arXiv subjects

Allan Peter Engsig-Karup

Publications and source records attributed to Allan Peter Engsig-Karup.

16 recordsLinked to original sources

arXiv2026

An unstructured second-order subgrid method for the shallow water equations

We propose a new unstructured numerical subgrid method for solving the shallow water equations using a finite volume method with enhanced bathymetry resolution. The method employs an unstructured triangular mesh with support for triangulation of elements to form finer subgrids locally. The bathymetry is represented on the fine mesh, allowing the incorporation of small-scale features, while velocities are defined on the coarse mesh. The governing equations are solved numerically on the coarse mesh, making the method computationally cheaper compared to a traditional fine mesh computation. To accurately represent the velocities, we employ a second-order accurate WENO method and for temporal integration an explicit second-order accurate Runge-Kutta method. Furthermore, we present a novel subgrid face value reconstruction that accounts for partially wet cells, where only some of the subgrid cells are wet. Together with a newly developed gravity source term discretization, we demonstrate that the scheme is well-balanced on the subgrid level. Finally, the subgrid method is validated on several test cases to show: i) that the scheme is well-balanced, ii) confirm the order of the accuracy of the scheme, and iii) demonstrate the capability of handling moving flood and dry boundaries. We also highlight when it is beneficial to increase the number of subgrid cells to improve the results of the finite volume method.

math.NA↗

arXiv2026

AURA: Angular Update Rate Adaptation for training complex-valued neural networks

Complex-valued neural networks (CVNNs) are increasingly adopted for complex-valued data; however, they are often trained with first-order optimizers inherited from the real-valued case. The efficiency of these methods depends largely on the step size, and their step-size rules ignore the angular information available in the complex plane. We address step-size adaptation in the complex domain by introducing AURA (Angular Update Rate Adaptation), a per-parameter step-size adaptation that can be added on top of any first-order optimizer, and removed from it, without altering its update direction. AURA measures the agreement between consecutive updates of each complex parameter, in length, alignment, and sense of rotation, and enlarges the step when they are consistent and reduces it when they are not. It requires no additional gradient evaluations and only inexpensive vector operations per step. We combine AURA with Adam and Muon and compare the resulting methods with well-known first-order optimizers on four test cases of increasing complexity, ranging from the approximation of scalar complex functions to physics-informed training. Fully connected neural networks are used throughout this work. All hyperparameters other than the step size are held fixed across test cases; for one case, we also tune the hyperparameters of each optimizer under the same budget. Our empirical tests show that AURA improves the convergence of its base optimizer in most cases with a small per-step overhead, and we identify the conditions under which it fails to do so.

cs.LG↗

arXiv2026

Estimating Hydrodynamic Coefficients for Floating Offshore Structures from Movement Data Using Physics-Informed Neural Networks

We present a method for estimating the hydrodynamic coefficients in the Cummins equations using time-series data from a moving body, such as a floating offshore structure. The proposed data-driven method is based on incorporating the Cummins equations governing the dynamics of a structural body interacting with water waves into a physics-informed neural network (PINN), along with available motion data. The proposed method first estimates the structure's state in terms of translational and rotational degrees of freedom, and then solves the inverse problem to determine the hydrodynamic forces acting on the body, expressed in terms of added mass, damping coefficients, and/or hydrostatic restoring. The Cummins equations are formulated as a first-order system, and both state and parameter estimation are performed using PINNs. The method is verified on the free decay of a sphere and a box. The results demonstrate that it is possible to estimate the state and hydrodynamic coefficients accurately, although accuracy depends on the volume and quality of the movement data.

physics.flu-dyn↗

arXiv2026

A holomorphic neural network framework for 3D boundary value problems governed by harmonic potentials

We present a neural-network-based framework for the solution of three-dimensional boundary value problems where the solution is expressible in terms of harmonic potentials. The approach leverages the Whittaker integral formula, which allows representing the solution through functions that are holomorphic with respect to a suitable complex variable. These functions are subsequently approximated using holomorphic neural networks, which guaranty fulfillment of the holomorphicity requirement. A key feature of the proposed formulation is that the governing partial differential equations (PDEs) are satisfied exactly by construction. Therefore, in contrast to standard physics-informed neural networks, no residual minimization of PDEs is required in the interior of the domain, and training is based exclusively on boundary collocation points. The method is validated against three-dimensional Laplace and linear elasticity problems, where, in the latter case, displacement and stress fields are expressed via the Papkovich-Neuber potentials. The numerical results show an accurate approximation of both scalar and vector fields, with errors remaining controlled throughout the domain. Overall, the work demonstrates that the incorporation of analytical structures into neural network architectures provides a natural and effective framework for the meshless approximation of three-dimensional boundary value problems while preserving the underlying properties of the governing equations.

math.NA↗

arXiv2026

An IMEX-DG solver with non-conforming mesh refinement for atmospheric dynamics with rotation

We present a high-order implicit-explicit discontinuous Galerkin (IMEX-DG) solver for the compressible Euler equations to account for rotational effects within a fully compressible atmospheric framework. Time integration follows a second-order additive Runge-Kutta scheme, treating stiff acoustic modes implicitly and advective terms explicitly. The solver is built on the deal.II finite element library, combining matrix-free operator evaluation, adaptive non-conforming meshes capabilities, and distributed-memory parallelism. Two alternative treatments of the rotational and gravitational source terms within the solution strategy, based on nonlinear fixed-point iterations, are introduced and compared in terms of accuracy, robustness, and computational efficiency. A discrete analysis of the rotational operator is also carried out in order to derive a formulation suitable for efficient matrix-free implementation and to avoid inconsistent naive discretisations. The proposed formulation is validated through convergence studies on rotating inertia-gravity wave benchmarks and further assessed in fully three-dimensional simulations of stratified flow over orography on both uniform and adaptive meshes. The numerical results show that the rotating IMEX-DG framework has the expected accuracy and stability properties while correctly capturing the asymmetry and wave structures induced by rotation in large-scale atmospheric flows.

math.NA↗

arXiv2026

Neural Code Translation of Legacy Code: APL to C#

Automatic translation between programming languages remains a challenging problem, particularly when the source language is highly concise and specialized. This paper investigates the translation of APL into C# using large language models. The task is difficult due to APL's sparse syntax, the scarcity of large-scale parallel corpora, and the requirement for specialized knowledge to interpret APL programs. To address these challenges, we introduce a novel framework for APL-to-C# translation by comparing three guided strategies, namely natural language description-mediated, retrieval-augmented, and iterative refinement, against a baseline direct translation model. We constructed multiple datasets of functionally equivalent code pairs spanning various levels of complexity, and to rigorously assess translation quality, we developed an automated evaluation pipeline that verifies both syntactic compilation and functional execution of the generated C# code. Our results demonstrate that neural code translation can successfully bridge the gap between APL and C# for a wide range of programs, and that incorporating additional context and guidance significantly improves model performance.

cs.SE↗

arXiv2026

Physics-Informed Tracking (PIT)

We propose Physics-Informed Tracking (PIT), a video-based framework for tracking a single particle from video, where a neural network autoencoder localizes a particle as a heatmap peak (landmark) and a differentiable physics module embedded in the autoencoder constrains several landmarks over time (a trajectory) to satisfy known dynamics. The novel Physics-Informed Landmark Loss (PILL) compares this predicted trajectory back against the landmarks, enforcing physical consistency without labels. Its supervised variant (PILLS) instead compares the prediction against ground-truth position, velocity, and bounce from simulation, enabling end-to-end backpropagation. To support supervised and unsupervised learning, we use an autoencoder with a split bottleneck that separates A) tracking-related structure via landmark heatmaps from B) background noise and subsequent image reconstruction. We evaluate a replicated 26 factorial design (n = 4 replicates, 64 configurations), showing that PILLS consistently achieves sub-pixel tracking accuracy for the bilinear and physics-refined decoder outputs under both clean and noisy conditions.

cs.CV↗

arXiv2025

A High-Order Spectral Element Solver for Steady-State Free Surface Flows

We present a spectral element solver for the steady incompressible Navier-Stokes equations subject to a free surface. Utilizing the kinematic behaviour of the free surface boundary, an iterative pseudo-time procedure is proposed to determine the a priori unknown free surface profile. The numerical model is implemented in the open-source finite element framework Firedrake, which enables the use of a high-order polynomial basis on unstructured meshes through weak formulations. Additionally, the curvature of the free surface and submerged bodies is incorporated through curvilinear elements obtained via transfinite linear blending, which conserves the high-order convergent properties of the overall scheme. The model is applied to several benchmark cases in two spatial dimensions. Initially, it addresses fixed-domain problems, including the lid-driven cavity flow and flows around bodies such as a cylinder and a NACA airfoil. Subsequently, with the presence of a free surface, it is extended to determine the flow around a bathymetry bump and a submerged NACA airfoil. The results confirm the high-order accuracy of the model through convergence studies and demonstrate a substantial speed-up over low-order numerical schemes.

math.NA↗

arXiv2025

Subspace Acceleration for Efficient Nonlinear Water Wave Simulation

Efficient simulation of nonlinear and dispersive free-surface flows governed by the incompressible Navier-Stokes equations remains a central challenge in ocean and coastal engineering. The computational bottleneck arises from solving a time-dependent discretized Poisson problem at every time step to enforce divergence free flow. This is crucial to ensure conservation of mass and requires solving long sequences of time-dependent linear systems typically using iterative methods, such as the preconditioned Krylov subspace methods. In this work, we investigate new subspace acceleration techniques for improving initial guesses to reduce the number of iterations required by iterative solvers, with a focus on nonlinear wave propagation problems. We extend the original subspace acceleration method by incorporating the complete history of previous solutions through an exponentially weighted formulation. This approach eliminates the need for repeated sketching and orthonormalization, resulting in a more efficient and scalable strategy to generate better initial guesses. Our method is implemented within a high-order finite-difference framework using a method-of-lines formulation and a low-storage Runge-Kutta time integration scheme. We demonstrate that subspace acceleration significantly reduces the number of GMRES iterations when solving the Poisson equation in nonlinear water wave simulations. Performance is evaluated on two benchmark problems: nonlinear stream function wave propagation and harmonic wave generation over a submerged bar. In both cases, the new approach achieves substantial improvements in computational efficiency without compromising accuracy. Although demonstrated using high-order finite difference methods, the technique is discretization independent and broadly applicable to incompressible free-surface flow solvers.

physics.flu-dyn↗

arXiv2025

Physics-Informed Regression: Parameter Estimation in Parameter-Linear Nonlinear Dynamic Models

We present a new efficient hybrid parameter estimation method based on the idea, that if nonlinear dynamic models are stated in terms of a system of equations that is linear in terms of the parameters, then regularized ordinary least squares can be used to estimate these parameters from time series data. We introduce the term "Physics-Informed Regression" (PIR) to describe the proposed data-driven hybrid technique as a way to bridge theory and data by use of ordinary least squares to efficiently perform parameter estimation of the model coefficients of different parameter-linear models; providing examples of models based on nonlinear ordinary equations (ODE) and partial differential equations (PDE). The focus is on parameter estimation on a selection of ODE and PDE models, each illustrating performance in different model characteristics. For two relevant epidemic models of different complexity and number of parameters, PIR is tested and compared against the related technique, physics-informed neural networks (PINN), both on synthetic data generated from known target parameters and on real public Danish time series data collected during the COVID-19 pandemic in Denmark. Both methods were able to estimate the target parameters, while PIR showed to perform noticeably better, especially on a compartment model with higher complexity. Given the difference in computational speed, it is concluded that the PIR method is superior to PINN for the models considered. It is also demonstrated how PIR can be applied to estimate the time-varying parameters of a compartment model that is fitted using real Danish data from the COVID-19 pandemic obtained during a period from 2020 to 2021. The study shows how data-driven and physics-informed techniques may support reliable and fast -- possibly real-time -- parameter estimation in parameter-linear nonlinear dynamic models.

cs.LG↗

arXiv2025

FNPF-SEM: A parallel spectral element model in Firedrake for fully nonlinear water wave simulations

We present a new parallel spectral element solver, FNPF-SEM, for simulating linear and fully nonlinear potential flow-based water waves and their interaction with offshore structures. The tool is designed as a general-purpose wave model for offshore engineering applications. Built within the open-source framework Firedrake, the new FNPF-SEM model is designed as a computational tool capable of capturing both linear and nonlinear wave phenomena with high accuracy and efficiency, with support for high-order (spectral) finite elements. Additionally, Firedrake provides native support for MPI-based parallelism, allowing for efficient multi-CPU distributed computations needed for large-scale simulations. We demonstrate the capabilities of the high-order spectral element model through h- and p-convergence studies, and weak and strong scaling tests. Validation is performed against analytical solutions and experimental data for several benchmark cases, including nonlinear high-order harmonic generation and linear and nonlinear wave interactions with a cylinder and a breakwater. The new FNPF-SEM model offers a numerical framework for simulating wave propagation and wave-structure interactions, with the following key features: i) the ability to represent complex geometries through flexible, unstructured finite element meshes; ii) reduced numerical diffusion and dispersion by using high-order polynomial expansions; and iii) scalability to full- and large-scale simulations over long time periods through a parallel implementation.

math.NA↗

arXiv2024

A stable decoupled perfectly matched layer for the 3D wave equation using the nodal discontinuous Galerkin method

In outdoor acoustics, the calculations of sound propagating in air can be computationally heavy if the domain is chosen large enough to fulfil the Sommerfeld radiation condition. By strategically truncating the computational domain with a efficient boundary treatment, the computational cost is lowered. One commonly used boundary treatment is the perfectly matched layer (PML) that dampens outgoing waves without polluting the computed solution in the inner domain. The purpose of this study is to propose and assess a new perfectly matched layer formulation for the 3D acoustic wave equation, using the nodal discontinuous Galerkin finite element method. The formulation is based on an efficient PML formulation that can be decoupled to further increase the computational efficiency and guarantee stability without sacrificing accuracy. This decoupled PML formulation is demonstrated to be long-time stable and an optimization procedure of the damping functions is proposed to enhance the performance of the formulation.

math.NA↗

arXiv2023

A spectral element solution of the Poisson equation with shifted boundary polynomial corrections: influence of the surrogate to true boundary mapping and an asymptotically preserving Robin formulation

We present a new high-order accurate spectral element solution to the two-dimensional scalar Poisson equation subject to a general Robin boundary condition. The solution is based on a simplified version of the shifted boundary method employing a continuous arbitrary order $hp$-Galerkin spectral element method as the numerical discretization procedure. The simplification relies on a polynomial correction to avoid explicitly evaluating high-order partial derivatives from the Taylor series expansion, which traditionally have been used within the shifted boundary method. In this setting, we apply an extrapolation and novel interpolation approach to project the basis functions from the true domain onto the approximate surrogate domain. The resulting solution provides a method that naturally incorporates curved geometrical features of the domain, overcomes complex and cumbersome mesh generation, and avoids problems with small-cut-cells. Dirichlet, Neumann, and general Robin boundary conditions are enforced weakly through: i) a generalized Nitsche's method and ii) a generalized Aubin's method. For this, a consistent asymptotic preserving formulation of the embedded Robin formulations is presented. We present several numerical experiments and analysis of the algorithmic properties of the different weak formulations. With this, we include convergence studies under polynomial, $p$, increase of the basis functions, mesh, $h$, refinement, and matrix conditioning to highlight the spectral and algebraic convergence features, respectively. This is done to assess the influence of errors across variational formulations, polynomial order, mesh size, and mappings between the true and surrogate boundaries.

math.NA↗

arXiv2022

Accelerating hydrodynamic simulations of urban drainage systems with physics-guided machine learning

We propose and demonstrate a new approach for fast and accurate surrogate modelling of urban drainage system hydraulics based on physics-guided machine learning. The surrogates are trained against a limited set of simulation results from a hydrodynamic (HiFi) model. Our approach reduces simulation times by one to two orders of magnitude compared to a HiFi model. It is thus slower than e.g. conceptual hydrological models, but it enables simulations of water levels, flows and surcharges in all nodes and links of a drainage network and thus largely preserves the level of detail provided by HiFi models. Comparing time series simulated by the surrogate and the HiFi model, R2 values in the order of 0.9 are achieved. Surrogate training times are currently in the order of one hour. However, they can likely be reduced through the application of transfer learning and graph neural networks. Our surrogate approach will be useful for interactive workshops in initial design phases of urban drainage systems, as well as for real time applications. In addition, our model formulation is generic and future research should investigate its application for simulating other water systems.

cs.LG↗

arXiv2020

The DeRisk database: Extreme Design Waves for Offshore Wind Turbines

The estimation of extreme loads from waves is an essential part of the design of an offshore wind turbine. Standard design codes suggest to either use simplified methods based on regular waves, or to perform fully nonlinear computations. The former might not provide an accurate representation of the extreme waves, while the latter is computationally too intensive for design iterations. We address these limitations by using the fully nonlinear solver OceanWave3D to establish the DeRisk database, a large dataset of extreme waves kinematics in a two-dimensional domain. From the database, which is open and freely available, a designer can extract fully-nonlinear wave kinematics for a wave condition and water depth of interest by identifying a suitable computation in the database and, if needed, by Froude-scaling the kinematics. The nonlinear solver is validated against the DeRisk model experiments at two different water depths, $33.0 [m]$ and $20.0 [m]$, and an excellent agreement is found for the analyzed cases. The experiments are used to calibrate OceanWave3D's numerical breaking filter constant, and the best agreement is found for $β=0.5$. We compare the experimental static force with predictions by the DeRisk database and the Rainey force model, and with state-of-the-art industrial practices. For milder storms, we find a good agreement in the predicted extreme force between the present methodology and the standard methodologies. At the deep location and for stronger storms, the largest loads are given by slamming loads due to breaking waves. In this condition, the database methodology is less accurate than the embedded stream function method and more accurate than the WiFi JIP methodology, providing generally nonconservative estimates. For strong storms at the shallower location, where wave breaking is less dominating, the database methodology is the most accurate overall.

physics.flu-dyn↗

arXiv2016

A Stabilised Nodal Spectral Element Method for Fully Nonlinear Water Waves

We present an arbitrary-order spectral element method for general-purpose simulation of non-overturning water waves, described by fully nonlinear potential theory. The method can be viewed as a high-order extension of the classical finite element method proposed by Cai et al (1998) \cite{CaiEtAl1998}, although the numerical implementation differs greatly. Features of the proposed spectral element method include: nodal Lagrange basis functions, a general quadrature-free approach and gradient recovery using global $L^2$ projections. The quartic nonlinear terms present in the Zakharov form of the free surface conditions can cause severe aliasing problems and consequently numerical instability for marginally resolved or very steep waves. We show how the scheme can be stabilised through a combination of over-integration of the Galerkin projections and a mild spectral filtering on a per element basis. This effectively removes any aliasing driven instabilities while retaining the high-order accuracy of the numerical scheme. The additional computational cost of the over-integration is found insignificant compared to the cost of solving the Laplace problem. The model is applied to several benchmark cases in two dimensions. The results confirm the high order accuracy of the model (exponential convergence), and demonstrate the potential for accuracy and speedup. The results of numerical experiments are in excellent agreement with both analytical and experimental results for strongly nonlinear and irregular dispersive wave propagation. The benefit of using a high-order -- possibly adapted -- spatial discretization for accurate water wave propagation over long times and distances is particularly attractive for marine hydrodynamics applications.

physics.comp-ph↗