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Steffen Schmidt

Publications and source records attributed to Steffen Schmidt.

13 recordsLinked to original sources
arXiv2026

Neptuna: A Comprehensive Machine Learning Framework for Benchmarking Complex Multiphase Flows

Compressible multiphase flows involving shocks and material interfaces arise in applications such as bubble collapse and droplet breakup, where strong nonlinear interactions produce complex interface deformation, mixing, and multiscale dynamics. Developing reliable machine learning surrogates for these flows remains challenging due to the simultaneous presence of compressibility, sharp discontinuities, and multiphase effects. In this work, we introduce the first large-scale benchmark specifically designed for shock-driven compressible multiphase flows, comprising 2.4 TB of high-fidelity 2D and 3D datasets featuring shock-induced bubble collapse and droplet breakup. We evaluate diverse surrogate model families on our benchmarking framework: Neptuna {https://github.com/tumaer/Neptuna}, including convolutional, spectral, transformer-based, and pre-trained PDE foundation models. Beyond standard MSE training, we investigate composite losses combining MSE with Sobolev, interface-aware, and structure-aware terms, together with adaptive loss balancing using SoftAdapt and GradNorm. Evaluation includes pointwise, spectral, feature-focused, structural, and physics-informed metrics. Results show that no single model performs best across all datasets and metrics, while composite losses significantly improve interface preservation and spectral fidelity. Among adaptive weighting strategies, SoftAdapt provides the most consistent improvements with almost no overhead compared to MSE-only training.

physics.flu-dyn↗
arXiv2026

Dirac operators for infinite-dimensional color Lie algebras

We construct cubic Dirac operators and relative cubic Dirac operators for infinite-dimensional quadratic $\mathbb{Z}$-graded color Lie algebras with finite-dimensional components. These operators are defined in completions of the quantum Weil algebra determined by the $\mathbb{Z}$-grading. The same grading fixes the normal-ordering convention. The failure of the normally ordered Casimir to be central, and of the normally ordered cubic Dirac operator to be $\mathfrak{g}$-invariant, is measured by a color analogue of the Kac-Peterson class. If this class is trivial, the Casimir admits a central correction and the cubic Dirac operator admits a corrected $\mathfrak{g}$-invariant form. For the corrected (relative) cubic Dirac operators, we establish Parthasarathy-type square formulas. We also extend the Chern-Weil homomorphism to completed $\mathfrak{g}$-differential algebras and identify the classical element whose quantization is the cubic Dirac operator with the Chern-Simons element associated with the quadratic invariant polynomial defined by $B$. As applications, we consider symmetrizable Kac-Moody superalgebras. In this setting the Kac-Peterson class is trivial, with primitive given by the Weyl vector. For the affine Kac-Moody superalgebra associated to $\mathfrak{osp}(1\vert 2n)$, we compute $\mathrm{ker}\operatorname{D}_{\mathfrak{g},\mathfrak{g}_{\bar{0}}}^{2}$ on integrable highest weight supermodules. We then apply the relative square formula to $\omega$-unitarizable highest weight supermodules and obtain a Dirac inequality giving necessary conditions for unitarity. Finally, under assumptions satisfied by Kac-Moody superalgebras such as $\widehat{\mathfrak{sl}}(m\vert n)$, we identify the Dirac kernel with Lie superalgebra cohomology.

math.RT↗
arXiv2026

Hybrid Fourier Neural Operator-Lattice Boltzmann Method

We propose an accelerated computational fluid dynamics framework based on a hybrid Fourier Neural Operator-Lattice Boltzmann Method (FNO-LBM) for steady and unsteady weakly compressible flows. FNO-based initialization significantly accelerates LBM in reaching steady-states of porous media flows across all macroscopic fields, achieving up to 70% speed-up in convergence of density and more than 40% of pressure-drop while preserving the final steady-state accuracy. Simulations of unsteady flows can be accelerated by hybrid coupling strategies that employ FNO rollouts embedded into LBM time advancement in a way of super-time-stepping. Global and time-resolved error metrics across 100 trajectories for generic 2D flows demonstrate that hybridization consistently improves accuracy and stabilizes long-horizon rollouts. Best efficiency is achieved for a lightweight 2.6M-parameter FNO, which diverges under pure autoregressive rollout but achieves 96-99.8% error reduction under hybrid coupling, matching the predictive capability of a much more expensive 11.2M-parameter model. The hybrid framework enhances predictive fidelity, suppresses error accumulation, and enables small and cheap surrogate models to operate effectively within the same error regime as larger surrogates. These results demonstrate that hybrid neural-operator coupling achieves robust and computationally efficient accelerated LBM while maintaining physically consistent flow evolution.

physics.flu-dyn↗
arXiv2026

Perturbations of Dirac Operators

We study perturbations of relative cubic Dirac operators for basic classical Lie superalgebras within the uniform formalism of the colour quantum Weil algebra. This perspective leads to three complementary classes of perturbations and resulting invariants. First, we define semisimple perturbations that assign to each finite-dimensional simple supermodule a finite collection of semisimple orbits, together with canonically defined vector spaces measuring the degree of atypicality. Second, we introduce nilpotent perturbations parametrized by the self-commuting variety of a quadratic Lie subsuperalgebra; the resulting family of cohomology theories combines Dirac cohomology and Duflo--Serganova cohomology. Third, we deform the cubic Dirac operator by a Weil-covariant differential built from the universal $1$-form in the colour quantum Weil algebra and the Weil differential, producing a Chern-type invariant that assigns to each finite-dimensional module a natural class in the cohomology of the Weil complex.

math.RT↗
arXiv2025

An index for unitarizable $\mathfrak{sl}(m\vert n)$-supermodules

The "superconformal index" is a character-valued invariant attached by theoretical physics to unitary representations of Lie superalgebras, such as $\mathfrak{su}(2,2\vert n)$, that govern certain quantum field theories. The index can be calculated as a supertrace over Hilbert space, and is constant in families induced by variation of physical parameters. This is because the index receives contributions only from "short" irreducible representations such that it is invariant under recombination at the boundary of the region of unitarity. The purpose of this paper is to develop these notions for unitarizable supermodules over the special linear Lie superalgebras $\mathfrak{sl}(m\vert n)$ with $m\ge 2$, $n\ge 1$. To keep it self-contained, we include a fair amount of background material on structure theory, unitarizable supermodules, the Duflo-Serganova functor, and elements of Harish-Chandra theory. Along the way, we provide a precise dictionary between various notions from theoretical physics and mathematical terminology. Our final result is a kind of "index theorem" that relates the counting of atypical constituents in a general unitarizable $\mathfrak{sl}(m\vert n)$-supermodule to the character-valued $Q$-Witten index, expressed as a supertrace over the full supermodule. The formal superdimension of holomorphic discrete series $\mathfrak{sl}(m\vert n)$-supermodules can also be formulated in this framework.

math.RT↗
arXiv2025

Cubic Dirac Operators and Dirac Cohomology for Basic Classical Lie Superalgebras

We study the Dirac cohomology of supermodules over basic classical Lie superalgebras, formulated in terms of cubic Dirac operators associated with parabolic subalgebras. Specifically, we establish a super-analog of the Casselman-Osborne theorem for supermodules with an infinitesimal character and use it to show that the Dirac cohomology of highest-weight supermodules is always non-trivial. In particular, we explicitly compute the Dirac cohomology of finite-dimensional simple supermodules for basic Lie superalgebras of type 1 with a typical highest weight, as well as of simple supermodules in the parabolic BGG category. We further investigate the relationship between Dirac cohomology and Kostant (co)homology, proving that, under suitable conditions, Dirac cohomology embeds into Kostant (co)homology. Moreover, we show that this embedding lifts to an isomorphism when the supermodule is unitarizable.

math.RT↗
arXiv2025

Dynamic Circuits for the Quantum Lattice-Boltzmann Method

We propose a quantum algorithm for the linear advection-diffusion equation (ADE) Lattice-Boltzmann method (LBM) that leverages dynamic circuits. Dynamic quantum circuits allow for an optimized collision-operator quantum algorithm, introducing partial measurements as an integral step. Efficient adaptation of the quantum circuit during execution based on digital information obtained through mid-circuit measurements is achieved. The proposed new collision algorithm is implemented as a fully unitary operator, which facilitates the computation of multiple time steps without state reinitialization. Unlike previous quantum collision operators that rely on linear combinations of unitaries, the proposed algorithm does not exhibit a probabilistic failure rate. Moreover, additional qubits no longer depend on the chosen velocity set, which reduces both qubit overhead and circuit complexity. Validation of the quantum collision algorithm is performed by comparing results with digital LBM in one and two dimensions, demonstrating excellent agreement. Performance analysis for multiple time steps highlights advantages compared to previous methods. As an additional variant, a hybrid quantum-digital approach is proposed, which reduces the number of mid-circuit measurements, therefore improving the efficiency of the quantum collision algorithm.

quant-ph↗
arXiv2024

Dirac Operators, Dirac Cohomology and Unitarity for $A(m\vert n)$

Dirac operators and Dirac cohomology for Lie superalgebras of Riemannian type, introduced by Huang and Pand\v{z}i\'{c}, provide an effective tool for the study of unitarizable supermodules. In this article, we study these objects for Lie superalgebras of type $A$ and relate them systematically to unitarity. In the first part, we establish the basic structure of the theory in this setting. We relate unitarity to the Dirac operator, derive the corresponding Dirac inequality, and show that Dirac cohomology determines unitarizable supermodules. We also determine explicitly the Dirac cohomology of unitarizable simple supermodules. In the second part, we turn to applications. We obtain a new characterization of unitarity, establish a relation with Kostant's cohomology, derive a formula for formal characters, and introduce a Dirac index.

math.RT↗
arXiv2024

Unitary Quantum Algorithm for the Lattice-Boltzmann Method

We present a quantum algorithm for computational fluid dynamics based on the Lattice-Boltzmann method. Our approach involves a novel encoding strategy and a modified collision operator, assuming full relaxation to the local equilibrium within a single time step. Our quantum algorithm enables the computation of multiple time steps in the linearized case, specifically for solving the advection-diffusion equation, before necessitating a full state measurement. Moreover, our formulation can be extended to compute the non-linear equilibrium distribution function for a single time step prior to measurement, utilizing the measurement as an essential algorithmic step. However, in the non-linear case, a classical postprocessing step is necessary for computing the moments of the distribution function. We validate our algorithm by solving the one dimensional advection-diffusion of a Gaussian hill. Our results demonstrate that our quantum algorithm captures non-linearity.

quant-ph↗
arXiv2016

Optical $π$ Phase Shift Created with a Single-Photon Pulse

A deterministic photon-photon quantum-logic gate is a long-standing goal. Building such a gate becomes possible if a light pulse containing only one photon imprints a phase shift of $π$ onto another light field. Here we experimentally demonstrate the generation of such a $π$ phase shift with a single-photon pulse. A first light pulse containing less than one photon on average is stored in an atomic gas. Rydberg blockade combined with electromagnetically induced transparency creates a phase shift for a second light pulse which propagates through the medium. Postselected on the detection of a retrieved photon from the first pulse, we measure a $π$ phase shift of the second pulse. This demonstrates a crucial step toward a photon-photon gate and offers a variety of applications in the field of quantum information processing.

quant-ph↗
arXiv2014

Neutrino Masses and Conformal Electro-Weak Symmetry Breaking

Dimensional transmutation in classically conformal invariant theories may explain the electro-weak scale and the fact that so far nothing but the Standard Model (SM) particles have been observed. We discuss in this paper implications of this type of symmetry breaking for neutrino mass generation.

hep-ph↗
arXiv2012

Generation of families of spectra in PT-symmetric quantum mechanics and scalar bosonic field theory

This paper explains the systematics of the generation of families of spectra for the PT-symmetric quantum-mechanical Hamiltonians $H=p^2+x^2(ix)^ε$, $H=p^2+(x^2)^δ$, and $H=p^2-(x^2)^μ$. In addition, it contrasts the results obtained with those found for a bosonic scalar field theory, in particular in one dimension, highlighting the similarities and differences to the quantum-mechanical case. It is shown that the number of families of spectra can be deduced from the number of noncontiguous pairs of Stokes' wedges that display PT-symmetry. To do so, simple arguments that use the WKB approximation are employed, and these imply that the eigenvalues are real. However, definitive results are in most cases presently only obtainable numerically, and not all eigenvalues in each family may be real. Within the approximations used, it is illustrated that the difference between the quantum-mechanical and the field-theoretical cases lies in the number of accessible regions in which the eigenfunctions decay exponentially. This paper reviews and implements well-known techniques in complex analysis and PT-symmetric quantum theory.

hep-th↗