arXiv subjects

Matthias Ehrhardt

Publications and source records attributed to Matthias Ehrhardt.

At least 19 recordsLinked to original sources

arXiv2026

Structure-Preserving Data-Driven Identification of Port-Hamiltonian Differential-Algebraic Systems

We present a data-driven approach to identifying linear index-1 differential-algebraic pH systems (pH-DAEs) based on input-output measurements. In comparison to the identification of port-Hamiltonian (pH) systems, the algebraic constraint and the index condition pose additional challenges. First, we establish a structure-preserving formulation of the considered pH-DAE class and derive an implicit midpoint discretization that preserves the algebraic constraints and discrete dissipation inequality. We formulate the identification problem as a regularized least-squares minimization problem subject to the pH-DAE dynamics. Exploiting the index-1 structure, we reduce the constrained problem to an unconstrained optimization problem over the system parameters while preserving the port-Hamiltonian structure. Next, we derive an adjoint-based formulation to efficiently evaluate the gradient of the resulting reduced cost functional. This enables us to use gradient-based optimization methods for parameter estimation. Under suitable assumptions on the admissible parameter set, the existence of a minimizer is established. Numerical experiments demonstrate that the proposed approach can identify surrogate pH-DAE systems that accurately reproduce the input-output behavior of reference systems. Further investigations show the approach's potential for identifying reduced-order surrogate models. Cross-validation with independent input signals confirms the predictive capability of the identified models.

math.NA↗

arXiv2026

Transparent Boundary Conditions for the Heat Equation on Metric Graphs

We study transparent boundary conditions (TBCs) for the time-dependent heat equation in a branching, quasi-one-dimensional domain modeled as a metric star graph. By combining the classical concept of TBCs with the theory of partial differential equations (PDEs) on networks, we derive an exact vertex condition that ensures unobstructed thermal flow across junctions. Specifically,we derive a sum rule for the diffusion coefficients that eliminates thermal backflow at the vertex.We demonstrate the validity of this analytical model numerically using a Crank-Nicolson finite-difference method, which confirms smooth, unobstructed heat propagation through the network.These results provide a practical mathematical framework for tunable control and optimization of thermal diffusion in low-dimensional structures. In particular, combining the well-known concept of the TBCs and theory for PDE on metric graphs, we propose a mathematical model providing a control tool for thermal diffusion in networks.

math-ph↗

arXiv2026

Transparent boundary conditions for the spatially discrete Schr\"odinger equation: Reflectionless quantum transport in 1D lattices

We construct exact transparent boundary conditions (TBCs) for a time-continuous, spatially discrete Schr\"odinger equation that models a one-dimensional quantum lattice. Using a recently developed exact solution for the discrete system, we derive the Dirichlet-to-Neumann maps analytically via Laplace transforms. This yields a convolution-type boundary condition governed by Bessel functions. We rigorously demonstrate the consistency of this discrete formulation with its continuous counterpart in the continuum limit. Additionally, we present an efficient time-discretization scheme based on the trapezoidal rule for practical implementation. Numerical experiments using a Crank-Nicolson solver verify that our proposed TBCs eliminate spurious backscattering entirely and preserve reflectionless propagation of a Gaussian wave packet exiting the computational domain

math-ph↗

arXiv2026

Structure-Preserving Neural ODEs via Nonstandard Finite Difference Discretization

Although neural ordinary differential equations (NODEs) are a powerful framework for learning continuous-time dynamics, they generally do not preserve essential qualitative properties, such as positivity. We propose a structure-preserving Neural ODE framework based on nonstandard finite difference (NSFD) discretization. The learned dynamics are parameterized by nonnegative production and destruction rates, yielding an explicit, differentiable update that integrates seamlessly into standard automatic differentiation pipelines. We prove that the resulting scheme unconditionally preserves positivity for arbitrary time-step sizes while retaining first-order consistency. We outline an extension based on Patankar-type discretizations that preserves conservation laws exactly. Numerical experiments on an SIR epidemic model show that our approach generates physically meaningful trajectories, remains robust under coarse discretizations, and outperforms conventional NODEs in preserving the qualitative structure of the learned dynamics.

math.NA↗

arXiv2026

A perfectly matched layer approach for the spectral split-step Pad\'e method

The split-step-Pad\'e (SSP) method is widely used to model wave phenomena in various applications, including radio physics, optics and acoustics. In this method, the propagator of the one-way counterpart of the Helmholtz equation is computed through its Pad\'e approximant and a finite-difference discretization of the transverse operator. This work develops and validates numerically a spectral counterpart of the SSP method. A key challenge in practical applications is inverting the transverse operator in the presence of perfectly matched layers (PMLs), which are commonly used to truncate the computational domain. Such inversion can be accomplished using Krylov subspace methods, which converge rapidly, provided that a suitable preconditioner is used. We also study the analytical properties of the spectral SSP marching scheme under periodicity conditions in the transverse variable. We validate the newly developed spectral SSP method numerically in two realistic test scenarios from radio physics and underwater acoustics.

math.NA↗

arXiv2026

A Nonstandard Finite Difference Scheme for a Nonlinear Parabolic Equation with p-Laplacian-Type Diffusion

We propose and analyze a nonstandard finite difference (NSFD) scheme for nonlinear parabolic equations involving a p-Laplacian-type diffusion operator in one- and two-dimensional spatial domains. Following Mickens' design principles, the proposed discretization employs a nonlinear denominator function phi(.) together with a nonlocal approximation of the nonlinear diffusion term Delta_p, yielding a structure-preserving discrete model. The scheme is designed to retain key qualitative properties of the continuous problem, including positivity, boundedness, and stability, which may be lost by standard finite difference methods (FDMs). We establish the well-posedness of the continuous model, derive the NSFD scheme, and investigate its consistency, convergence, and local truncation error. Numerical experiments confirm the theoretical results and demonstrate that, unlike the standard explicit FDM, the proposed NSFD scheme avoids spurious oscillations and nonphysical negative solutions even for relatively large time-step sizes.

math.NA↗

arXiv2026

A Fractional-Memory Physics-Informed Neural Network with Fast History Compression for Tempered Fractional Coupled Phase-Field Systems

Tempered time-fractional coupled phase-field (tTFCP) systems are used to model interfacial phenomena involving memory-dependent transport and relaxation mechanisms. Numerical solutions to these systems are challenging due to the simultaneous presence of nonlocal temporal operators, weak initial singularities, moving diffuse interfaces, and strongly coupled multiphysics dynamics. In this work, we introduce FM-tfPINN (fractional-memory physics-informed neural network), which is used for forward simulation and inverse parameter identification in tempered fractional coupled phase-field systems. Unlike conventional fractional PINNs, which enforce memory effects solely through residual constraints, our framework incorporates tempered fractional memory directly into the neural representation via latent memory-source functions and a tempered fractional integral operator. We develop a fast shifted residual formulation based on graded temporal meshes and sum-of-exponentials (SOE) history compression to efficiently evaluate the tempered fractional operators. This framework combines interface-aware and residual-adaptive collocation strategies, improving resolution near evolving diffuse interfaces. A unified, physics-informed loss formulation allows for the forward prediction and inverse recovery of unknown physical parameters from sparse observations. We assess the proposed method on a class of tempered fractional corrosion phase-field models, including one-dimensional corrosion-front propagation, activation- and diffusion-controlled regimes, two-dimensional pitting corrosion, and inverse mobility identification problems. The numerical results demonstrate the accurate recovery of coupled phase and concentration fields, the robust prediction of physically relevant interface diagnostics, and the reliable estimation of parameters from limited data.

math.NA↗

arXiv2026

Structure-Preserving Schemes for a Fractional SVIR Epidemic Model with a Hybrid Mittag-Leffler-Caputo-Fabrizio Operator

This paper proposes and analyzes a fractional-order SVIR epidemic model based on a hybrid Mittag-Leffler-Caputo-Fabrizio (MLCF) fractional operator with a nonsingular kernel. This model captures short- and long-term memory effects in epidemic transmission dynamics. The positivity and boundedness of the solutions are proven through an integrated formulation of the MLCF operator and a fractional Gronwall inequality. The basic reproduction number $\mathcal{R}_0$, equilibrium points, and their local and global stability properties are rigorously investigated through Jacobian analysis, logarithmic Lyapunov functionals, and a fractional LaSalle invariance principle. To approximate the model, a $\theta$-weighted nonstandard finite difference (NSFD) method is developed. This method preserves the continuous system's key qualitative properties, including positivity and boundedness, and is unconditionally stable in the fully implicit case. Consistency and first-order convergence are also proven. Numerical experiments, together with sensitivity and bifurcation analyses, illustrate the impact of fractional memory parameters on epidemic evolution and demonstrate the effectiveness of the proposed approach.

math.NA↗

arXiv2026

A Koopman-PINN Framework for Epidemic Models: Parameter Inference and Forecasting

We propose a Koopman-enhanced physics-informed neural network (K--PINN) framework for parameter inference and forecasting in nonlinear epidemic models. This method combines Koopman operator theory and physics-informed learning. It maps epidemic states into a latent observable space where the dynamics evolve approximately linearly while satisfying the governing epidemic equations through automatic differentiation. This integration improves interpretability, parameter identifiability, and long-term predictive stability. We apply the proposed framework to a normalized SEIRSD epidemic model and evaluate it using synthetic monkeypox (Mpox) data and real-world datasets from Germany, Morocco, and Sweden for the SARS-CoV-2 virus. Synthetic trajectories are generated using a structure-preserving, nonstandard finite difference scheme to ensure reliable training data. Numerical results demonstrate that K--PINN achieves more accurate parameter estimation, trajectory reconstruction, and long-term forecasting than classical PINNs and Koopman-EDMD approaches. These results suggest that K--PINN is an effective machine learning framework for epidemic modeling that can be extended to more complex systems.

stat.AP↗

arXiv2026

A Generalized Hacker Dynamics Model with Nonlinear Incidence: Analysis and Positivity-Preserving Numerical Simulation

We propose and analyze a generalized compartment model for hacker dynamics in cybersecurity systems. This model is an extension of a recently introduced framework, replacing the bilinear interaction term with a broad class of nonlinear incidence functions. This provides greater modeling flexibility and allows for the description of a wider range of cyber-propagation and information-spreading processes. First, we investigate the qualitative dynamics of the model. We establish the positivity and boundedness of solutions, derive the basic reproduction number, and characterize the existence of hacker-free and hacker-present equilibria. We obtain local and global asymptotic stability results, yielding a complete description of the global dynamics in terms of the basic reproduction number. Next, we develop a second-order, positivity-preserving, nonstandard finite difference (NSFD) scheme for numerical simulations. Unlike many existing NSFD approaches, which are typically first-order accurate, the proposed method achieves second-order convergence while preserving positivity for arbitrary step sizes. Furthermore, the proposed method reproduces the asymptotic stability properties of the continuous model, making it suitable for long-time simulations. Numerical experiments validate the theoretical results and demonstrate the superior accuracy and qualitative performance of the proposed scheme compared to several first-order methods.

math.NA↗

arXiv2026

Constraint-Aware Physics-Informed Neural Networks for SEIR Reaction-Diffusion Epidemic Models with Vital Dynamics

Reaction-diffusion epidemic models with vital dynamics are an important framework for describing the spatial and temporal spread of infectious diseases. In this work, we present a constraint-aware, physics-informed neural network (PINN) approach to an SEIR reaction-diffusion system with homogeneous Neumann boundary conditions. Due to the scarcity of spatial epidemiological datasets, we generate synthetic benchmark data using structure-preserving implicit-explicit nonstandard finite difference (NSFD) schemes that ensure positivity, boundedness, and numerical stability. The PINN framework integrates PDE residuals, observational data, boundary conditions, and epidemiological constraints within a unified optimization procedure. Specifically, the loss function incorporates the non-negativity of compartment populations and the admissibility of epidemiological parameters. We apply the method to forward simulation and inverse parameter estimation in one- and two-dimensional settings. Numerical experiments demonstrate the framework's ability to accurately reconstruct spatiotemporal epidemic dynamics and reliably identify parameters, even when data is sparse or noisy. These results underscore the potential of constraint-aware PINNs as a robust, data-driven methodology for spatial epidemic modeling.

math.DS↗

arXiv2026

Alikhanov-XfPINNs: Adaptive Physics-Informed Learning for Nonlinear Fractional PDEs on Nonuniform Meshes

To address the initial singularity inherent in solutions to fractional partial differential equations (fPDEs), we propose an accelerated Alikhanov discretization formulation implemented on nonuniform time grids. Based on the physics-informed neural networks (PINNs) framework, we introduce an Alikhanov-extended fractional PINNs (XfPINNs) architecture that combines high-order temporal discretization and deep learning. The nonlocal memory term in fPDEs leads to high computational cost, while the weak singularity near $t\to 0^+$ can deteriorate accuracy on uniform meshes. To separate temporal discretization effects from optimization and sampling errors, we further develop an auxiliary time-marching configuration that enables auditable temporal-convergence studies under controlled training tolerances. This architecture can solve general nonlinear fPDEs. The XfPINNs approach is designed for forward and inverse problems, allowing for data-driven solution reconstruction and parameter estimation. First, the neural network approximates the solution of nonlinear fPDEs; then, an adaptive activation function accelerates convergence and enhances training efficiency. The optimization framework embeds a variational loss function constructed from the Alikhanov scheme, where the initial and boundary conditions are imposed using a combination of hard and soft constraints. Numerical experiments, including cases with known and unknown exact solutions which demonstrate the robustness, computational efficiency, and significant CPU time savings of the Alikhanov-XfPINNs method.

math.NA↗

arXiv2026

Stochastic Port-Hamiltonian Neural Networks: Universal Approximation with Passivity Guarantees

Stochastic port-Hamiltonian systems represent open dynamical systems with dissipation, inputs, and stochastic forcing in an energy based form. We introduce stochastic port-Hamiltonian neural networks, SPH-NNs, which parameterize the Hamiltonian with a feedforward network and enforce skew symmetry of the interconnection matrix and positive semidefiniteness of the dissipation matrix. For It\^o dynamics we establish a weak passivity inequality in expectation under an explicit generator condition, stated for a stopped process on a compact set. We also prove a universal approximation result showing that, on any compact set and finite horizon, SPH-NNs approximate the coefficients of a target stochastic port-Hamiltonian system with $C^2$ accuracy of the Hamiltonian and yield coupled solutions that remain close in mean square up to the exit time. Experiments on noisy mass spring, Duffing, and Van der Pol oscillators show improved long horizon rollouts and reduced energy error relative to a multilayer perceptron baseline.

cs.LG↗

arXiv2025

Structure-Preserving Coupling and Decoupling of Port-Hamiltonian Systems

The port-Hamiltonian framework is a structure-preserving modeling approach that preserves key physical properties such as energy conservation and dissipation. When subsystems are modeled as port-Hamiltonian systems (pHS) with linearly related inputs and outputs, their interconnection remains port-Hamiltonian. This paper introduces a systematic method for transforming coupled port Hamiltonian ordinary differential equations systems (pHODE) into a single monolithic formulation, and for decomposing a monolithic system into weakly coupled subsystems. The monolithic representation ensures stability and structural integrity, whereas the decoupled form enables efficient distributed simulation via operator splitting or dynamic iteration.

math.DS↗

arXiv2025

A Spectral Split-Step Pad\'e Method for Guided Wave Propagation

In this study, a Fourier-based, split-step Pad\'e (SSP) method for solving the parabolic wave equation with applications in guided wave propagation in ocean acoustics is presented. Traditional SSP implementations rely in finite-difference discretizations of the depth-dependent differential operator. This approach limits accuracy in coarse discretizations as well as computational efficiency in dense discretizations since it does not significantly benefit from parallelization. In contrast, our proposed method replaces finite differences with a spectral representation using the discrete sine transform (DST). This enables an exact treatment of the vertical operator under homogeneous boundary conditions. For non-constant sound speed, we use a Neumann series expansion to treat inhomogeneities as perturbations. Numerical experiments demonstrate the method's accuracy in range-independent media and rage-dependent scenarios, including propagation in deep ocean with Munk profile and in the presence of a parametrized synoptic eddy. Compared to finite-difference SSP methods, the Fourier-based approach achieves higher accuracy with fewer depth discretization points and avoids the resolution bottleneck associated with sharp field features, making it well-suited for large-scale, high-frequency wave propagation problems in ocean environments.

math.NA↗

arXiv2025

A Generalized Second-Order Positivity-Preserving Numerical Method for Non-Autonomous Dynamical Systems with Applications

In this work, we propose a generalized, second-order, nonstandard finite difference (NSFD) method for non-autonomous dynamical systems. The proposed method combines the NSFD framework with a new non-local approximation of the right-hand side function. This method achieves second-order convergence and unconditionally preserves the positivity of solutions for all step sizes. Especially, it avoids the restrictive conditions required by many existing positivity-preserving, second-order NSFD methods. The method is easy to implement and computationally efficient. Numerical experiments, including an improved NSFD scheme for an SIR epidemic model, confirm the theoretical results. Additionally, we demonstrate the method's applicability to nonlinear partial differential equations and boundary value problems with positive solutions, showcasing its versatility in real-world modeling.

math.NA↗

arXiv2025

Port-Hamiltonian Neural Networks: From Theory to Simulation of Interconnected Stochastic Systems

This work introduces a new framework integrating port-Hamiltonian systems (PHS) and neural network architectures. This framework bridges the gap between deterministic and stochastic modeling of complex dynamical systems. We introduce new mathematical formulations and computational methods that expand the geometric structure of PHS to account for uncertainty, environmental noise, and random perturbations. Building on these advances, we introduce stochastic port-Hamiltonian neural networks (pHNNs), which facilitate the accurate learning and prediction of non-autonomous and interconnected stochastic systems. Our proposed framework generalizes passivity concepts to the stochastic regime, ensuring stability while maintaining the system's energy-consistent structure. Extensive simulations, including those involving damped mass-spring systems, Duffing oscillators, and robotic control tasks, demonstrate the capability of pHNNs to capture complex dynamics with high fidelity, even under noise and uncertainty. This unified approach establishes a foundation for the robust, data-driven modeling and control of nonlinear stochastic systems.

math-ph↗

arXiv2025

A Koopman Operator Framework for Nonlinear Epidemic Dynamics: Application to an SIRSD Model

We develop and analyze an SIRSD epidemic model, which extends the classical SIR framework by incorporating waning immunity and disease-induced mortality. A rigorous well-posedness analysis ensures the existence, uniqueness, positivity, and boundedness of solutions, guaranteeing the model's epidemiological feasibility. To facilitate theoretical investigations and data-driven modeling, we reformulated the system in normalized variables. To capture and predict complex nonlinear epidemic dynamics, we use the Koopman operator framework with extended dynamic mode decomposition (EDMD) and an epidemiologically informed dictionary of observables. We compare two Koopman approximations: one based on a minimal epidemiological dictionary and another enriched with nonlinear and cross terms. We generate synthetic data using a nonstandard finite difference (NSFD) scheme for four representative epidemics: SARS-CoV-2, seasonal influenza, Ebola, and measles. Numerical experiments demonstrate that the Koopman-based approach effectively identifies dominant epidemic modes and accurately predicts key outbreak characteristics, including peak infection dynamics.

math.DS↗