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In backward error analysis, an approximate solution to an equation is compared to the exact solution to a nearby "modified" equation. In numerical ordinary differential equations, the two agree up to any power of the step size. If the differential equation has a geometric property then the modified equation may share it. In this way, known properties of differential equations can be applied to the approximation. But for partial differential equations, the known modified equations are of higher order, limiting applicability of the theory. Therefore, we study symmetric solutions of discretized partial differential equations that arise from a discrete variational principle. These symmetric solutions obey infinite-dimensional functional equations. We show that these equations admit second-order modified equations which are Hamiltonian and also possess first-order Lagrangians in modified coordinates. The modified equation and its associated structures are computed explicitly for the case of rotating travelling waves in the nonlinear wave equation.

In this paper we study some theoretical and numerical issues of the Boussinesq/Full dispersion system. This is a a three-parameter system of pde's that models the propagation of internal waves along the interface of two-fluid layers with rigid lid condition for the upper layer, and under a Boussinesq regime for the upper layer and a full dispersion regime for the lower layer. We first discretize in space the periodic initial-value problem with a Fourier-Galerkin spectral method and prove error estimates for several ranges of values of the parameters. Solitary waves of the model systems are then studied numerically in several ways. The numerical generation is analyzed by approximating the ode system with periodic boundary conditions for the solitary-wave profiles with a Fourier spectral scheme, implemented in a collocation form, and solving iteratively the corresponding algebraic system in Fourier space with the Petviashvili method accelerated with the minimal polynomial extrapolation technique. Motivated by the numerical results, a new result of existence of solitary waves is proved. In the last part of the paper, the dynamics of these solitary waves is studied computationally, To this end, the semidiscrete systems obtained from the Fourier-Galerkin discretization in space are integrated numerically in time by a Runge-Kutta Composition method of order four. The fully discrete scheme is used to explore numerically the stability of solitary waves, their collisions, and the resolution of other initial conditions into solitary waves.

The mathematical forces at work behind Generative Adversarial Networks raise challenging theoretical issues. Motivated by the important question of characterizing the geometrical properties of the generated distributions, we provide a thorough analysis of Wasserstein GANs (WGANs) in both the finite sample and asymptotic regimes. We study the specific case where the latent space is univariate and derive results valid regardless of the dimension of the output space. We show in particular that for a fixed sample size, the optimal WGANs are closely linked with connected paths minimizing the sum of the squared Euclidean distances between the sample points. We also highlight the fact that WGANs are able to approach (for the 1-Wasserstein distance) the target distribution as the sample size tends to infinity, at a given convergence rate and provided the family of generative Lipschitz functions grows appropriately. We derive in passing new results on optimal transport theory in the semi-discrete setting.

With the fast development of modern deep learning techniques, the study of dynamic systems and neural networks is increasingly benefiting each other in a lot of different ways. Since uncertainties often arise in real world observations, SDEs (stochastic differential equations) come to play an important role. To be more specific, in this paper, we use a collection of SDEs equipped with neural networks to predict long-term trend of noisy time series which has big jump properties and high probability distribution shift. Our contributions are, first, we explored SDEs driven by $\alpha$-stable L\'evy motion to model the time series data and solved the problem through neural network approximation. Second, we theoretically proved the convergence of the model and obtained the convergence rate. Finally, we illustrated our method by applying it to stock marketing time series prediction and found the convergence order of error.

This paper focuses on stochastic saddle point problems with decision-dependent distributions in both the static and time-varying settings. These are problems whose objective is the expected value of a stochastic payoff function, where random variables are drawn from a distribution induced by a distributional map. For general distributional maps, the problem of finding saddle points is in general computationally burdensome, even if the distribution is known. To enable a tractable solution approach, we introduce the notion of equilibrium points -- which are saddle points for the stationary stochastic minimax problem that they induce -- and provide conditions for their existence and uniqueness. We demonstrate that the distance between the two classes of solutions is bounded provided that the objective has a strongly-convex-strongly-concave payoff and Lipschitz continuous distributional map. We develop deterministic and stochastic primal-dual algorithms and demonstrate their convergence to the equilibrium point. In particular, by modeling errors emerging from a stochastic gradient estimator as sub-Weibull random variables, we provide error bounds in expectation and in high probability that hold for each iteration; moreover, we show convergence to a neighborhood in expectation and almost surely. Finally, we investigate a condition on the distributional map -- which we call opposing mixture dominance -- that ensures the objective is strongly-convex-strongly-concave. Under this assumption, we show that primal-dual algorithms converge to the saddle points in a similar fashion.

We consider a moving boundary problem with kinetic condition that describes the diffusion of solvent into rubber and study semi-discrete finite element approximations of the corresponding weak solutions. We report on both a priori and a posteriori error estimates for the mass concentration of the diffusants, and respectively, for the a priori unknown position of the moving boundary. Our working techniques include integral and energy-based estimates for a nonlinear parabolic problem posed in a transformed fixed domain combined with a suitable use of the interpolation-trace inequality to handle the interface terms. Numerical illustrations of our FEM approximations are within the experimental range and show good agreement with our theoretical investigation. This work is a preliminary investigation necessary before extending the current moving boundary modeling to account explicitly for the mechanics of hyperelastic rods to capture a directional swelling of the underlying elastomer.

In transient simulations of particulate Stokes flow, to accurately capture the interaction between the constituent particles and the confining wall, the discretization of the wall often needs to be locally refined in the region approached by the particles. Consequently, standard fast direct solvers lose their efficiency since the linear system changes at each time step. This manuscript presents a new computational approach that avoids this issue by pre-constructing a fast direct solver for the wall ahead of time, computing a low-rank factorization to capture the changes due to the refinement, and solving the problem on the refined discretization via a Woodbury formula. Numerical results illustrate the efficiency of the solver in accelerating particulate Stokes simulations.

Although Deep Neural Networks (DNNs) have shown incredible performance in perceptive and control tasks, several trustworthy issues are still open. One of the most discussed topics is the existence of adversarial perturbations, which has opened an interesting research line on provable techniques capable of quantifying the robustness of a given input. In this regard, the Euclidean distance of the input from the classification boundary denotes a well-proved robustness assessment as the minimal affordable adversarial perturbation. Unfortunately, computing such a distance is highly complex due the non-convex nature of NNs. Despite several methods have been proposed to address this issue, to the best of our knowledge, no provable results have been presented to estimate and bound the error committed. This paper addresses this issue by proposing two lightweight strategies to find the minimal adversarial perturbation. Differently from the state-of-the-art, the proposed approach allows formulating an error estimation theory of the approximate distance with respect to the theoretical one. Finally, a substantial set of experiments is reported to evaluate the performance of the algorithms and support the theoretical findings. The obtained results show that the proposed strategies approximate the theoretical distance for samples close to the classification boundary, leading to provable robustness guarantees against any adversarial attacks.

This paper deals with the following important research question. Traditionally, the neural network employs non-linear activation functions concatenated with linear operators to approximate a given physical phenomenon. They "fill the space" with the concatenations of the activation functions and linear operators and adjust their coefficients to approximate the physical phenomena. We claim that it is better to "fill the space" with linear combinations of smooth higher-order B-splines base functions as employed by isogeometric analysis and utilize the neural networks to adjust the coefficients of linear combinations. In other words, the possibilities of using neural networks for approximating the B-spline base functions' coefficients and by approximating the solution directly are evaluated. Solving differential equations with neural networks has been proposed by Maziar Raissi et al. in 2017 by introducing Physics-informed Neural Networks (PINN), which naturally encode underlying physical laws as prior information. Approximation of coefficients using a function as an input leverages the well-known capability of neural networks being universal function approximators. In essence, in the PINN approach the network approximates the value of the given field at a given point. We present an alternative approach, where the physcial quantity is approximated as a linear combination of smooth B-spline basis functions, and the neural network approximates the coefficients of B-splines. This research compares results from the DNN approximating the coefficients of the linear combination of B-spline basis functions, with the DNN approximating the solution directly. We show that our approach is cheaper and more accurate when approximating smooth physical fields.

In this work, we consider the distributed optimization of non-smooth convex functions using a network of computing units. We investigate this problem under two regularity assumptions: (1) the Lipschitz continuity of the global objective function, and (2) the Lipschitz continuity of local individual functions. Under the local regularity assumption, we provide the first optimal first-order decentralized algorithm called multi-step primal-dual (MSPD) and its corresponding optimal convergence rate. A notable aspect of this result is that, for non-smooth functions, while the dominant term of the error is in $O(1/\sqrt{t})$, the structure of the communication network only impacts a second-order term in $O(1/t)$, where $t$ is time. In other words, the error due to limits in communication resources decreases at a fast rate even in the case of non-strongly-convex objective functions. Under the global regularity assumption, we provide a simple yet efficient algorithm called distributed randomized smoothing (DRS) based on a local smoothing of the objective function, and show that DRS is within a $d^{1/4}$ multiplicative factor of the optimal convergence rate, where $d$ is the underlying dimension.