arXiv subjects

Daniel Ruprecht

Publications and source records attributed to Daniel Ruprecht.

At least 19 recordsLinked to original sources

arXiv2026

Fourier Neural Operators for Rayleigh-B\'enard Convection

We propose an improved Fourier Neural Operator (FNO) for modeling two-dimensional Rayleigh-B\'enard convection by predicting time increments instead of full solutions, achieving higher accuracy than a standard FNO baseline. The resulting model is compact (314k parameters, 1.26 MB) and fast (7 ms inference), while maintaining similar accuracy as demonstrated in previous benchmarks. We show that although FNOs generalize to finer meshes, accuracy remains limited by the resolution of the training data.

cs.LG↗

arXiv2026

Tracking in-silico Lagrangian sensors in a lab-scale stirred tank reactor

Lagrangian sensors have shown promise to improve operator awareness of conditions inside a chemical reactor but three-dimensional tracking remains a mostly unsolved challenge. We explore a setup where in-silico sensors, based on a recently proposed real-world design, are tracked using data from an accelerometer and magnetometer available from a built-in inertial measurement unit. Filtering algorithms, using a bespoke dynamical model, are used to process these readings into position estimates. We compare tracking performance of an extended Kalman filter, a particle filter and the unscented Kalman filter implemented in the pykalman library. Our numerical experiments track in-silico particles moving in an analytically given three dimensional vortex as well as in the experimentally measured flow-field of a lab-scale stirred tank reactor. Using the Maxey-Riley-Gatignol equations for the movement of inertial particles as ground-truth, we demonstrate that trajectories can be reconstructed from noisy synthetic data with errors below 10%.

math.NA↗

arXiv2026

Approximation of the Basset force in the Maxey-Riley-Gatignol equations via universal differential equations

The Maxey-Riley-Gatignol equations (MaRGE) model the motion of spherical inertial particles in a fluid. They contain the Basset force, an integral term which models history effects due to the formation of wakes and boundary layer effects. This causes the force that acts on a particle to depend on its past trajectory and complicates the numerical solution of MaRGE. Therefore, the Basset force is often neglected, despite substantial evidence that it has both quantitative and qualitative impact on the movement patterns of modelled particles. Using the concept of universal differential equations, we propose an approximation of the history term via neural networks which approximates MaRGE by a system of ordinary differential equations that can be solved with standard numerical solvers like Runge-Kutta methods.

cs.LG↗

arXiv2026

Bathymetry Reconstruction by Bayesian Inference

Bathymetry reconstruction is an important problem in various fields, including oceanography and environmental monitoring. This paper presents a Bayesian inference approach to reconstructing bathymetries from point measurements of the water height. We test the method for parameterized and discretized bathymetries with synthetic data to evaluate its performance and limitations. Our results indicate that the Bayesian framework provides a robust approach to bathymetry reconstruction. Finally, we use the framework to reconstruct a real-world bathymetry in a wave flume from experimental measurements and compare its performance to an adjoint optimization method. The Bayesian approach improves the normalized root mean squared error (NRMSE) of the reconstruction and provides better qualitative features, while also quantifying uncertainty.

stat.AP↗

arXiv2026

Spectral Deferred Corrections in the framework of Runge-Kutta methods

We interpret a wide range of flavors of Spectral Deferred Corrections (SDC) as Runge-Kutta methods (RKM). Using Butcher series, we show that the considered class of SDC methods achieve at least order p after p iterations compared to the underlying RKM, independently of the error discretisation chosen and the choice of nodes. For all collocation RKM, we analyse the phenomenon of order jumps in SDC iterations, where the order is increased by two at each iteration. We prove that it can be obtained by using appropriate inconsistent, implicit, parallelisable error discretisations. We also investigate the stability properties of the new SDC methods which can in general reduce to that of explicit RKM, but it can be improved by suitable combinations of error discretisations. We confirm the convergence analysis with numerical experiments and we apply relaxation RKM to derive SDC variants that conserve quadratic invariants.

math.NA↗

arXiv2025

Enforcing boundary conditions for physics-informed neural operators

Machine-learning based methods like physics-informed neural networks and physics-informed neural operators are becoming increasingly adept at solving even complex systems of partial differential equations. Boundary conditions can be enforced either weakly by penalizing deviations in the loss function or strongly by training a solution structure that inherently matches the prescribed values and derivatives. The former approach is easy to implement but the latter can provide benefits with respect to accuracy and training times. However, previous approaches to strongly enforcing Neumann or Robin boundary conditions require a domain with a fully $C^1$ boundary and, as we demonstrate, can lead to instability if those boundary conditions are posed on a segment of the boundary that is piecewise $C^1$ but only $C^0$ globally. We introduce a generalization of the approach by Sukumar \& Srivastava (doi: 10.1016/j.cma.2021.114333), and a new approach based on orthogonal projections that overcome this limitation. The performance of these new techniques is compared against weakly and semi-weakly enforced boundary conditions for the scalar Darcy flow equation and the stationary Navier-Stokes equations.

math.NA↗

arXiv2025

Fast-wave slow-wave spectral deferred correction methods applied to the compressible Euler equations

This paper investigates the application of a fast-wave slow-wave spectral deferred correction time-stepping method (FWSW-SDC) to the compressible Euler equations. The resulting model achieves arbitrary order accuracy in time, demonstrating robust performance in standard benchmark idealised test cases for dynamical cores used for numerical weather prediction. The model uses a compatible finite element spatial discretisation, achieving good linear wave dispersion properties without spurious computational modes. A convergence test confirms the model's high temporal accuracy. Arbitrarily high spatial-temporal convergence is demonstrated using a gravity wave test case. The model is further extended to include the parametrisation of a simple physics process by adding two phases of moisture and its validity is demonstrated for a rising thermal problem. Finally, a baroclinic wave in simulated in a Cartesian domain.

math.NA↗

arXiv2024

Resilience Against Soft Faults through Adaptivity in Spectral Deferred Correction

As supercomputers grow in hardware complexity, their susceptibility to faults increases and measures need to be taken to ensure the correctness of results. Some numerical algorithms have certain characteristics that allow them to recover from some types of faults. It has been demonstrated that adaptive Runge-Kutta methods provide resilience against transient faults without adding computational cost. Using recent advances in adaptive step size selection for spectral deferred correction (SDC), an iterative numerical time stepping scheme that can produce methods of arbitrary order, we show that adaptive SDC can also detect and correct transient faults. Its performance is found to be comparable to that of the dedicated resilience strategy Hot Rod.

cs.DC↗

arXiv2024

Relevance of the Basset history term for Lagrangian particle dynamics

The movement of small but finite spherical particles in a fluid can be described by the Maxey-Riley equation (MRE) if they are too large to be considered passive tracers. The MRE contains an integral "history term" modeling wake effects, which causes the force acting on a particle at some given time to depend on its full past trajectory. The history term causes complications in the numerical solution of the MRE and is therefore often neglected, despite both numerical and experimental evidence that its effects are generally not negligible. By numerically computing trajectories with and without the history term of a large number of particles in different flow fields, we investigate its impact on the large-scale Lagrangian dynamics of simulated particles. We show that for moderate to large Stokes numbers, ignoring the history term leads to significant differences in clustering patterns. Furthermore, we compute finite-time Lyapunov exponents and show that, even for small particles, the differences in the resulting scalar field from ignoring the BHT can be significant, in particular if the underlying flow is turbulent.

physics.flu-dyn↗

arXiv2024

Dynamic Deep Learning Based Super-Resolution For The Shallow Water Equations

Using the nonlinear shallow water equations as benchmark, we demonstrate that a simulation with the ICON-O ocean model with a 20km resolution that is frequently corrected by a U-net-type neural network can achieve discretization errors of a simulation with 10km resolution. The network, originally developed for image-based super-resolution in post-processing, is trained to compute the difference between solutions on both meshes and is used to correct the coarse mesh every 12h. Our setup is the Galewsky test case, modeling transition of a barotropic instability into turbulent flow. We show that the ML-corrected coarse resolution run correctly maintains a balance flow and captures the transition to turbulence in line with the higher resolution simulation. After 8 day of simulation, the $L_2$-error of the corrected run is similar to a simulation run on the finer mesh. While mass is conserved in the corrected runs, we observe some spurious generation of kinetic energy.

cs.LG↗

arXiv2024

Bathymetry reconstruction from experimental data using PDE-constrained optimisation

Knowledge of the bottom topography, also called bathymetry, of rivers, seas or the ocean is important for many areas of maritime science and civil engineering. While direct measurements are possible, they are time consuming and expensive. Therefore, many approaches have been proposed how to infer the bathymetry from measurements of surface waves. Mathematically, this is an inverse problem where an unknown system state needs to be reconstructed from observations with a suitable model for the flow as constraint. In many cases, the shallow water equations can be used to describe the flow. While theoretical studies of the efficacy of such a PDE-constrained optimisation approach for bathymetry reconstruction exist, there seem to be few publications that study its application to data obtained from real-world measurements. This paper shows that the approach can, at least qualitatively, reconstruct a Gaussian-shaped bathymetry in a wave flume from measurements of the water height at up to three points. Achieved normalized root mean square errors (NRMSE) are in line with other approaches.

math.NA↗

arXiv2024

Space-time parallel scaling of Parareal with a physics-informed Fourier Neural Operator coarse propagator applied to the Black-Scholes equation

Iterative parallel-in-time algorithms like Parareal can extend scaling beyond the saturation of purely spatial parallelization when solving initial value problems. However, they require the user to build coarse models to handle the inevitably serial transport of information in time.This is a time consuming and difficult process since there is still only limited theoretical insight into what constitutes a good and efficient coarse model. Novel approaches from machine learning to solve differential equations could provide a more generic way to find coarse level models for parallel-in-time algorithms. This paper demonstrates that a physics-informed Fourier Neural Operator (PINO) is an effective coarse model for the parallelization in time of the two-asset Black-Scholes equation using Parareal. We demonstrate that PINO-Parareal converges as fast as a bespoke numerical coarse model and that, in combination with spatial parallelization by domain decomposition, it provides better overall speedup than both purely spatial parallelization and space-time parallelizaton with a numerical coarse propagator.

math.NA↗

arXiv2024

Parallel performance of shared memory parallel spectral deferred corrections

We investigate the parallel performance of Parallel Spectral Deferred corrections, a numerical approach that provides small-scale parallelism for the numerical solution of initial value problems. The scheme is applied to the shallow-water equation and uses an implicit-explicit splitting that, in order to be efficient, integrates fast modes implicitly and slow modes explicitly. We describe parallel \OpenMP-based implementations of parallel Spectral Deferred Corrections for two well established simulation codes: the finite volume based operational ocean model \ICON and the spherical harmonics based research code \SWEET. We also develop a performance model and benchmark our implementations on a single node of the JUSUF (\SWEET) and JUWELS (\ICON) system at J\"ulich Supercomputing Centre. A reduction of time-to-solution across a range of accuracies is demonstrated. For \ICON, we show speedup over the currently used Adams--Bashforth-2 integrator with \OpenMP loop parallelization. For \SWEET, we show speedup over serial Spectral Deferred Corrections and a second order implicit-explicit integrator.

cs.CE↗

arXiv2024

Improving Efficiency of Parallel Across the Method Spectral Deferred Corrections

Parallel-across-the method time integration can provide small scale parallelism when solving initial value problems. Spectral deferred corrections (SDC) with a diagonal sweeper, which is closely related to iterated Runge-Kutta methods proposed by Van der Houwen and Sommeijer, can use a number of threads equal to the number of quadrature nodes in the underlying collocation method. However, convergence speed, efficiency and stability depends critically on the used coefficients. Previous approaches have used numerical optimization to find good parameters. Instead, we propose an ansatz that allows to find optimal parameters analytically. We show that the resulting parallel SDC methods provide stability domains and convergence order very similar to those of well established serial SDC variants. Using a model for computational cost that assumes 80% efficiency of an implementation of parallel SDC we show that our variants are competitive with serial SDC, previously published parallel SDC coefficients as well as Picard iteration, explicit RKM-4 and an implicit fourth-order diagonally implicit Runge-Kutta method.

math.NA↗

arXiv2024

Adaptive time step selection for Spectral Deferred Correction

Spectral Deferred Correction (SDC) is an iterative method for the numerical solution of ordinary differential equations. It works by refining the numerical solution for an initial value problem by approximately solving differential equations for the error, and can be interpreted as a preconditioned fixed-point iteration for solving the fully implicit collocation problem. We adopt techniques from embedded Runge-Kutta Methods (RKM) to SDC in order to provide a mechanism for adaptive time step size selection and thus increase computational efficiency of SDC. We propose two SDC-specific estimates of the local error that are generic and do not rely on problem specific quantities. We demonstrate a gain in efficiency over standard SDC with fixed step size and compare efficiency favorably against state-of-the-art adaptive RKM.

math.NA↗

arXiv2024

Efficient numerical methods for the Maxey-Riley-Gatignol equations with Basset history term

The Maxey-Riley-Gatignol equations (MRGE) describe the motion of a finite-sized, spherical particle in a fluid. Because of wake effects, the force acting on a particle depends on its past trajectory. This is modelled by an integral term in the MRGE, also called Basset force, that makes its numerical solution challenging and memory intensive. A recent approach proposed by Prasath et al. exploits connections between the integral term and fractional derivatives to reformulate the MRGE as a time-dependent partial differential equation on a semi-infinite pseudo-space. They also propose a numerical algorithm based on polynomial expansions. This paper develops a numerical approach based on finite difference instead, by adopting techniques by Koleva et al. and Fazio et al. to cope with the issues of having an unbounded spatial domain. We compare convergence order and computational efficiency for particles of varying size and density of the polynomial expansion by Prasath et al., our finite difference schemes and a direct integrator for the MRGE based on multi-step methods proposed by Daitche. While all methods achieve their theoretical convergence order for neutrally buoyant particles with zero initial relative velocity, they suffer from various degrees of order reduction if the initial relative velocity is non-zero or the particle has a different density than the fluid.

math.NA↗

arXiv2023

Spectral deferred correction methods for second-order problems

Spectral deferred corrections (SDC) are a class of iterative methods for the numerical solution of ordinary differential equations. SDC can be interpreted as a Picard iteration to solve a fully implicit collocation problem, preconditioned with a low-order method. It has been widely studied for first-order problems, using explicit, implicit or implicit-explicit Euler and other low-order methods as preconditioner. For first-order problems, SDC achieves arbitrary order of accuracy and possesses good stability properties. While numerical results for SDC applied to the second-order Lorentz equations exist, no theoretical results are available for SDC applied to second-order problems. We present an analysis of the convergence and stability properties of SDC using velocity-Verlet as the base method for general second-order initial value problems. Our analysis proves that the order of convergence depends on whether the force in the system depends on the velocity. We also demonstrate that the SDC iteration is stable under certain conditions. Finally, we show that SDC can be computationally more efficient than a simple Picard iteration or a fourth-order Runge-Kutta-Nystr\"om method.

math.NA↗

arXiv2023

Parareal with a physics-informed neural network as coarse propagator

Parallel-in-time algorithms provide an additional layer of concurrency for the numerical integration of models based on time-dependent differential equations. Methods like Parareal, which parallelize across multiple time steps, rely on a computationally cheap and coarse integrator to propagate information forward in time, while a parallelizable expensive fine propagator provides accuracy. Typically, the coarse method is a numerical integrator using lower resolution, reduced order or a simplified model. Our paper proposes to use a physics-informed neural network (PINN) instead. We demonstrate for the Black-Scholes equation, a partial differential equation from computational finance, that Parareal with a PINN coarse propagator provides better speedup than a numerical coarse propagator. Training and evaluating a neural network are both tasks whose computing patterns are well suited for GPUs. By contrast, mesh-based algorithms with their low computational intensity struggle to perform well. We show that moving the coarse propagator PINN to a GPU while running the numerical fine propagator on the CPU further improves Parareal's single-node performance. This suggests that integrating machine learning techniques into parallel-in-time integration methods and exploiting their differences in computing patterns might offer a way to better utilize heterogeneous architectures.

math.NA↗