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Norbert Hoffmann

Publications and source records attributed to Norbert Hoffmann.

At least 19 recordsLinked to original sources
arXiv2025

Bridging ocean wave physics and deep learning: Physics-informed neural operators for nonlinear wavefield reconstruction in real-time

Accurate real-time prediction of phase-resolved ocean wave fields remains a critical yet largely unsolved problem, primarily due to the absence of practical data assimilation methods for reconstructing initial conditions from sparse or indirect wave measurements. While recent advances in supervised deep learning have shown potential for this purpose, they require large labelled datasets of ground truth wave data, which are infeasible to obtain in real-world scenarios. To overcome this limitation, we propose a Physics-Informed Neural Operator (PINO) framework for reconstructing spatially and temporally phase-resolved, nonlinear ocean wave fields from sparse measurements, without the need for ground truth data during training. This is achieved by embedding residuals of the free surface boundary conditions of ocean gravity waves into the loss function of the PINO, constraining the solution space in a soft manner. After training, we validate our approach using highly realistic synthetic wave data and demonstrate the accurate reconstruction of nonlinear wave fields from both buoy time series and radar snapshots. Our results indicate that PINOs enable accurate, real-time reconstruction and generalize robustly across a wide range of wave conditions, thereby paving the way for operational, data-driven wave reconstruction and prediction in realistic marine environments.

cs.LG↗
arXiv2025

Physics-informed neural networks for phase-resolved data assimilation and prediction of nonlinear ocean waves

The assimilation and prediction of phase-resolved surface gravity waves are critical challenges in ocean science and engineering. Potential flow theory (PFT) has been widely employed to develop wave models and numerical techniques for wave prediction. However, traditional wave prediction methods are often limited. For example, most simplified wave models have a limited ability to capture strong wave nonlinearity, while fully nonlinear PFT solvers often fail to meet the speed requirements of engineering applications. This computational inefficiency also hinders the development of effective data assimilation techniques, which are required to reconstruct spatial wave information from sparse measurements to initialize the wave prediction. To address these challenges, we propose a novel solver method that leverages physics-informed neural networks (PINNs) that parameterize PFT solutions as neural networks. This provides a computationally inexpensive way to assimilate and predict wave data. The proposed PINN framework is validated through comparisons with analytical linear PFT solutions and experimental data collected in a laboratory wave flume. The results demonstrate that our approach accurately captures and predicts irregular, nonlinear, and dispersive wave surface dynamics. Moreover, the PINN can infer the fully nonlinear velocity potential throughout the entire fluid volume solely from surface elevation measurements, enabling the calculation of fluid velocities that are difficult to measure experimentally.

cs.LG↗
arXiv2024

Exploring localization in nonlinear oscillator systems through network-based predictions

Localized vibrations, arising from nonlinearities or symmetry breaking, pose a challenge in engineering, as the resulting high-amplitude vibrations may result in component failure due to fatigue. During operation, the emergence of localization is difficult to predict, partly because of changing parameters over the life cycle of a system. This work proposes a novel, network-based approach to detect an imminent localized vibration. Synthetic measurement data is used to generate a functional network, which captures the dynamic interplay of the machine parts, complementary to their geometric coupling. Analysis of these functional networks reveals an impending localized vibration and its location. The method is demonstrated using a model system for a bladed disk, a ring composed of coupled nonlinear Duffing oscillators. Results indicate that the proposed method is robust against small parameter uncertainties, added measurement noise, and the length of the measurement data samples. The source code for this work is available.

math.DS↗
arXiv2024

Data assimilation and parameter identification for water waves using the nonlinear Schr\"{o}dinger equation and physics-informed neural networks

The measurement of deep water gravity wave elevations using in-situ devices, such as wave gauges, typically yields spatially sparse data. This sparsity arises from the deployment of a limited number of gauges due to their installation effort and high operational costs. The reconstruction of the spatio-temporal extent of surface elevation poses an ill-posed data assimilation problem, challenging to solve with conventional numerical techniques. To address this issue, we propose the application of a physics-informed neural network (PINN), aiming to reconstruct physically consistent wave fields between two designated measurement locations several meters apart. Our method ensures this physical consistency by integrating residuals of the hydrodynamic nonlinear Schr\"{o}dinger equation (NLSE) into the PINN's loss function. Using synthetic wave elevation time series from distinct locations within a wave tank, we initially achieve successful reconstruction quality by employing constant, predetermined NLSE coefficients. However, the reconstruction quality is further improved by introducing NLSE coefficients as additional identifiable variables during PINN training. The results not only showcase a technically relevant application of the PINN method but also represent a pioneering step towards improving the initialization of deterministic wave prediction methods.

physics.flu-dyn↗
arXiv2023

Machine learning for phase-resolved reconstruction of nonlinear ocean wave surface elevations from sparse remote sensing data

Accurate short-term predictions of phase-resolved water wave conditions are crucial for decision-making in ocean engineering. However, the initialization of remote-sensing-based wave prediction models first requires a reconstruction of wave surfaces from sparse measurements like radar. Existing reconstruction methods either rely on computationally intensive optimization procedures or simplistic modelling assumptions that compromise the real-time capability or accuracy of the subsequent prediction process. We therefore address these issues by proposing a novel approach for phase-resolved wave surface reconstruction using neural networks based on the U-Net and Fourier neural operator (FNO) architectures. Our approach utilizes synthetic yet highly realistic training data on uniform one-dimensional grids, that is generated by the high-order spectral method for wave simulation and a geometric radar modelling approach. The investigation reveals that both models deliver accurate wave reconstruction results and show good generalization for different sea states when trained with spatio-temporal radar data containing multiple historic radar snapshots in each input. Notably, the FNO demonstrates superior performance in handling the data structure imposed by wave physics due to its global approach to learn the mapping between input and output in Fourier space.

physics.ao-ph↗
arXiv2022

Extreme wave excitation from localized phase-shift perturbations

The modulation instability is a focusing mechanism responsible for the formation of strong wave localizations not only on the water surface, but also in a variety of nonlinear dispersive media. Such dynamics is initiated from the injection of side-bands, which translate into an amplitude modulation of the wave field. The nonlinear stage of unstable wave evolution can be described by exact solutions of the nonlinear Schr\"odinger equation (NLSE). In that case, the amplitude modulation of such coherent extreme wave structures is connected to a particular phase-shift seed in the carrier wave. In this letter, we show that phase-shifts localization applied to the background, excluding any amplitude modulation excitation, can indeed trigger extreme events. Such rogue waves can be for instance generated by considering the parametrization of fundamental breathers and thus, by seeding only the local phase-shift information to the regular carrier wave. Our wave tank experiments show an excellent agreement with the expected NLSE hydrodynamics and confirm that even though delayed in their evolution, breather-type extreme waves can be generated from a purely regular wave train. Such novel focusing mechanism awaits experimental confirmation in other nonlinear media, such optics, plasma, and Bose-Einstein condensates.

physics.flu-dyn↗
arXiv2022

Surface Similarity Parameter: A New Machine Learning Loss Metric for Oscillatory Spatio-Temporal Data

Supervised machine learning approaches require the formulation of a loss functional to be minimized in the training phase. Sequential data are ubiquitous across many fields of research, and are often treated with Euclidean distance-based loss functions that were designed for tabular data. For smooth oscillatory data, those conventional approaches lack the ability to penalize amplitude, frequency and phase prediction errors at the same time, and tend to be biased towards amplitude errors. We introduce the surface similarity parameter (SSP) as a novel loss function that is especially useful for training machine learning models on smooth oscillatory sequences. Our extensive experiments on chaotic spatio-temporal dynamics systems indicate that the SSP is beneficial for shaping gradients, thereby accelerating the training process, reducing the final prediction error, increasing weight initialization robustness, and implementing a stronger regularization effect compared to using classical loss functions. The results indicate the potential of the novel loss metric particularly for highly complex and chaotic data, such as data stemming from the nonlinear two-dimensional Kuramoto-Sivashinsky equation and the linear propagation of dispersive surface gravity waves in fluids.

cs.LG↗
arXiv2022

Galilean-transformed solitons and supercontinuum generation in dispersive media

The Galilean transformation is a universal operation connecting the coordinates of a dynamical system, which move relative to each other with a constant speed. In the context of exact solutions of the universal nonlinear Schr\"odinger equation (NLSE), inducing a Galilean velocity (GV) to the pulse involves a frequency shift to satisfy the symmetry of the wave equation. As such, the Galilean transformation has been deemed to be not applicable to wave groups in nonlinear dispersive media. In this paper, we demonstrate that in a wave tank generated Galilean transformed envelope and Peregrine solitons show clear variations from their respective pure dynamics on the water surface. The type of deviations depends on the sign of the GV and can be captured by the modified NLSE or the Euler equations. Moreover, we show that positive Galilean-translated envelope soliton pulses exhibit self-modulation. While designated GS and wave steepness values expedite multi-soliton dynamics, the strong focusing of such higher-order coherent waves inevitably lead to the generation of supercontinua as a result of soliton fission. We anticipate that kindred experimental and numerical studies might be implemented in other dispersive wave guides governed by nonlinearity.

nlin.PS↗
arXiv2021

G-torsors and universal torsors over nonsplit del Pezzo surfaces

Let S be a smooth del Pezzo surface that is defined over a field K and splits over a Galois extension L. Let G be either the split reductive group given by the root system of $S_L$ in Pic $S_L$, or a form of it containing the N\'eron-Severi torus. Let $\mathcal{G}$ be the G-torsor over $S_L$ obtained by extension of structure group from a universal torsor $\mathcal{T}$ over $S_L$. We prove that $\mathcal{G}$ does not descend to S unless $\mathcal{T}$ does. This is in contrast to a result of Friedman and Morgan that such $\mathcal{G}$ always descend to singular del Pezzo surfaces over $\mathbb{C}$ from their desingularizations.

math.AG↗
arXiv2020

The Peregrine breather on the zero-background limit as the two-soliton degenerate solution: An experimental study

Solitons are coherent structures that describe the nonlinear evolution of wave localizations in hydrodynamics, optics, plasma and Bose-Einstein condensates. While the Peregrine breather is known to amplify a single localized perturbation of a carrier wave of finite amplitude by a factor of three, there is a counterpart solution on zero background known as the degenerate two-soliton which also leads to high amplitude maxima. In this study, we report several observations of such multi-soliton with doubly-localized peaks in a water wave flume. The data collected in this experiment confirm the distinctive attainment of wave amplification by a factor of two in good agreement with the dynamics of the nonlinear Schr\"odinger equation solution. Advanced numerical simulations solving the problem of nonlinear free water surface boundary conditions of an ideal fluid quantify the physical limitations of the degenerate two-soliton in hydrodynamics.

physics.flu-dyn↗
arXiv2020

Nonlinear vibration localisation in a symmetric system of two coupled beams

We report nonlinear vibration localisation in a system of two symmetric weakly coupled nonlinear oscillators. A two degree-of-freedom model with piecewise linear stiffness shows bifurcations to localised solutions. An experimental investigation employing two weakly coupled beams touching against stoppers for large vibration amplitudes confirms the nonlinear localisation.

nlin.PS↗
arXiv2019

A birational Torelli theorem with a Brauer class

Let $\text{M}_C( 2, \mathcal{O}_C) \cong \mathbb{P}^3$ denote the coarse moduli space of semistable vector bundles of rank $2$ with trivial determinant over a smooth projective curve $C$ of genus $2$ over $\mathbb{C}$. Let $\beta_C$ denote the natural Brauer class over the stable locus. We prove that if $f^*( \beta_{C'}) = \beta_C$ for some birational map $f$ from $\text{M}_C( 2, \mathcal{O}_C)$ to $\text{M}_{C'}( 2, \mathcal{O}_{C'})$, then the Jacobians of $C$ and of $C'$ are isomorphic as abelian varieties. If moreover these Jacobians do not admit real multiplication, then the curves $C$ and $C'$ are isomorphic. Similar statements hold for Kummer surfaces in $\mathbb{P}^3$ and for quadratic line complexes.

math.AG↗
arXiv2019

Energy harvesting below the onset of flutter

This work demonstrates preliminary results on energy harvesting from a linearly stable flutter-type system with circulatory friction forces. Harmonic external forcing is applied to study the energy flow in the steady sliding configuration. In certain parameter ranges negative excitation work is observed where the external forcing allows to pull part of the friction energy out of the system and thus makes energy harvesting possible. Studies reveal that this behavior is largely independent of the flutter point and thus that it is primarily controlled by the excitation. Contrary to existing energy harvesting approaches for such systems, this approach uses external forcing in the linearly stable regime of the oscillator which allows to control vibrations and harvest energy on demand.

physics.app-ph↗
arXiv2018

Directional Soliton and Breather Beams

Solitons and breathers are nonlinear modes that exist in a wide range of physical systems. They are fundamental solutions of a number of nonlinear wave evolution equations, including the uni-directional nonlinear Schr\"odinger equation (NLSE). We report the observation of slanted solitons and breathers propagating at an angle with respect to the direction of propagation of the wave field. As the coherence is diagonal, the scale in the crest direction becomes finite, consequently, a beam dynamics forms. Spatio-temporal measurements of the water surface elevation are obtained by stereo-reconstructing the positions of the floating markers placed on a regular lattice and recorded with two synchronized high-speed cameras. Experimental results, based on the predictions obtained from the (2D+1) hyperbolic NLSE equation, are in excellent agreement with the theory. Our study proves the existence of such unique and coherent wave packets and has serious implications for practical applications in optical sciences and physical oceanography. Moreover, unstable wave fields in this geometry may explain the formation of directional large amplitude rogue waves with a finite crest length within a wide range of nonlinear dispersive media, such as Bose-Einstein condensates, plasma, hydrodynamics and optics.

nlin.PS↗
arXiv2018

Dissipative solitons in forced cyclic and symmetric structures

The emergence of localised vibrations in cyclic and symmetric rotating structures, such as bladed disks of aircraft engines, has challenged engineers in the past few decades. In the linear regime, localised states may arise due to a lack of symmetry, as for example induced by inhomogeneities. However, when structures deviate from the linear behaviour, e.g. due to material nonlinearities, geometric nonlinearities like large deformations, or other nonlinear elements like joints or friction interfaces, localised states may arise even in perfectly symmetric structures. In this paper, a system consisting of coupled Duffing oscillators with linear viscous damping is subjected to external travelling wave forcing. The system may be considered a minimal model for bladed disks in turbomachinery operating in the nonlinear regime, where such excitation may arise due to imbalance or aerodynamic excitation. We demonstrate that near the resonance, in this non-conservative regime, localised vibration states bifurcate from the travelling waves. Complex bifurcation diagrams result, comprising stable and unstable dissipative solitons. The localised solutions can also be continued numerically to a conservative limit, where solitons bifurcate from the backbone curves of the travelling waves at finite amplitudes.

nlin.PS↗
arXiv2017

On the rationality problem for forms of moduli spaces of stable marked curves of positive genus

Let $M_{g, n}$ (respectively, $\overline{M_{g, n}}$) be the moduli space of smooth (respectively stable) curves of genus $g$ with $n$ marked points. Over the field of complex numbers, it is a classical problem in algebraic geometry to determine whether or not $M_{g, n}$ (or equivalently, $\overline{M_{g, n}}$) is a rational variety. Theorems of J. Harris, D. Mumford, D. Eisenbud and G. Farkas assert that $M_{g, n}$ is not unirational for any $n \geqslant 0$ if $g \geqslant 22$. Moreover, P. Belorousski and A. Logan showed that $M_{g, n}$ is unirational for only finitely many pairs $(g, n)$ with $g \geqslant 1$. Finding the precise range of pairs $(g, n)$, where $M_{g, n}$ is rational, stably rational or unirational, is a problem of ongoing interest. In this paper we address the rationality problem for twisted forms of $\overline{M_{g, n}}$ defined over an arbitrary field $F$ of characteristic $\neq 2$. We show that all $F$-forms of $\overline{M_{g, n}}$ are stably rational for $g = 1$ and $3 \leqslant n \leqslant 4$, $g = 2$ and $2 \leqslant n \leqslant 3$, $g = 3$ and $1 \leqslant n \leqslant 14$, $g = 4$ and $1 \leqslant n \leqslant 9$, $g = 5$ and $1 \leqslant n \leqslant 12$.

math.AG↗
arXiv2017

Stability of the Poincar\'e bundle

Let X be an irreducible smooth projective curve, of genus at least two, over an algebraically closed field k. Let $\mathcal{M}^d_G$ denote the moduli stack of principal G-bundles over X of fixed topological type $d \in \pi_1(G)$, where G is any almost simple affine algebraic group over k. We prove that the universal bundle over $X \times \mathcal{M}^d_G$ is stable with respect to any polarization on $X \times \mathcal{M}^d_G$. A similar result is proved for the Poincar\'e adjoint bundle over $X \times M_G^{d, rs}$, where $M_G^{d, rs}$ is the coarse moduli space of regularly stable principal G-bundles over X of fixed topological type d.

math.AG↗