CyberSec Research

Can regularization terms in the training of invertible neural networks lead to known Bayesian point estimators in reconstruction? Invertible networks are attractive for inverse problems due to their inherent stability and interpretability. Recently, optimization strategies for invertible neural networks that approximate either a reconstruction map or the forward operator have been studied from a Bayesian perspective, but each has limitations. To address this, we introduce and analyze two regularization terms for the network training that, upon inversion of the network, recover properties of classical Bayesian point estimators: while the first can be connected to the posterior mean, the second resembles the MAP estimator. Our theoretical analysis characterizes how each loss shapes both the learned forward operator and its inverse reconstruction map. Numerical experiments support our findings and demonstrate how these loss-term regularizers introduce data-dependence in a stable and interpretable way.

We propose and justify a new approach for fast calculation of the electrostatic interaction energy of clusters of charged particles in constrained energy minimization in the framework of rigid protein-ligand docking. Our ``blind search'' docking technique is based on the low-rank range-separated (RS) tensor-based representation of the free-space electrostatic potential of the biomolecule represented on large $n\times n\times n$ 3D grid. We show that both the collective electrostatic potential of a complex protein-ligand system and the respective electrostatic interaction energy can be calculated by tensor techniques in $O(n)$-complexity, such that the numerical cost for energy calculation only mildly (logarithmically) depends on the number of particles in the system. Moreover, tensor representation of the electrostatic potential enables usage of large 3D Cartesian grids (of the order of $n^3 \sim 10^{12}$), which could allow the accurate modeling of complexes with several large proteins. In our approach selection of the correct geometric pose predictions in the localized posing process is based on the control of van der Waals distance between the target molecular clusters. Here, we confine ourselves by constrained minimization of the energy functional by using only fast tensor-based free-space electrostatic energy recalculation for various rotations and translations of both clusters. Numerical tests of the electrostatic energy-based ``protein-ligand docking'' algorithm applied to synthetic and realistic input data present a proof of concept for rather complex particle configurations. The method may be used in the framework of the traditional stochastic or deterministic posing/docking techniques.

We develop an accelerated algorithm for computing an approximate eigenvalue decomposition of bistochastic normalized kernel matrices. Our approach constructs a low rank approximation of the original kernel matrix by the pivoted partial Cholesky algorithm and uses it to compute an approximate decomposition of its bistochastic normalization without requiring the formation of the full kernel matrix. The cost of the proposed algorithm depends linearly on the size of the employed training dataset and quadratically on the rank of the low rank approximation, offering a significant cost reduction compared to the naive approach. We apply the proposed algorithm to the kernel based extraction of spatiotemporal patterns from chaotic dynamics, demonstrating its accuracy while also comparing it with an alternative algorithm consisting of subsampling and Nystroem extension.

Coarse spaces are essential to ensure robustness w.r.t. the number of subdomains in two-level overlapping Schwarz methods. Robustness with respect to the coefficients of the underlying partial differential equation (PDE) can be achieved by adaptive (or spectral) coarse spaces involving the solution of local eigenproblems. The solution of these eigenproblems, although scalable, entails a large setup cost which may exceed the cost for the iteration phase. In this paper we present and analyse a new variant of the GenEO (Generalised Eigenproblems in the Overlap) coarse space which involves solving eigenproblems only in a strip connected to the boundary of the subdomain. This leads to a significant reduction of the setup cost while the method satisfies a similar coefficient-robust condition number estimate as the original method, albeit with a possibly larger coarse space.

It is well known that phase formation by electrodeposition yields films of poorly controllable morphology. This typically leads to a range of technological issues in many fields of electrochemical technology. Presently, a particularly relevant case is that of high-energy density next-generation batteries with metal anodes, that cannot yet reach practical cyclability targets, owing to uncontrolled elelctrode shape evolution. In this scenario, mathematical modelling is a key tool to lay the knowledge-base for materials-science advancements liable to lead to concretely stable battery material architectures. In this work, we introduce the Evolving Surface DIB (ESDIB) model, a reaction-diffusion system posed on a dynamically evolving electrode surface. Unlike previous fixed-surface formulations, the ESDIB model couples surface evolution to the local concentration of electrochemical species, allowing the geometry of the electrode itself to adapt in response to deposition. To handle the challenges related to the coupling between surface motion and species transport, we numerically solve the system by proposing an extension of the Lumped Evolving Surface Finite Element Method (LESFEM) for spatial discretisation, combined with an IMEX Euler scheme for time integration. The model is validated through six numerical experiments, each compared with laboratory images of electrodeposition. Results demonstrate that the ESDIB framework accurately captures branching and dendritic growth, providing a predictive and physically consistent tool for studying metal deposition phenomena in energy storage devices.

In this paper, we develop a geometric, structure-preserving semi-discrete formulation of Maxwell's equations in both three- and two-dimensional settings within the framework of discrete exterior calculus. This approach preserves the intrinsic geometric and topological structures of the continuous theory while providing a consistent spatial discretization. We analyze the essential properties of the proposed semi-discrete model and compare them with those of the classical Maxwell's equations. As a special case, the model is illustrated on a combinatorial two-dimensional torus, where the semi-discrete Maxwell's equations take the form of a system of first-order linear ordinary differential equations. An explicit expression for the general solution of this system is also derived.

We derive explicit a priori consistency error estimates for a standard finite element discretization of the Poisson equation on convex domains, where the domain is approximated by an internal convex polyhedron. The obtained explicit estimates depend only on global geometric parameters and are applicable to general convex domains and arbitrary families of simplicial meshes.

We consider the minimization of integral functionals in one dimension and their approximation by $r$-adaptive finite elements. Including the grid of the FEM approximation as a variable in the minimization, we are able to show that the optimal grid configurations have a well-defined limit when the number of nodes in the grid is being sent to infinity. This is done by showing that the suitably renormalized energy functionals possess a limit in the sense of $\Gamma$-convergence. We provide numerical examples showing the closeness of the optimal asymptotic mesh obtained as a minimizer of the $\Gamma$-limit to the optimal finite meshes.

In this paper, local H\"older regularization is incorporated into a physics-informed neural networks (PINNs) framework for solving elliptic partial differential equations (PDEs). Motivated by the interior regularity properties of linear elliptic PDEs, a modified loss function is constructed by introducing local H\"older regularization term. To approximate this term effectively, a variable-distance discrete sampling strategy is developed. Error estimates are established to assess the generalization performance of the proposed method. Numerical experiments on a range of elliptic problems demonstrate notable improvements in both prediction accuracy and robustness compared to standard physics-informed neural networks.

This paper deals with the numerical simulation of the 2D magnetic time-dependent Ginzburg-Landau (TDGL) equations in the regime of small but finite (inverse) Ginzburg-Landau parameter $\epsilon$ and constant (order $1$ in $\epsilon$) applied magnetic field. In this regime, a well-known feature of the TDGL equation is the appearance of quantized vortices with core size of order $\epsilon$. Moreover, in the singular limit $\epsilon \searrow 0$, these vortices evolve according to an explicit ODE system. In this work, we first introduce a new numerical method for the numerical integration of this limiting ODE system, which requires to solve a linear second order PDE at each time step. We also provide a rigorous theoretical justification for this method that applies to a general class of 2D domains. We then develop and analyze a numerical strategy based on the finite-dimensional ODE system to efficiently simulate the infinite-dimensional TDGL equations in the presence of a constant external magnetic field and for small, but finite, $\epsilon$. This method allows us to avoid resolving the $\epsilon$-scale when solving the TDGL equations, where small values of $\epsilon$ typically require very fine meshes and time steps. We provide numerical examples on a few test cases and justify the accuracy of the method with numerical investigations.

Inverse source localization from Helmholtz boundary data collected over a narrow aperture is highly ill-posed and severely undersampled, undermining classical solvers (e.g., the Direct Sampling Method). We present a modular framework that significantly improves multi-source localization from extremely sparse single-frequency measurements. First, we extend a uniqueness theorem for the inverse source problem, proving that a unique solution is guaranteed under limited viewing apertures. Second, we employ a Deep Operator Network (DeepONet) with a branch-trunk architecture to interpolate the sparse measurements, lifting six to ten samples within the narrow aperture to a sufficiently dense synthetic aperture. Third, the super-resolved field is fed into the Direct Sampling Method (DSM). For a single source, we derive an error estimate showing that sparse data alone can achieve grid-level precision. In two- and three-source trials, localization from raw sparse measurements is unreliable, whereas DeepONet-reconstructed data reduce localization error by about an order of magnitude and remain effective with apertures as small as $\pi/4$. By decoupling interpolation from inversion, the framework allows the interpolation and inversion modules to be swapped with neural operators and classical algorithms, respectively, providing a practical and flexible design that improves localization accuracy compared with standard baselines.

We analyze single-core and split-core defect structures in nematic liquid crystals within the Landau-de Gennes framework by studying minimizers of the associated energy functional. A bifurcation occurs at a critical temperature threshold, below which both split-core and single-core configurations are solutions to the Euler-Lagrange equation, with the split-core defect possessing lower energy. Above the threshold, the split-core configuration vanishes, leaving the single-core defect as the only stable solution. We analyze the dependence of such temperature threshold on the domain size and characterize the nature of the transition between the two defect types. We carry out a quantitative study of defect core sizes as functions of temperature and domain size for both single and split core defects.

To address the magnetization dynamics in ferromagnetic materials described by the Landau-Lifshitz-Gilbert equation under large damping parameters, a third-order accurate numerical scheme is developed by building upon a second-order method \cite{CaiChenWangXie2022} and leveraging its efficiency. This method boasts two key advantages: first, it only involves solving linear systems with constant coefficients, enabling the use of fast solvers and thus significantly enhancing numerical efficiency over existing first or second-order approaches. Second, it achieves third-order temporal accuracy and fourth-order spatial accuracy, while being unconditionally stable for large damping parameters. Numerical tests in 1D and 3D scenarios confirm both its third-order accuracy and efficiency gains. When large damping parameters are present, the method demonstrates unconditional stability and reproduces physically plausible structures. For domain wall dynamics simulations, it captures the linear relationship between wall velocity and both the damping parameter and external magnetic field, outperforming lower-order methods in this regard.

This paper is dedicated to enhancing the computational efficiency of traditional parallel-in-time methods for solving stochastic initial-value problems. The standard parareal algorithm often suffers from slow convergence when applied to problems with stochastic inputs, primarily due to the poor quality of the initial guess. To address this issue, we propose a hybrid parallel algorithm, termed KLE-CGC, which integrates the Karhunen-Lo\`{e}ve (KL) expansion with the coarse grid correction (CGC). The method first employs the KL expansion to achieve a low-dimensional parameterization of high-dimensional stochastic parameter fields. Subsequently, a generalized Polynomial Chaos (gPC) spectral surrogate model is constructed to enable rapid prediction of the solution field. Utilizing this prediction as the initial value significantly improves the initial accuracy for the parareal iterations. A rigorous convergence analysis is provided, establishing that the proposed framework retains the same theoretical convergence rate as the standard parareal algorithm. Numerical experiments demonstrate that KLE-CGC maintains the same convergence order as the original algorithm while substantially reducing the number of iterations and improving parallel scalability.

This study presents a meshfree two-dimensional fractional-order Element-Free Galerkin (2D f-EFG) method as a viable alternative to conventional mesh-based FEM for a numerical solution of (spatial) fractional-order differential equations (FDEs). The previously developed one-dimensional f-EFG solver offers a limited demonstration of the true efficacy of EFG formulations for FDEs, as it is restricted to simple 1D line geometries. In contrast, the 2D f-EFG solver proposed and developed here effectively demonstrates the potential of meshfree approaches for solving FDEs. The proposed solver can handle complex and irregular 2D domains that are challenging for mesh-based methods. As an example, the developed framework is employed to investigate nonlocal elasticity governed by fractional-order constitutive relations in a square and circular plate. Furthermore, the proposed approach mitigates key drawbacks of FEM, including high computational cost, mesh generation, and reduced accuracy in irregular domains. The 2D f-EFG employs 2D Moving Least Squares (MLS) approximants, which are particularly effective in approximating fractional derivatives from nodal values. The 2D f-EFG solver is employed here for the numerical solution of fractional-order linear and nonlinear partial differential equations corresponding to the nonlocal elastic response of a plate. The solver developed here is validated with the benchmark results available in the literature. While the example chosen here focuses on nonlocal elasticity, the numerical method can be extended for diverse applications of fractional-order derivatives in multiscale modeling, multiphysics coupling, anomalous diffusion, and complex material behavior.

A domain decomposition method for the solution of general variable-coefficient elliptic partial differential equations on regular domains is introduced. The method is based on tessellating the domain into overlapping thin slabs or shells, and then explicitly forming a reduced linear system that connects the different domains. Rank-structure ('H-matrix structure') is exploited to handle the large dense blocks that arise in the reduced linear system. Importantly, the formulation used is well-conditioned, as it converges to a second kind Fredholm equation as the precision in the local solves is refined. Moreover, the dense blocks that arise are far more data-sparse than in existing formulations, leading to faster and more efficient H-matrix arithmetic. To form the reduced linear system, black-box randomized compression is used, taking full advantage of the fact that sparse direct solvers are highly efficient on the thin sub-domains. Numerical experiments demonstrate that our solver can handle oscillatory 2D and 3D problems with as many as 28 million degrees of freedom.

In this study, the capabilities of the Physics-Informed Neural Network (PINN) method are investigated for three major tasks: modeling, simulation, and optimization in the context of the heat conduction problem. In the modeling phase, the governing equation of heat transfer by conduction is reconstructed through equation discovery using fractional-order derivatives, enabling the identification of the fractional derivative order that best describes the physical behavior. In the simulation phase, the thermal conductivity is treated as a physical parameter, and a parametric simulation is performed to analyze its influence on the temperature field. In the optimization phase, the focus is placed on the inverse problem, where the goal is to infer unknown physical properties from observed data. The effectiveness of the PINN approach is evaluated across these three fundamental engineering problem types and compared against conventional numerical methods. The results demonstrate that although PINNs may not yet outperform traditional numerical solvers in terms of speed and accuracy for forward problems, they offer a powerful and flexible framework for parametric simulation, optimization, and equation discovery, making them highly valuable for inverse and data-driven modeling applications.

The main ideas behind a research plan to use the Wigner formulation as a bridge between classical and quantum probabilistic algorithms are presented, focusing on a particular case: the Quantum analog of Stochastic Gradient Descent in its continuous-time limit based on the Wigner formulation of Open Quantum Systems.