Search

We investigate the performance of multigrid preconditioners for solving linear systems arising from finite element discretizations of elliptic interface problems using the Fictitious Domain with Distributed Lagrange Multipliers (FD-DLM) formulation. Numerical experiments are conducted using continuous and discontinuous finite element spaces for the Lagrange multiplier. Results indicate that multigrid is a promising preconditioner for problems in the FD-DLM formulation.

Physics Playground is an immersive Virtual Reality (VR) application designed for educational purposes, featuring a virtual laboratory where users interact with various physics phenomena through guided experiments. This study aims to evaluate the application's design and educational content to facilitate its integration into classroom settings. A quantitative data collection investigated learning outcomes, related confidence, user experience, and perceived cognitive load, through a 2x2 within-between subjects setup, with participants divided into two conditions (VR vs. slideshow) and knowledge levels assessed twice (pre- and post-tests). A qualitative approach included interviews and a focus group to explore education experts' opinions on the overall experience and didactic content. Results showed an improvement in physics knowledge and confidence after the learning experience compared to baseline, regardless of the condition. Despite comparable perceived cognitive load, slideshow learning was slightly more effective in enhancing physics knowledge. However, both qualitative and quantitative results highlighted the immersive advantage of VR in enhancing user satisfaction. This approach pointed out limitations and advantages of VR-based learning, but more research is needed to understand how it can be implemented into broader teaching strategies., Comment: Presented at Intelligent Human Computer Interaction (IHCI) 2024

Active systems comprise a class of nonequilibrium dynamics in which individual components autonomously dissipate energy. Efforts towards understanding the role played by activity have centered on computation of the entropy production rate (EPR), which quantifies the breakdown of time reversal symmetry. A fundamental difficulty in this program is that high dimensionality of the phase space renders traditional computational techniques infeasible for estimating the EPR. Here, we overcome this challenge with a novel deep learning approach that estimates probability currents directly from stochastic system trajectories. We derive a new physical connection between the probability current and two local definitions of the EPR for inertial systems, which we apply to characterize the departure from equilibrium in a canonical model of flocking. Our results highlight that entropy is produced and consumed on the spatial interface of a flock as the interplay between alignment and fluctuation dynamically creates and annihilates order. By enabling the direct visualization of when and where a given system is out of equilibrium, we anticipate that our methodology will advance the understanding of a broad class of complex nonequilibrium dynamics.

Diffusion-based generative models provide a powerful framework for learning to sample from a complex target distribution. The remarkable empirical success of these models applied to high-dimensional signals, including images and video, stands in stark contrast to classical results highlighting the curse of dimensionality for distribution recovery. In this work, we take a step towards understanding this gap through a careful analysis of learning diffusion models over the Barron space of single layer neural networks. In particular, we show that these shallow models provably adapt to simple forms of low dimensional structure, thereby avoiding the curse of dimensionality. We combine our results with recent analyses of sampling with diffusion models to provide an end-to-end sample complexity bound for learning to sample from structured distributions. Importantly, our results do not require specialized architectures tailored to particular latent structures, and instead rely on the low-index structure of the Barron space to adapt to the underlying distribution.

The Special Theory of Relativity, introduced by Albert Einstein in the early 20th century, marked a radical shift in our understanding of space and time. Nevertheless, the theory's non-intuitive implications continue to pose conceptual challenges for novice physicists. In this thesis, we propose a virtual reality solution based on the development of a Unity package capable of simulating the effects of relativity in a digital environment. The current implementation includes the representation of space contraction, time dilation and relativistic Doppler effect. The primary focus lies in the accurate representation of relativistic laws, as well as in computational efficiency and in the modeling of a user interface specifically crafted to enhance understanding and interactivity. The package significantly reduces developer workload through a streamlined API, enabling maximum freedom in the development of virtual scenarios. Design goals are validated by a testing phase conducted through dedicated probe scenes. To showcase the potential of this work, we also present the deployment of a VR application built on top of the package, that transports users in experiencing relativistic effects in real-life scenarios. The application is scheduled to be used by Master's students in Physical Engineering at Politecnico di Milano. In the meanwhile, other educational areas of expansion are being considered, suggesting a promising future in the direction of this work.

A posteriori error estimator is derived for an elliptic interface problem in the fictitious domain formulation with distributed Lagrange multiplier considering a discontinuous Lagrange multiplier finite element space. A posteriori error estimation plays a pivotal role in assessing the accuracy and reliability of computational solutions across various domains of science and engineering. This study delves into the theoretical underpinnings and computational considerations of a residual-based estimator. Theoretically, the estimator is studied for cases with constant coefficients which jump across an interface as well as generalized scenarios with smooth coefficients that jump across an interface. Theoretical findings demonstrate the reliability and efficiency of the proposed estimators under all considered cases. Numerical experiments are conducted to validate the theoretical results, incorporating various immersed geometries and instances of high coefficients jumps at the interface. Leveraging an adaptive algorithm, the estimator identifies regions with singularities and applies refinement accordingly. Results substantiate the theoretical findings, highlighting the reliability and efficiency of the estimators. Furthermore, numerical solutions exhibit optimal convergence properties, demonstrating resilience against geometric singularities or coefficients jumps., Comment: 32 pages, 36 figures

Generative models based on dynamical transport of measure, such as diffusion models, flow matching models, and stochastic interpolants, learn an ordinary or stochastic differential equation whose trajectories push initial conditions from a known base distribution onto the target. While training is cheap, samples are generated via simulation, which is more expensive than one-step models like GANs. To close this gap, we introduce flow map matching -- an algorithm that learns the two-time flow map of an underlying ordinary differential equation. The approach leads to an efficient few-step generative model whose step count can be chosen a-posteriori to smoothly trade off accuracy for computational expense. Leveraging the stochastic interpolant framework, we introduce losses for both direct training of flow maps and distillation from pre-trained (or otherwise known) velocity fields. Theoretically, we show that our approach unifies many existing few-step generative models, including consistency models, consistency trajectory models, progressive distillation, and neural operator approaches, which can be obtained as particular cases of our formalism. With experiments on CIFAR-10 and ImageNet 32x32, we show that flow map matching leads to high-quality samples with significantly reduced sampling cost compared to diffusion or stochastic interpolant methods.

We consider a fictitious domain formulation for fluid-structure interaction problems based on a distributed Lagrange multiplier to couple the fluid and solid behaviors. How to deal with the coupling term is crucial since the construction of the associated finite element matrix requires the integration of functions defined over non-matching grids: the exact computation can be performed by intersecting the involved meshes, whereas an approximate coupling matrix can be evaluated on the original meshes by introducing a quadrature error. The purpose of this paper is twofold: we prove that the discrete problem is well-posed also when the coupling term is constructed in approximate way and we discuss quadrature error estimates over non-matching grids., Comment: 27 pages, 6 figures

We propose a framework for probabilistic forecasting of dynamical systems based on generative modeling. Given observations of the system state over time, we formulate the forecasting problem as sampling from the conditional distribution of the future system state given its current state. To this end, we leverage the framework of stochastic interpolants, which facilitates the construction of a generative model between an arbitrary base distribution and the target. We design a fictitious, non-physical stochastic dynamics that takes as initial condition the current system state and produces as output a sample from the target conditional distribution in finite time and without bias. This process therefore maps a point mass centered at the current state onto a probabilistic ensemble of forecasts. We prove that the drift coefficient entering the stochastic differential equation (SDE) achieving this task is non-singular, and that it can be learned efficiently by square loss regression over the time-series data. We show that the drift and the diffusion coefficients of this SDE can be adjusted after training, and that a specific choice that minimizes the impact of the estimation error gives a F\"ollmer process. We highlight the utility of our approach on several complex, high-dimensional forecasting problems, including stochastically forced Navier-Stokes and video prediction on the KTH and CLEVRER datasets.

This paper focuses on the numerical solution of elliptic partial differential equations (PDEs) with Dirichlet and mixed boundary conditions, specifically addressing the challenges arising from irregular domains. Both finite element method (FEM) and finite difference method (FDM), face difficulties in dealing with arbitrary domains. The paper introduces a novel nodal symmetric ghost {method based on a variational formulation}, which combines the advantages of FEM and FDM. The method employs bilinear finite elements on a structured mesh and provides a detailed implementation description. A rigorous a priori convergence rate analysis is also presented. The convergence rates are validated with many numerical experiments, in both one and two space dimensions.

We present Scalable Interpolant Transformers (SiT), a family of generative models built on the backbone of Diffusion Transformers (DiT). The interpolant framework, which allows for connecting two distributions in a more flexible way than standard diffusion models, makes possible a modular study of various design choices impacting generative models built on dynamical transport: learning in discrete or continuous time, the objective function, the interpolant that connects the distributions, and deterministic or stochastic sampling. By carefully introducing the above ingredients, SiT surpasses DiT uniformly across model sizes on the conditional ImageNet 256x256 and 512x512 benchmark using the exact same model structure, number of parameters, and GFLOPs. By exploring various diffusion coefficients, which can be tuned separately from learning, SiT achieves an FID-50K score of 2.06 and 2.62, respectively., Comment: ECCV 2024; Code available: https://github.com/willisma/SiT

In this paper we introduce an abstract setting for the convergence analysis of the virtual element approximation of an acoustic vibration problem. We discuss the effect of the stabilization parameters and remark that in some cases it is possible to achieve optimal convergence without the need of any stabilization. This statement is rigorously proved for lowest order triangular element and supported by several numerical experiments., Comment: 36 pages, 7 figures, 14 tables

Two-photon microscopy (2PM) has become an important tool in biology to study the structure and function of intact tissues in-vivo. However, adult mammalian tissues such as the mouse brain are highly scattering, thereby putting fundamental limits on the achievable imaging depth, which typically resides around 600-800um. In principle, shifting both the excitation as well as (fluorescence) emission light to the shortwave near-infrared (SWIR, 1000-1700 nm) region promises substantially deeper imaging in 2PM, yet has proven challenging in the past due to the limited availability of detectors and probes in this wavelength region. To overcome these limitations and fully capitalize on the SWIR region, in this work we introduce a novel array of superconducting nanowire single-photon detectors (SNSPDs) and associated custom detection electronics for the use in near-infrared 2PM. The SNSPD array exhibits high efficiency and dynamic range, as well as low dark-count rates over a wide wavelength range. Additionally, the electronics and software permit seamless integration into typical 2PM systems. Together with a fluorescent dye emitting at 1105 nm, we report imaging depth of > 1.1mm in the in-vivo mouse brain, limited only by available labeling density and laser power. Our work further establishes SWIR 2PM approaches and SNSPDs as promising technologies for deep tissue biological imaging., Comment: 17 pages, 5+1 figures

In this paper we consider the finite element approximation of Maxwell's problem and analyse the prescription of essential boundary conditions in a weak sense using Nitsche's method. To avoid indefiniteness of the problem, the original equations are augmented with the gradient of a scalar field that allows one to impose the zero divergence of the magnetic induction, even if the exact solution for this scalar field is zero. Two finite element approximations are considered, namely, one in which the approximation spaces are assumed to satisfy the appropriate inf-sup condition that render the standard Galerkin method stable, and another augmented and stabilised one that permits the use of finite element interpolations of arbitrary order. Stability and convergence results are provided for the two finite element formulations considered.

Generative models inspired by dynamical transport of measure -- such as flows and diffusions -- construct a continuous-time map between two probability densities. Conventionally, one of these is the target density, only accessible through samples, while the other is taken as a simple base density that is data-agnostic. In this work, using the framework of stochastic interpolants, we formalize how to \textit{couple} the base and the target densities, whereby samples from the base are computed conditionally given samples from the target in a way that is different from (but does preclude) incorporating information about class labels or continuous embeddings. This enables us to construct dynamical transport maps that serve as conditional generative models. We show that these transport maps can be learned by solving a simple square loss regression problem analogous to the standard independent setting. We demonstrate the usefulness of constructing dependent couplings in practice through experiments in super-resolution and in-painting., Comment: ICML 2024

Given a set of $K$ probability densities, we consider the multimarginal generative modeling problem of learning a joint distribution that recovers these densities as marginals. The structure of this joint distribution should identify multi-way correspondences among the prescribed marginals. We formalize an approach to this task within a generalization of the stochastic interpolant framework, leading to efficient learning algorithms built upon dynamical transport of measure. Our generative models are defined by velocity and score fields that can be characterized as the minimizers of simple quadratic objectives, and they are defined on a simplex that generalizes the time variable in the usual dynamical transport framework. The resulting transport on the simplex is influenced by all marginals, and we show that multi-way correspondences can be extracted. The identification of such correspondences has applications to style transfer, algorithmic fairness, and data decorruption. In addition, the multimarginal perspective enables an efficient algorithm for reducing the dynamical transport cost in the ordinary two-marginal setting. We demonstrate these capacities with several numerical examples.

Active matter systems, from self-propelled colloids to motile bacteria, are characterized by the conversion of free energy into useful work at the microscopic scale. They involve physics beyond the reach of equilibrium statistical mechanics, and a persistent challenge has been to understand the nature of their nonequilibrium states. The entropy production rate and the probability current provide quantitative ways to do so by measuring the breakdown of time-reversal symmetry. Yet, their efficient computation has remained elusive, as they depend on the system's unknown and high-dimensional probability density. Here, building upon recent advances in generative modeling, we develop a deep learning framework to estimate the score of this density. We show that the score, together with the microscopic equations of motion, gives access to the entropy production rate, the probability current, and their decomposition into local contributions from individual particles. To represent the score, we introduce a novel, spatially-local transformer network architecture that learns high-order interactions between particles while respecting their underlying permutation symmetry. We demonstrate the broad utility and scalability of the method by applying it to several high-dimensional systems of active particles undergoing motility-induced phase separation (MIPS). We show that a single network trained on a system of 4096 particles at one packing fraction can generalize to other regions of the phase diagram, including systems with as many as 32768 particles. We use this observation to quantify the spatial structure of the departure from equilibrium in MIPS as a function of the number of particles and the packing fraction.

We address a benchmark task in agile robotics: catching objects thrown at high-speed. This is a challenging task that involves tracking, intercepting, and cradling a thrown object with access only to visual observations of the object and the proprioceptive state of the robot, all within a fraction of a second. We present the relative merits of two fundamentally different solution strategies: (i) Model Predictive Control using accelerated constrained trajectory optimization, and (ii) Reinforcement Learning using zeroth-order optimization. We provide insights into various performance trade-offs including sample efficiency, sim-to-real transfer, robustness to distribution shifts, and whole-body multimodality via extensive on-hardware experiments. We conclude with proposals on fusing "classical" and "learning-based" techniques for agile robot control. Videos of our experiments may be found at https://sites.google.com/view/agile-catching, Comment: L4DC 2023

We discuss the approximation of the eigensolutions associated with the Maxwell eigenvalues problem in the framework of least-squares finite elements. We write the Maxwell curl curl equation as a system of two first order equation and design a novel least-squares formulation whose minimum is attained at the solution of the system. The eigensolution are then approximated by considering the eigenmodes of the underlying solution operator. We study the convergence of the finite element approximation and we show several numerical tests confirming the good behavior of the method. It turns out that nodal elements can be successfully employed for the approximation of our problem also in presence of singular solutions.

We perform a systematic comparison of various numerical schemes for the approximation of interface problems. We consider unfitted approaches in view of their application to possibly moving configurations. Particular attention is paid to the implementation aspects and to the analysis of the costs related to the different phases of the simulations., Comment: 25 pages, 15 figures, 15 tables

We consider nodal-based Lagrangian interpolations for the finite element approximation of the Maxwell eigenvalue problem. The first approach introduced is a standard Galerkin method on Powell-Sabin meshes, which has recently been shown to yield convergent approximations in two dimensions, whereas the other two are stabilized formulations that can be motivated by a variational multiscale approach. For the latter, a mixed formulation equivalent to the original problem is used, in which the operator has a saddle point structure. The Lagrange multiplier introduced to enforce the divergence constraint vanishes in an appropriate functional setting. The first stabilized method we consider consists of an augmented formulation with the introduction of a mesh dependent term that can be regarded as the Laplacian of the multiplier of the divergence constraint. The second formulation is based on orthogonal projections, which can be recast as a residual based stabilization technique. We rely on the classical spectral theory to analyze the approximating methods for the eigenproblem. The stability and convergence aspects are inherited from the associated source problems. We investigate the numerical performance of the proposed formulations and provide some convergence results validating the theoretical ones for several benchmark tests, including ones with smooth and singular solutions.

We use a Gaussian Process Regression (GPR) strategy that was recently developed [3,16,17] to analyze different types of curves that are commonly encountered in parametric eigenvalue problems. We employ an offline-online decomposition method. In the offline phase, we generate the basis of the reduced space by applying the proper orthogonal decomposition (POD) method on a collection of pre-computed, full-order snapshots at a chosen set of parameters. Then, we generate our GPR model using four different Mat\'{e}rn covariance functions. In the online phase, we use this model to predict both eigenvalues and eigenvectors at new parameters. We then illustrate how the choice of each covariance function influences the performance of GPR. Furthermore, we discuss the connection between Gaussian Process Regression and spline methods and compare the performance of the GPR method against linear and cubic spline methods. We show that GPR outperforms other methods for functions with a certain regularity.

In this article we apply reduced order techniques for the approximation of parametric eigenvalue problems. The effect of the choice of sampling points is investigated. Here we use the standard proper orthogonal decomposition technique to obtain the basis of the reduced space and Galerking orthogonal technique is used to get the reduced problem. We present some numerical results and observe that the use of sparse sampling is a good idea for sampling if the dimension of parameter space is high., Comment: 6 pages

A finite element discretization is developed for the Cai-Hu model, describing the formation of biological networks. The model consists of a non linear elliptic equation for the pressure $p$ and a non linear reaction-diffusion equation for the conductivity tensor $\mathbb{C}$. The problem requires high resolution due to the presence of multiple scales, the stiffness in all its components and the non linearities. We propose a low order finite element discretization in space coupled with a semi-implicit time advancing scheme. The code is {verified} with several numerical tests performed with various choices for the parameters involved in the system. In absence of the exact solution, we apply Richardson extrapolation technique to estimate the order of the method., Comment: 11 pages, 3 figures, 18 plots, 2 tables

A class of generative models that unifies flow-based and diffusion-based methods is introduced. These models extend the framework proposed in Albergo & Vanden-Eijnden (2023), enabling the use of a broad class of continuous-time stochastic processes called `stochastic interpolants' to bridge any two arbitrary probability density functions exactly in finite time. These interpolants are built by combining data from the two prescribed densities with an additional latent variable that shapes the bridge in a flexible way. The time-dependent probability density function of the stochastic interpolant is shown to satisfy a first-order transport equation as well as a family of forward and backward Fokker-Planck equations with tunable diffusion coefficient. Upon consideration of the time evolution of an individual sample, this viewpoint immediately leads to both deterministic and stochastic generative models based on probability flow equations or stochastic differential equations with an adjustable level of noise. The drift coefficients entering these models are time-dependent velocity fields characterized as the unique minimizers of simple quadratic objective functions, one of which is a new objective for the score of the interpolant density. We show that minimization of these quadratic objectives leads to control of the likelihood for generative models built upon stochastic dynamics, while likelihood control for deterministic dynamics is more stringent. We also discuss connections with other methods such as score-based diffusion models, stochastic localization processes, probabilistic denoising techniques, and rectifying flows. In addition, we demonstrate that stochastic interpolants recover the Schr\"odinger bridge between the two target densities when explicitly optimizing over the interpolant. Finally, algorithmic aspects are discussed and the approach is illustrated on numerical examples.

In this paper we discuss reduced order models for the approximation of parametric eigenvalue problems. In particular, we are interested in the presence of intersections or clusters of eigenvalues. The singularities originating by these phenomena make it hard a straightforward generalization of well known strategies normally used for standards PDEs. We investigate how the known results extend (or not) to higher order frequencies.

We present and analyze a parallel solver for the solution of fluid structure interaction problems described by a fictitious domain approach. In particular, the fluid is modeled by the non-stationary incompressible Navier-Stokes equations, while the solid evolution is represented by the elasticity equations. The parallel implementation is based on the PETSc library and the solver has been tested in terms of robustness with respect to mesh refinement and weak scalability by running simulations on a Linux cluster., Comment: Contribution to the 5th African Conference on Computational Mechanics

In this article, we propose a data-driven reduced basis (RB) method for the approximation of parametric eigenvalue problems. The method is based on the offline and online paradigms. In the offline stage, we generate snapshots and construct the basis of the reduced space, using a POD approach. Gaussian process regressions (GPR) are used for approximating the eigenvalues and projection coefficients of the eigenvectors in the reduced space. All the GPR corresponding to the eigenvalues and projection coefficients are trained in the offline stage, using the data generated in the offline stage. The output corresponding to new parameters can be obtained in the online stage using the trained GPR. The proposed algorithm is used to solve affine and non-affine parameter-dependent eigenvalue problems. The numerical results demonstrate the robustness of the proposed non-intrusive method.

The aim of this work is to present a parallel solver for a formulation of fluid-structure interaction (FSI) problems which makes use of a distributed Lagrange multiplier in the spirit of the fictitious domain method. The fluid subproblem, consisting of the non-stationary Stokes equations, is discretized in space by $\mathcal{Q}_2$-$\mathcal{P}_1$ finite elements, whereas the structure subproblem, consisting of the linear or finite incompressible elasticity equations, is discretized in space by $\mathcal{Q}_1$ finite elements. A first order semi-implicit finite difference scheme is employed for time discretization. The resulting linear system at each time step is solved by a parallel GMRES solver, accelerated by block diagonal or triangular preconditioners. The parallel implementation is based on the PETSc library. Several numerical tests have been performed on Linux clusters to investigate the effectiveness of the proposed FSI solver., Comment: 29 pages, 8 figures, 14 tables

In this paper, new unfitted mixed finite elements are presented for elliptic interface problems with jump coefficients. Our model is based on a fictitious domain formulation with distributed Lagrange multiplier. The relevance of our investigations is better seen when applied to the framework of fluid structure interaction problems. Two finite elements schemes with piecewise constant Lagrange multiplier are proposed and their stability is proved theoretically. Numerical results compare the performance of those elements, confirming the theoretical proofs and verifying that the schemes converge with optimal rate., Comment: 28 pages, 24 figures. Numer Methods Partial Differential Eq. 2023

We compare the solutions of two systems of partial differential equations (PDE), seen as two different interpretations of the same model that describes formation of complex biological networks. Both approaches take into account the time evolution of the medium flowing through the network, and we compute the solution of an elliptic-parabolic PDE system for the conductivity vector $m$, the conductivity tensor $\mathbb{C}$ and the pressure $p$. We use finite differences schemes in a uniform Cartesian grid in the spatially two-dimensional setting to solve the two systems, where the parabolic equation is solved by a semi-implicit scheme in time. Since the conductivity vector and tensor appear also in the Poisson equation for the pressure $p$, the elliptic equation depends implicitly on time. For this reason we compute the solution of three linear systems in the case of the conductivity vector $m\in\mathbb{R}^2$, and four linear systems in the case of the symmetric conductivity tensor $\mathbb{C}\in\mathbb{R}^{2\times 2}$, at each time step. To accelerate the simulations, we make use of the Alternating Direction Implicit (ADI) method. The role of the parameters is important for obtaining detailed solutions. We provide numerous tests with various values of the parameters involved, to see the differences in the solutions of the two systems., Comment: 22 pages, 8 figures, 6 tables

In this paper we introduce an algorithm based on a sparse grid adaptive refinement, for the approximation of the eigensolutions to parametric problems arising from elliptic partial differential equations. In particular, we are interested in detecting the crossing of the hypersurfaces describing the eigenvalues as a function of the parameters. The a priori matching is followed by an a posteriori verification, driven by a suitably defined error indicator. At a given refinement level, a sparse grid approach is adopted for the construction of the grid of the next level, by using the marking given by the a posteriori indicator. Various numerical tests confirm the good performance of the scheme.

In this paper a novel numerical approximation of parametric eigenvalue problems is presented. We motivate our study with the analysis of a POD reduced order model for a simple one dimensional example. In particular, we introduce a new algorithm capable to track the matching of eigenvalues when the parameters vary.

We study a non standard mixed formulation of the Poisson problem, sometimes known as dual mixed formulation. For reasons related to the equilibration of the flux, we use finite elements that are conforming in H(div) for the approximation of the gradients, even if the formulation would allow for discontinuous finite elements. The scheme is not uniformly inf-sup stable, but we can show existence and uniqueness of the solution, as well as optimal error estimates for the gradient variable when suitable regularity assumptions are made. Several additional remarks complete the paper, shedding some light on the sources of instability for mixed formulations.

The method of choice for integrating the time-dependent Fokker-Planck equation in high-dimension is to generate samples from the solution via integration of the associated stochastic differential equation. Here, we study an alternative scheme based on integrating an ordinary differential equation that describes the flow of probability. Acting as a transport map, this equation deterministically pushes samples from the initial density onto samples from the solution at any later time. Unlike integration of the stochastic dynamics, the method has the advantage of giving direct access to quantities that are challenging to estimate from trajectories alone, such as the probability current, the density itself, and its entropy. The probability flow equation depends on the gradient of the logarithm of the solution (its "score"), and so is a-priori unknown. To resolve this dependence, we model the score with a deep neural network that is learned on-the-fly by propagating a set of samples according to the instantaneous probability current. We show theoretically that the proposed approach controls the KL divergence from the learned solution to the target, while learning on external samples from the stochastic differential equation does not control either direction of the KL divergence. Empirically, we consider several high-dimensional Fokker-Planck equations from the physics of interacting particle systems. We find that the method accurately matches analytical solutions when they are available as well as moments computed via Monte-Carlo when they are not. Moreover, the method offers compelling predictions for the global entropy production rate that out-perform those obtained from learning on stochastic trajectories, and can effectively capture non-equilibrium steady-state probability currents over long time intervals.

We study a recent formulation for fluid-structure interaction problems based on the use of a distributed Lagrange multiplier in the spirit of the fictitious domain approach. In this paper, we focus our attention on a crucial computational aspect regarding the interface matrix for the finite element discretization: it involves integration of functions supported on two different meshes. Several numerical tests show that accurate computation of the interface matrix has to be performed in order to ensure the optimal convergence of the method., Comment: 30 pages, 18 figures

We prove convergence of the Maxwell eigenvalue problem using quadratic or higher Lagrange finite elements on Worsey-Farin splits in three dimensions. To do this, we construct two Fortin-like operators to prove uniform convergence of the corresponding source problem. We present numerical experiments to illustrate the theoretical results.

In this paper we consider a reduced order method for the approximation of the eigensolutions of the Laplace problem with Dirichlet boundary condition. We use a time continuation technique that consists in the introduction of a fictitious time parameter. We use a POD approach and we present some theoretical results showing how to choose the optimal dimension of the POD basis. The results of our computations, related to the first eigenvalue, confirm the optimal behavior of our approximate solution.

Motivated by bridging the simulation to reality gap in the context of safety-critical systems, we consider learning adversarially robust stability certificates for unknown nonlinear dynamical systems. In line with approaches from robust control, we consider additive and Lipschitz bounded adversaries that perturb the system dynamics. We show that under suitable assumptions of incremental stability on the underlying system, the statistical cost of learning an adversarial stability certificate is equivalent, up to constant factors, to that of learning a nominal stability certificate. Our results hinge on novel bounds for the Rademacher complexity of the resulting adversarial loss class, which may be of independent interest. To the best of our knowledge, this is the first characterization of sample-complexity bounds when performing adversarial learning over data generated by a dynamical system. We further provide a practical algorithm for approximating the adversarial training algorithm, and validate our findings on a damped pendulum example.

Stokes flows are a type of fluid flow where convective forces are small in comparison with viscous forces, and momentum transport is entirely due to viscous diffusion. Besides being routinely used as benchmark test cases in numerical fluid dynamics, Stokes flows are relevant in several applications in science and engineering including porous media flow, biological flows, microfluidics, microrobotics, and hydrodynamic lubrication. The present study concerns the discretization of the equations of motion of Stokes flows in three dimensions utilizing the MINI mixed finite element, focusing on the superconvergence of the method which was investigated with numerical experiments using five purpose-made benchmark test cases with analytical solution. Despite the fact that the MINI element is only linearly convergent according to standard mixed finite element theory, a recent theoretical development proves that, for structured meshes in two dimensions, the pressure superconverges with order 1.5, as well as the linear part of the computed velocity with respect to the piecewise-linear nodal interpolation of the exact velocity. The numerical experiments documented herein suggest a more general validity of the superconvergence in pressure, possibly to unstructured tetrahedral meshes and even up to quadratic convergence which was observed with one test problem, thereby indicating that there is scope to further extend the available theoretical results on convergence.