vacuum. Relaxation is demonstrated on a finite-$β$ configuration, where we study $h$ refinement and the targeted formation of magnetic island chains at rational surfaces following an energy criterion. An inexact Newton method enables the computation of high-resolution finite-$β$ equilibria with islands and chaos in minutes on a single GPU.

physics.comp-ph↗

arXiv2026

Stochastic Lifting for Generating Trajectories of Stochastic Physical Systems

Many stochastic physical systems evolve smoothly over time in the sense that the distribution of states changes regularly across time steps. The transition from current state to the next state can often be modeled as the combination of a smooth map and an explicit source of randomness. Stochastic Lifting exploits this structure by attaching an independent, high-dimensional random label to each state transition in the training data and fitting a transition map from the current state and label to the next state using a standard regression loss. The labels act as auxiliary coordinates that let the model represent multiple plausible next states from similar current states, avoiding collapse to a mean prediction in the finite-sample size regime. At inference, fresh labels are sampled at each time step and the learned map is rolled forward autoregressively, generating diverse trajectories with a single network evaluation per time step.

cs.LG↗

arXiv2026

Two-Parameter Flows for Learning Population Dynamics of Physical Systems

This work addresses the problem of learning the dynamics of high-dimensional probability densities over time using unlabeled samples, without assuming access to trajectory information. We introduce two-parameter flows that learn only sampling-time transports from a base distribution to each marginal and then extract a physics-time velocity by regressing on coupled synthetic trajectories. We prove that the resulting physics-time dynamics are unique and inherit regularity from the sampling-time transports. Because we can build on standard, well-developed conditional flow matching techniques for learning the base-to-marginal transports, our approach scales to high dimensions and avoids per-step optimal-transport couplings, while allowing admissible non-gradient dynamics that can naturally explain rotational or circulating physics phenomena.

cs.LG↗

arXiv2026

Leveraging Gauge Freedom for Learning Non-Gradient Population Dynamics of Stochastic Systems

Existing work on population dynamics inference often focuses on flows arising from vector fields that are the gradients of scalar potentials. Among all admissible flows that are compatible with the population dynamics, gradient flows are optimal in a specific sense: they minimize kinetic energy. The selection of fields based on different criteria corresponds to a gauge freedom when determining population dynamics, which we leverage in this work. We propose Non-Gradient Inference Flows (NGIF), an algorithm to infer non-gradient population dynamics using a weak formulation of the continuity equation. This allows us to parameterize general vector fields and choose other selection criteria beyond minimal kinetic energy. We demonstrate on a variety of low- and high-dimensional physics problems that this more general approach improves distributional accuracy over gradient-restricted baselines and better captures non-potential transport.

cs.LG↗

arXiv2026

Computational boundary specification in 3D fixed-boundary magnetohydrodynamic equilibrium modeling

Outside the core of the plasma, the plasma current and pressure rapidly transition to zero in a scrape-off or edge region or plasma-vacuum interface. However, existing tools for fixed-boundary magnetohydrodynamic equilibria in 2D and 3D domains $Ω$ typically prescribe the computational boundary $\partialΩ$ interior to this transition layer. We (1) argue that a more realistic and robust assumption is to define the computational boundary exterior to this transition layer, in a vacuum-like region where $J|_{\partialΩ} \sim p|_{\partialΩ} \sim 0$, (2) show that, without this boundary change, existing coil optimization routines for 3D toroidal equilibria (stellarators) should be changed to match free-boundary equilibrium requirements, and (3) derive an algorithm for a fixed-boundary 3D equilibrium solver compatible with a very general computational boundary, with conditions $B \cdot n|_{\partialΩ} \neq 0$ (not necessarily a flux surface), $p|_{\partialΩ} \neq \text{const.}$ (not necessarily an isobar), and $J \times n|_{\partialΩ} \neq 0$.

physics.plasm-ph↗

arXiv2025

DICE: Discrete inverse continuity equation for learning population dynamics

We introduce the Discrete Inverse Continuity Equation (DICE) method, a generative modeling approach that learns the evolution of a stochastic process from given sample populations at a finite number of time points. Models learned with DICE capture the typically smooth and well-behaved population dynamics, rather than the dynamics of individual sample trajectories that can exhibit complex or even chaotic behavior. The DICE loss function is developed specifically to be invariant, even in discrete time, to spatially constant but time-varying spurious constants that can emerge during training; this invariance increases training stability and robustness. Generating a trajectory of sample populations with DICE is fast because samples evolve directly in the time interval over which the stochastic process is formulated, in contrast to approaches that condition on time and then require multiple sampling steps per time step. DICE is stable to train, in situations where other methods for learning population dynamics fail, and DICE generates representative samples with orders of magnitude lower costs than methods that have to condition on time. Numerical experiments on a wide range of problems from random waves, Vlasov-Poisson instabilities and high-dimensional chaos are included to justify these assertions.

cs.LG↗

arXiv2024

Parametric model reduction of mean-field and stochastic systems via higher-order action matching

The aim of this work is to learn models of population dynamics of physical systems that feature stochastic and mean-field effects and that depend on physics parameters. The learned models can act as surrogates of classical numerical models to efficiently predict the system behavior over the physics parameters. Building on the Benamou-Brenier formula from optimal transport and action matching, we use a variational problem to infer parameter- and time-dependent gradient fields that represent approximations of the population dynamics. The inferred gradient fields can then be used to rapidly generate sample trajectories that mimic the dynamics of the physical system on a population level over varying physics parameters. We show that combining Monte Carlo sampling with higher-order quadrature rules is critical for accurately estimating the training objective from sample data and for stabilizing the training process. We demonstrate on Vlasov-Poisson instabilities as well as on high-dimensional particle and chaotic systems that our approach accurately predicts population dynamics over a wide range of parameters and outperforms state-of-the-art diffusion-based and flow-based modeling that simply condition on time and physics parameters.

stat.ML↗

arXiv2023

A registration method for reduced basis problems using linear optimal transport

We present a registration method for model reduction of parametric partial differential equations with dominating advection effects and moving features. Registration refers to the use of a parameter-dependent mapping to make the set of solutions to these equations more amicable for approximation using classical reduced basis methods. The proposed approach utilizes concepts from optimal transport theory, as we utilize Monge embeddings to construct these mappings in a purely data-driven way. The method relies on one interpretable hyper-parameter. We discuss how our approach relates to existing works that combine model order reduction and optimal transport theory. Numerical results are provided to demonstrate the effect of the registration. This includes a model problem where the solution is itself a probability density and one where it is not.

math.NA↗

arXiv2022

Waserstein model reduction approach for parametrized flow problems in porous media

The aim of this work is to build a reduced-order model for parametrized porous media equations. The main challenge of this type of problems is that the Kolmogorov width of the solution manifold typically decays quite slowly and thus makes usual linear model-order reduction methods inappropriate. In this work, we investigate an adaptation of the methodology proposed in a previous work, based on the use of Wasserstein barycenters, to the case of non-conservative problems. Numerical examples in one-dimensional test cases illustrate the advantages and limitations of this approach and suggest further research directions that we intend to explore in the future.

math.NA↗