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In recent years, deep learning has gained increasing popularity in the fields of Partial Differential Equations (PDEs) and Reduced Order Modeling (ROM), providing domain practitioners with new powerful data-driven techniques such as Physics-Informed Neural Networks (PINNs), Neural Operators, Deep Operator Networks (DeepONets) and Deep-Learning based ROMs (DL-ROMs). In this context, deep autoencoders based on Convolutional Neural Networks (CNNs) have proven extremely effective, outperforming established techniques, such as the reduced basis method, when dealing with complex nonlinear problems. However, despite the empirical success of CNN-based autoencoders, there are only a few theoretical results supporting these architectures, usually stated in the form of universal approximation theorems. In particular, although the existing literature provides users with guidelines for designing convolutional autoencoders, the subsequent challenge of learning the latent features has been barely investigated. Furthermore, many practical questions remain unanswered, e.g., the number of snapshots needed for convergence or the neural network training strategy. In this work, using recent techniques from sparse high-dimensional function approximation, we fill some of these gaps by providing a new practical existence theorem for CNN-based autoencoders when the parameter-to-solution map is holomorphic. This regularity assumption arises in many relevant classes of parametric PDEs, such as the parametric diffusion equation, for which we discuss an explicit application of our general theory.

Structural identifiability is an important property of parametric ODE models. When conducting an experiment and inferring the parameter value from the time-series data, we want to know if the value is globally, locally, or non-identifiable. Global identifiability of the parameter indicates that there exists only one possible solution to the inference problem, local identifiability suggests that there could be several (but finitely many) possibilities, while non-identifiability implies that there are infinitely many possibilities for the value. Having this information is useful since, one would, for example, only perform inferences for the parameters which are identifiable. Given the current significance and widespread research conducted in this area, we decided to create a database of linear compartment models and their identifiability results. This facilitates the process of checking theorems and conjectures and drawing conclusions on identifiability. By only storing models up to symmetries and isomorphisms, we optimize memory efficiency and reduce query time. We conclude by applying our database to real problems. We tested a conjecture about deleting one leak of the model states in the paper 'Linear compartmental models: Input-output equations and operations that preserve identifiability' by E. Gross et al., and managed to produce a counterexample. We also compute some interesting statistics related to the identifiability of linear compartment model parameters.

We propose a high order discontinuous Galerkin (DG) scheme with subcell finite volume (FV) limiter to solve a monolithic first--order hyperbolic BSSNOK formulation of the coupled Einstein--Euler equations. The numerical scheme runs with adaptive mesh refinement (AMR) in three space dimensions, is endowed with time-accurate local time stepping (LTS) and is able to deal with both conservative and non-conservative hyperbolic systems. The system of governing partial differential equations was shown to be strongly hyperbolic and is solved in a monolithic fashion with one numerical framework that can be simultaneously applied to both the conservative matter subsystem as well as the non-conservative subsystem for the spacetime. Since high order unlimited DG schemes are well-known to produce spurious oscillations in the presence of discontinuities and singularities, our subcell finite volume limiter is crucial for the robust discretization of shock waves arising in the matter as well as for the stable treatment of puncture black holes. We test the new method on a set of classical test problems of numerical general relativity, showing good agreement with available exact or numerical reference solutions. In particular, we perform the first long term evolution of the inspiralling merger of two puncture black holes with a high order ADER-DG scheme.

Our main result is a succinct counterpoint to Courcelle's meta-theorem as follows: every arborescent monadic second-order (MSO) property is either NP-hard or coNP-hard over graphs given by succinct representations. Succint representations are Boolean circuits computing the adjacency relation. Arborescent properties are those which have infinitely many models and countermodels with bounded treewidth. Moreover, we explore what happens when the arborescence condition is dropped and show that, under a reasonable complexity assumption, the previous dichotomy fails, even for questions expressible in first-order logic.

We extend the laminate based framework of direct Deep Material Networks (DMNs) to treat suspensions of rigid fibers in a non-Newtonian solvent. To do so, we derive two-phase homogenization blocks that are capable of treating incompressible fluid phases and infinite material contrast. In particular, we leverage existing results for linear elastic laminates to identify closed form expressions for the linear homogenization functions of two-phase layered emulsions. To treat infinite material contrast, we rely on the repeated layering of two-phase layered emulsions in the form of coated layered materials. We derive necessary and sufficient conditions which ensure that the effective properties of coated layered materials with incompressible phases are non-singular, even if one of the phases is rigid. With the derived homogenization blocks and non-singularity conditions at hand, we present a novel DMN architecture, which we name the Flexible DMN (FDMN) architecture. We build and train FDMNs to predict the effective stress response of shear-thinning fiber suspensions with a Cross-type matrix material. For 31 fiber orientation states, six load cases, and over a wide range of shear rates relevant to engineering processes, the FDMNs achieve validation errors below 4.31% when compared to direct numerical simulations with Fast-Fourier-Transform based computational techniques. Compared to a conventional machine learning approach introduced previously by the consortium of authors, FDMNs offer better accuracy at an increased computational cost for the considered material and flow scenarios.

We studied the dynamical properties of Rabi oscillations driven by an alternating Rashba field applied to a two-dimensional (2D) harmonic confinement system. We solve the time-dependent (TD) Schr\"{o}dinger equation numerically and rewrite the resulting TD wavefunction onto the Bloch sphere (BS) using two BS parameters of the zenith ($\theta_B$) and azimuthal ($\phi_B$) angles, extracting the phase information $\phi_B$ as well as the mixing ratio $\theta_B$ between the two BS-pole states. We employed a two-state rotating wave (TSRW) approach and studied the fundamental features of $\theta_B$ and $\phi_B$ over time. The TSRW approach reveals a triangular wave formation in $\theta_B$. Moreover, at each apex of the triangular wave, the TD wavefunction passes through the BS pole, and the state is completely replaced by the opposite spin state. The TSRW approach also elucidates a linear change in $\phi_B$. The slope of $\phi_B$ vs. time is equal to the difference between the dynamical terms, leading to a confinement potential in the harmonic system. The TSRW approach further demonstrates a jump in the phase difference by $\pi$ when the wavefunction passes through the BS pole. The alternating Rashba field causes multiple successive Rabi transitions in the 2D harmonic system. We then introduce the effective BS (EBS) and transform these complicated transitions into an equivalent "single" Rabi one. Consequently, the EBS parameters $\theta_B^{\mathrm{eff}}$ and $\phi_B^{\mathrm{eff}}$ exhibit mixing and phase difference between two spin states $\alpha$ and $\beta$, leading to a deep understanding of the TD features of multi-Rabi oscillations. Furthermore, the combination of the BS representation with the TSRW approach successfully reveals the dynamical properties of the Rabi oscillation, even beyond the TSRW approximation.

The statistical modeling of discrete extremes has received less attention than their continuous counterparts in the Extreme Value Theory (EVT) literature. One approach to the transition from continuous to discrete extremes is the modeling of threshold exceedances of integer random variables by the discrete version of the generalized Pareto distribution. However, the optimal choice of thresholds defining exceedances remains a problematic issue. Moreover, in a regression framework, the treatment of the majority of non-extreme data below the selected threshold is either ignored or separated from the extremes. To tackle these issues, we expand on the concept of employing a smooth transition between the bulk and the upper tail of the distribution. In the case of zero inflation, we also develop models with an additional parameter. To incorporate possible predictors, we relate the parameters to additive smoothed predictors via an appropriate link, as in the generalized additive model (GAM) framework. A penalized maximum likelihood estimation procedure is implemented. We illustrate our modeling proposal with a real dataset of avalanche activity in the French Alps. With the advantage of bypassing the threshold selection step, our results indicate that the proposed models are more flexible and robust than competing models, such as the negative binomial distribution

We prove the well posedness in weighted Sobolev spaces of certain linear and nonlinear elliptic boundary value problems posed on convex domains and under singular forcing. It is assumed that the weights belong to the Muckenhoupt class $A_p$ with $p \in (1,\infty$). We also propose and analyze a convergent finite element discretization for the nonlinear elliptic boundary value problems mentioned above. As an instrumental result, we prove that the discretization of certain linear problems are well posed in weighted spaces.

We propose a novel and simple spectral method based on the semi-discrete Fourier transforms to discretize the fractional Laplacian $(-\Delta)^\frac{\alpha}{2}$. Numerical analysis and experiments are provided to study its performance. Our method has the same symbol $|\boldsymbol\xi|^\alpha$ as the fractional Laplacian $(-\Delta)^\frac{\alpha}{2}$ at the discrete level, and thus it can be viewed as the exact discrete analogue of the fractional Laplacian. This {\it unique feature} distinguishes our method from other existing methods for the fractional Laplacian. Note that our method is different from the Fourier pseudospectral methods in the literature which are usually limited to periodic boundary conditions (see Remark \ref{remark0}). Numerical analysis shows that our method can achieve a spectral accuracy. The stability and convergence of our method in solving the fractional Poisson equations were analyzed. Our scheme yields a multilevel Toeplitz stiffness matrix, and thus fast algorithms can be developed for efficient matrix-vector multiplications. The computational complexity is ${\mathcal O}(2N\log(2N))$, and the memory storage is ${\mathcal O}(N)$ with $N$ the total number of points. Extensive numerical experiments verify our analytical results and demonstrate the effectiveness of our method in solving various problems.