arXiv subjects

Georg Maierhofer

Publications and source records attributed to Georg Maierhofer.

At least 19 recordsLinked to original sources

arXiv2026

Physics-informed reduced-order modelling with equivariant spectral submanifolds

Spectral submanifold (SSM) reduction has emerged as a mathematically principled route to reliable nonlinear reduced-order models, capturing dynamics beyond the reach of linear techniques such as Dynamic Mode Decomposition (DMD). The computation of SSMs, however, remains computationally expensive, particularly for high-dimensional systems. In this work, we introduce equivariant spectral submanifold (eSSM) reduction, a novel extension of the SSM framework that explicitly incorporates symmetries of the full-order model into the reduction process. We establish the mathematical foundations of this approach by showing that SSMs are naturally equivariant submanifolds and that the associated charts and reduced dynamics inherit the appropriate induced group actions. Building on this framework, we develop a novel equivariant SSM reduction algorithm that exploits these symmetries to achieve substantially faster computations while also improving model robustness. We demonstrate the advantages of this approach on several benchmark problems including a test from the Common Task Framework for Science.

cs.LG↗

arXiv2026

CTF4Nuclear: Common Task Framework for Nuclear Fission and Fusion Models

The demand for clean energy is ever increasing, with new nuclear technologies presenting a complementary solution to renewable energies. However, designing and operating these systems is exceptionally difficult, given the complexity of the physical phenomena that interact to form the system dynamics. While high-fidelity simulations help to understand the non-linear, multi-physics interactions within a reactor, they are computationally expensive and rarely suitable for real-time applications. Furthermore, model-based approaches are inherently sensitive to simplifying assumptions required to derive their governing equations and parameters, leading to inevitable discrepancies with real-world measurements. In contrast, Machine Learning (ML) methods have the potential to generate reliable surrogate models which may be able to quickly predict the system's behaviour. However, the number of data-driven methods that can potentially be used for this task is large and diverse. In a safety-critical setting such as nuclear engineering, a fair comparison of different ML methods, and a clear understanding of their advantages and limitations, is of paramount importance. To address this, we introduce a Common Task Framework (CTF) for ML in nuclear engineering, building upon previous efforts in dynamical systems and seismology. This CTF considers a curated set of datasets from different nuclear and nuclear-adjacent systems. The CTF evaluates the performance of a method on 12 established metrics, alongside a new paradigm focused on system monitoring from sparse measurements only. We illustrate the framework by benchmarking standard ML baselines against these datasets, revealing current method limitations. Our vision is to replace ad hoc comparisons with standardized evaluations on hidden test sets, raising the bar for rigour and reproducibility in scientific ML for the nuclear industry.

cs.LG↗

arXiv2026

Stable Hermite transforms via the Golub-Welsch algorithm

We introduce an efficient stable algorithm for transforms associated with expansions in Hermite functions interpolated at Hermite polynomial roots. The Hermite transform matrix can be factorised into a diagonal component and an orthogonal matrix, leading to a form which allows both the forward and inverse Hermite transforms to be computed stably. Our novel algorithm computes this factorisation based on the eigendecomposition of the Jacobi matrix associated with Hermite functions. Through numerical experiments, we demonstrate the stability and efficiency gains of this novel method over prior work. Numerical experiments show that the new approach matches or improves on the accuracy of existing stabilized methods, is substantially faster in practice, and enables reliable use of large Hermite expansions in downstream PDE computations. We also provide an open-source implementation, together with reference implementations of previous methods, to facilitate adoption by the community.

math.NA↗

arXiv2026

The Seismic Wavefield Common Task Framework

Seismology faces fundamental challenges in state forecasting and reconstruction (e.g., earthquake early warning and ground motion prediction) and managing the parametric variability of source locations, mechanisms, and Earth models (e.g., subsurface structure and topography effects). Addressing these with simulations is hindered by their massive scale, both in synthetic data volumes and numerical complexity, while real-data efforts are constrained by models that inadequately reflect the Earth's complexity and by sparse sensor measurements from the field. Recent machine learning (ML) efforts offer promise, but progress is obscured by a lack of proper characterization, fair reporting, and rigorous comparisons. To address this, we introduce a Common Task Framework (CTF) for ML for seismic wavefields, demonstrated here on three distinct wavefield datasets. Our CTF features a curated set of datasets at various scales (global, crustal, and local) and task-specific metrics spanning forecasting, reconstruction, and generalization under realistic constraints such as noise and limited data. Inspired by CTFs in fields like natural language processing, this framework provides a structured and rigorous foundation for head-to-head algorithm evaluation. We evaluate various methods for reconstructing seismic wavefields from sparse sensor measurements, with results illustrating the CTF's utility in revealing strengths, limitations, and suitability for specific problem classes. Our vision is to replace ad hoc comparisons with standardized evaluations on hidden test sets, raising the bar for rigor and reproducibility in scientific ML.

cs.LG↗

arXiv2026

On scattering for NLS: rigidity properties and numerical simulations via the lens transform

We analyse the scattering operator associated with the defocusing nonlinear Schr{ö}dinger equation which captures the evolution of solutions over an infinite time-interval under the nonlinear flow of this equation. The asymptotic nature of the scattering operator (involving unbounded time) makes its computation particularly challenging. We overcome this by exploiting the space-time compactification provided by the lens transform, marking the first use of this technique in numerical simulations. This results in a highly efficient and reliable methodology for computing the scattering operator in various regimes. In developing this approach we introduce and prove several new identities and theoretical properties of the scattering operator. We support our construction with several numerical experiments which we show to agree with known analytical properties of the scattering operator, and also address the case of long-range scattering for the one-dimensional cubic Schr{ö}dinger equation. Our simulations permit us to further explore regimes beyond current analytical understanding, and lead us to formulate new conjectures concerning fixed and rotating points of the operator, as well as its existence in the long-range setting for both defocusing and focusing cases.

math.AP↗

arXiv2025

Computing nonlinear Schr\"odinger equations with Hermite functions beyond harmonic traps

Hermite basis functions are a classical tool for the spatial discretisation of Schr\"odinger equations with harmonic potential. In this work, we prove that their favourable stability properties extend to Schr\"odinger equations without a trap: the free Schr\"odinger flow is stable in the weighted Sobolev spaces $\Sigma^k$ which govern the convergence of Hermite spectral methods. This makes the Hermite basis a natural discretisation for a larger class of nonlinear Schr\"odinger equations posed on the full space $\mathbb{R}^d$, avoiding artificial periodisation and the associated distortion of the dynamics incurred by domain truncation in Fourier methods. Within this framework we provide a rigorous fully discrete convergence analysis of a splitting method for the cubic nonlinear Schr\"odinger equation. In addition, by combining the Hermite basis with a gauge transform, we introduce a novel, fully explicit, unconditionally stable numerical method for the derivative nonlinear Schr\"odinger equation. Our theoretical results are supported with numerical examples across various nonlinearities and dimensions, which showcase the accuracy, stability and resulting efficiency of this Hermite basis approach.

math.NA↗

arXiv2025

Common Task Framework For a Critical Evaluation of Scientific Machine Learning Algorithms

Machine learning (ML) is transforming modeling and control in the physical, engineering, and biological sciences. However, rapid development has outpaced the creation of standardized, objective benchmarks - leading to weak baselines, reporting bias, and inconsistent evaluations across methods. This undermines reproducibility, misguides resource allocation, and obscures scientific progress. To address this, we propose a Common Task Framework (CTF) for scientific machine learning. The CTF features a curated set of datasets and task-specific metrics spanning forecasting, state reconstruction, and generalization under realistic constraints, including noise and limited data. Inspired by the success of CTFs in fields like natural language processing and computer vision, our framework provides a structured, rigorous foundation for head-to-head evaluation of diverse algorithms. As a first step, we benchmark methods on two canonical nonlinear systems: Kuramoto-Sivashinsky and Lorenz. These results illustrate the utility of the CTF in revealing method strengths, limitations, and suitability for specific classes of problems and diverse objectives. Next, we are launching a competition around a global real world sea surface temperature dataset with a true holdout dataset to foster community engagement. Our long-term vision is to replace ad hoc comparisons with standardized evaluations on hidden test sets that raise the bar for rigor and reproducibility in scientific ML.

cs.CE↗

arXiv2025

Privacy-Preserving Generative Modeling and Clinical Validation of Longitudinal Health Records for Chronic Disease

Data privacy is a critical challenge in modern medical workflows as the adoption of electronic patient records has grown rapidly. Stringent data protection regulations limit access to clinical records for training and integrating machine learning models that have shown promise in improving diagnostic accuracy and personalized care outcomes. Synthetic data offers a promising alternative; however, current generative models either struggle with time-series data or lack formal privacy guaranties. In this paper, we enhance a state-of-the-art time-series generative model to better handle longitudinal clinical data while incorporating quantifiable privacy safeguards. Using real data from chronic kidney disease and ICU patients, we evaluate our method through statistical tests, a Train-on-Synthetic-Test-on-Real (TSTR) setup, and expert clinical review. Our non-private model (Augmented TimeGAN) outperforms transformer- and flow-based models on statistical metrics in several datasets, while our private model (DP-TimeGAN) maintains a mean authenticity of 0.778 on the CKD dataset, outperforming existing state-of-the-art models on the privacy-utility frontier. Both models achieve performance comparable to real data in clinician evaluations, providing robust input data necessary for developing models for complex chronic conditions without compromising data privacy.

cs.LG↗

arXiv2025

Symmetric resonance based integrators and forest formulae

In the present work we introduce a unified framework that allows for the very first systematic construction of symmetric resonance-based integrators to approximate a wide class of nonlinear dispersive equations at low-regularity. The inclusion of symmetries in the construction of resonance-based schemes presents serious challenges and induces a need for a significant extension of prior approaches to allow for sufficient number of degrees of freedom in the resulting schemes while preserving the favorable low-regularity convergence properties of prior constructions. Motivated by recent work arXiv:2005.01649, we achieve this by introducing a novel formalism based on forest formulae that allows us to encode a wider range of possibilities of iterating Duhamel's formula and interpolatory approximations of lower order parts in the construction of these time-stepping methods. The forest formulae allow for a simple characterisation of symmetric schemes and provides a fascinating algebraic structure in its own right which echo those used in Quantum Field Theory for renormalising Feynman diagrams and those used for the renormalisation of singular SPDEs via the theory of Regularity Structures. Our constructions lead to the development of several new symmetric low regularity integrators that exhibit remarkable structure preservation and convergence properties which are witnessed in numerical experiments.

math.NA↗

arXiv2025

G-Adaptivity: optimised graph-based mesh relocation for finite element methods

We present a novel, and effective, approach to achieve optimal mesh relocation in finite element methods (FEMs). The cost and accuracy of FEMs is critically dependent on the choice of mesh points. Mesh relocation (r-adaptivity) seeks to optimise the mesh geometry to obtain the best solution accuracy at given computational budget. Classical r-adaptivity relies on the solution of a separate nonlinear "meshing" PDE to determine mesh point locations. This incurs significant cost at remeshing, and relies on estimates that relate interpolation- and FEM-error. Recent machine learning approaches have focused on the construction of fast surrogates for such classical methods. Instead, our new approach trains a graph neural network (GNN) to determine mesh point locations by directly minimising the FE solution error from the PDE system Firedrake to achieve higher solution accuracy. Our GNN architecture closely aligns the mesh solution space to that of classical meshing methodologies, thus replacing classical estimates for optimality with a learnable strategy. This allows for rapid and robust training and results in an extremely efficient and effective GNN approach to online r-adaptivity. Our method outperforms both classical, and prior ML, approaches to r-adaptive meshing. In particular, it achieves lower FE solution error, whilst retaining the significant speed-up over classical methods observed in prior ML work.

cs.LG↗

arXiv2025

A Wong--Zakai resonance-based integrator for nonlinear Schrödinger equation with white noise dispersion

We introduce a novel approach to numerical approximation of nonlinear Schrödinger equation with white noise dispersion in the regime of low-regularity solutions. Approximating such solutions in the stochastic setting is particularly challenging due to randomized frequency interactions and presents a compelling challenge for the construction of tailored schemes. In particular, we design the first resonance-based schemes for this equation, which achieve provable convergence for solutions of much lower regularity than previously required. A crucial ingredient in this construction is the Wong--Zakai approximation of stochastic dispersive system, which introduces piecewise linear phases that capture nonlinear frequency interactions and can subsequently be approximated to construct resonance-based schemes. We prove the well-posedness of the Wong--Zakai approximated equation and establish its proximity to the original full stochastic dispersive system. Based on this approximation, we demonstrate an improved strong convergence rate for our new scheme, which exploits the stochastic nature of the dispersive terms. Finally, we provide numerical experiments underlining the favourable performance of our novel method in practice.

math.NA↗

arXiv2025

Fully discrete backward error analysis for the midpoint rule applied to the nonlinear Schroedinger equation

The use of symplectic numerical schemes on Hamiltonian systems is widely known to lead to favorable long-time behaviour. While this phenomenon is thoroughly understood in the context of finite-dimensional Hamiltonian systems, much less is known in the context of Hamiltonian PDEs. In this work we provide the first dimension-independent backward error analysis for a Runge-Kutta-type method, the midpoint rule, which shows the existence of a modified energy for this method when applied to nonlinear Schroedinger equations regardless of the level of spatial discretisation. We use this to establish long-time stability of the numerical flow for the midpoint rule.

math.AP↗

arXiv2025

Explicit symmetric low-regularity integrators for the nonlinear Schrödinger equation

The numerical approximation of low-regularity solutions to the nonlinear Schrödinger equation is notoriously difficult and even more so if structure-preserving schemes are sought. Recent works have been successful in establishing symmetric low-regularity integrators for this equation. However, so far, all prior symmetric low-regularity algorithms are fully implicit, and therefore require the solution of a nonlinear equation at each time step, leading to significant numerical cost in the iteration. In this work, we introduce the first fully explicit (multi-step) symmetric low-regularity integrators for the nonlinear Schrödinger equation. We demonstrate the construction of an entire class of such schemes which notably can be used to symmetrise (in explicit form) a large amount of existing low-regularity integrators. We provide rigorous convergence analysis of our schemes and numerical examples demonstrating both the favourable structure preservation properties obtained with our novel schemes, and the significant reduction in computational cost over implicit methods.

math.NA↗

arXiv2025

An accelerated Levin-Clenshaw-Curtis method for the evaluation of highly oscillatory integrals

The efficient approximation of highly oscillatory integrals plays an important role in a wide range of applications. Whilst traditional quadrature becomes prohibitively expensive in the high-frequency regime, Levin methods provide a way to approximate these integrals in many settings at uniform cost. In this work, we present an accelerated version of Levin methods that can be applied to a wide range of physically important oscillatory integrals, by exploiting the banded action of certain differential operators on a Chebyshev polynomial basis. Our proposed version of the Levin method can be computed essentially in just $\mathcal{O}(ν\logν)$ operations, where $ν$ is the number of quadrature points and the dependence of the cost on a number of additional parameters is made explicit in the manuscript. This presents a significant speed-up over the direct computation of the Levin method in current state-of-the-art. We outline the construction of this accelerated method for a fairly broad class of integrals and support our theoretical description with a number of illustrative numerical examples.

math.NA↗

arXiv2025

A fast neural hybrid Newton solver adapted to implicit methods for nonlinear dynamics

The use of implicit time-stepping schemes for the numerical approximation of solutions to stiff nonlinear time-evolution equations brings well-known advantages including, typically, better stability behaviour and corresponding support of larger time steps, and better structure preservation properties. However, this comes at the price of having to solve a nonlinear equation at every time step of the numerical scheme. In this work, we propose a novel deep learning based hybrid Newton's method to accelerate this solution of the nonlinear time step system for stiff time-evolution nonlinear equations. We propose a targeted learning strategy which facilitates robust unsupervised learning in an offline phase and provides a highly efficient initialisation for the Newton iteration leading to consistent acceleration of Newton's method. A quantifiable rate of improvement in Newton's method achieved by improved initialisation is provided and we analyse the upper bound of the generalisation error of our unsupervised learning strategy. These theoretical results are supported by extensive numerical results, demonstrating the efficiency of our proposed neural hybrid solver both in one- and two-dimensional cases.

math.NA↗

arXiv2024

Bridging the gap: symplecticity and low regularity in Runge-Kutta resonance-based schemes

Recent years have seen an increasing amount of research devoted to the development of so-called resonance-based methods for dispersive nonlinear partial differential equations. In many situations, this new class of methods allows for approximations in a much more general setting (e.g. for rough data) than, for instance, classical splitting or exponential integrator methods. However, they lack one important property: the preservation of geometric properties of the flow. This is particularly drastic in the case of the Korteweg-de Vries (KdV) equation and the nonlinear Schrödinger equation (NLSE) which are fundamental models in the broad field of dispersive infinite-dimensional Hamiltonian systems, possessing infinitely many conserved quantities, an important property which we wish to capture - at least up to some degree - also on the discrete level. Nowadays, a wide range of structure preserving integrators for Hamiltonian systems are available, however, typically these existing algorithms can only approximate highly regular solutions efficiently. State-of-the-art low-regularity integrators, on the other hand, poorly preserve the geometric structure of the underlying PDE. In this work we introduce a novel framework, so-called Runge-Kutta resonance-based methods, for a large class of dispersive nonlinear equations which incorporate a much larger amount of degrees of freedom than prior resonance-based schemes while featuring similarly favourable low-regularity convergence properties. In particular, for the KdV and NLSE case, we are able to bridge the gap between low regularity and structure preservation by characterising a large class of symplectic (in the Hamiltonian picture) resonance-based methods for both equations that allow for low-regularity approximations to the solution while preserving the underlying geometric structure of the continuous problem on the discrete level.

math.NA↗

arXiv2023

Numerical integration of Schrödinger maps via the Hasimoto transform

We introduce a numerical approach to computing the Schrödinger map (SM) based on the Hasimoto transform which relates the SM flow to a cubic nonlinear Schrödinger (NLS) equation. In exploiting this nonlinear transform we are able to introduce the first fully explicit unconditionally stable symmetric integrators for the SM equation. Our approach consists of two parts: an integration of the NLS equation followed by the numerical evaluation of the Hasimoto transform. Motivated by the desire to study rough solutions to the SM equation, we also introduce a new symmetric low-regularity integrator for the NLS equation. This is combined with our novel fast low-regularity Hasimoto (FLowRH) transform, based on a tailored analysis of the resonance structures in the Magnus expansion and a fast realisation based on block-Toeplitz partitions, to yield an efficient low-regularity integrator for the SM equation. This scheme in particular allows us to obtain approximations to the SM in a more general regime (i.e. under lower regularity assumptions) than previously proposed methods. The favorable properties of our methods are exhibited both in theoretical convergence analysis and in numerical experiments.

math.NA↗

arXiv2023

Long-time error bounds of low-regularity integrators for nonlinear Schrödinger equations

We introduce a new non-resonant low-regularity integrator for the cubic nonlinear Schrödinger equation (NLSE) allowing for long-time error estimates which are optimal in the sense of the underlying PDE. The main idea thereby lies in treating the zeroth mode exactly within the discretization. For long-time error estimates, we rigorously establish the long-time error bounds of different low-regularity integrators for the nonlinear Schrödinger equation (NLSE) with small initial data characterized by a dimensionless parameter $\varepsilon \in (0, 1]$. We begin with the low-regularity integrator for the quadratic NLSE in which the integral is computed exactly and the improved uniform first-order convergence in $H^r$ is proven at $O(\varepsilon τ)$ for solutions in $H^r$ with $r > 1/2$ up to the time $T_{\varepsilon } = T/\varepsilon $ with fixed $T > 0$. Then, the improved uniform long-time error bound is extended to a symmetric second-order low-regularity integrator in the long-time regime. For the cubic NLSE, we design new non-resonant first-order and symmetric second-order low-regularity integrators which treat the zeroth mode exactly and rigorously carry out the error analysis up to the time $T_{\varepsilon } = T/\varepsilon ^2$. With the help of the regularity compensation oscillation (RCO) technique, the improved uniform error bounds are established for the new non-resonant low-regularity schemes, which further reduce the long-time error by a factor of $\varepsilon^2$ compared with classical low-regularity integrators for the cubic NLSE. Numerical examples are presented to validate the error estimates and compare with the classical time-splitting methods in the long-time simulations.

math.NA↗