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We present an opinion model founded upon the principles of the bounded confidence interaction among agents. Our objective is to explain the polarization effects inherent to vector-valued opinions. The evolutionary process adheres to the rule where each agent aspires to increase polarization through communication with a single friend during each discrete time step. The dynamics ensure that agents' ultimate (temporal) configuration will encompass a finite number of outlier states. We introduce deterministic and stochastic models, accompanied by a comprehensive mathematical analysis of their inherent properties. Additionally, we provide compelling illustrative examples and introduce a stochastic solver tailored for scenarios featuring an extensive set of agents. Furthermore, in the context of smaller agent populations, we scrutinize the suitability of neural networks for the rapid inference of limit configurations.

In order to support the advancement of machine learning methods for predicting time-series data, we present a comprehensive dataset designed explicitly for long-term time-series forecasting. We incorporate a collection of datasets obtained from diverse, dynamic systems and real-life records. Each dataset is standardized by dividing it into training and test trajectories with predetermined lookback lengths. We include trajectories of length up to $2000$ to ensure a reliable evaluation of long-term forecasting capabilities. To determine the most effective model in diverse scenarios, we conduct an extensive benchmarking analysis using classical and state-of-the-art models, namely LSTM, DeepAR, NLinear, N-Hits, PatchTST, and LatentODE. Our findings reveal intriguing performance comparisons among these models, highlighting the dataset-dependent nature of model effectiveness. Notably, we introduce a custom latent NLinear model and enhance DeepAR with a curriculum learning phase. Both consistently outperform their vanilla counterparts.

We raise concerns about controllers' robustness in simple reinforcement learning benchmark problems. We focus on neural network controllers and their low neuron and symbolic abstractions. A typical controller reaching high mean return values still generates an abundance of persistent low-return solutions, which is a highly undesirable property, easily exploitable by an adversary. We find that the simpler controllers admit more persistent bad solutions. We provide an algorithm for a systematic robustness study and prove existence of persistent solutions and, in some cases, periodic orbits, using a computer-assisted proof methodology., Comment: accepted to ECAI23

We study the overparametrization bounds required for the global convergence of stochastic gradient descent algorithm for a class of one hidden layer feed-forward neural networks, considering most of the activation functions used in practice, including ReLU. We improve the existing state-of-the-art results in terms of the required hidden layer width. We introduce a new proof technique combining nonlinear analysis with properties of random initializations of the network. First, we establish the global convergence of continuous solutions of the differential inclusion being a nonsmooth analogue of the gradient flow for the MSE loss. Second, we provide a technical result (working also for general approximators) relating solutions of the aforementioned differential inclusion to the (discrete) stochastic gradient descent sequences, hence establishing linear convergence towards zero loss for the stochastic gradient descent iterations.

In this paper we introduce a new approach to compute rigorously solutions of Cauchy problems for a class of semi-linear parabolic partial differential equations. Expanding solutions with Chebyshev series in time and Fourier series in space, we introduce a zero finding problem $F(a)=0$ on a Banach algebra $X$ of Fourier-Chebyshev sequences, whose solution solves the Cauchy problem. The challenge lies in the fact that the linear part $\mathcal{L} = DF(0)$ has an infinite block diagonal structure with blocks becoming less and less diagonal dominant at infinity. We introduce analytic estimates to show that $\mathcal{L}$ is an invertible linear operator on $X$, and we obtain explicit, rigorous and computable bounds for the operator norm $\| \mathcal{L}^{-1}\|_{B(X)}$. These bounds are then used to verify the hypotheses of a Newton-Kantorovich type argument which shows that the (Newton-like) operator $\mathcal{T}(a)=a - \mathcal{L}^{-1} F(a)$ is a contraction on a small ball centered at a numerical approximation of the Cauchy problem. The contraction mapping theorem yields a fixed point which corresponds to a classical (strong) solution of the Cauchy problem. The approach is simple to implement, numerically stable and is applicable to a class of PDE models, which include for instance Fisher's equation and the Swift-Hohenberg equation. We apply our approach to each of these models.

We show that Neural ODEs, an emerging class of time-continuous neural networks, can be verified by solving a set of global-optimization problems. For this purpose, we introduce Stochastic Lagrangian Reachability (SLR), an abstraction-based technique for constructing a tight Reachtube (an over-approximation of the set of reachable states over a given time-horizon), and provide stochastic guarantees in the form of confidence intervals for the Reachtube bounds. SLR inherently avoids the infamous wrapping effect (accumulation of over-approximation errors) by performing local optimization steps to expand safe regions instead of repeatedly forward-propagating them as is done by deterministic reachability methods. To enable fast local optimizations, we introduce a novel forward-mode adjoint sensitivity method to compute gradients without the need for backpropagation. Finally, we establish asymptotic and non-asymptotic convergence rates for SLR., Comment: 12 pages, 2 figures

We introduce LRT-NG, a set of techniques and an associated toolset that computes a reachtube (an over-approximation of the set of reachable states over a given time horizon) of a nonlinear dynamical system. LRT-NG significantly advances the state-of-the-art Langrangian Reachability and its associated tool LRT. From a theoretical perspective, LRT-NG is superior to LRT in three ways. First, it uses for the first time an analytically computed metric for the propagated ball which is proven to minimize the ball's volume. We emphasize that the metric computation is the centerpiece of all bloating-based techniques. Secondly, it computes the next reachset as the intersection of two balls: one based on the Cartesian metric and the other on the new metric. While the two metrics were previously considered opposing approaches, their joint use considerably tightens the reachtubes. Thirdly, it avoids the "wrapping effect" associated with the validated integration of the center of the reachset, by optimally absorbing the interval approximation in the radius of the next ball. From a tool-development perspective, LRT-NG is superior to LRT in two ways. First, it is a standalone tool that no longer relies on CAPD. This required the implementation of the Lohner method and a Runge-Kutta time-propagation method. Secondly, it has an improved interface, allowing the input model and initial conditions to be provided as external input files. Our experiments on a comprehensive set of benchmarks, including two Neural ODEs, demonstrates LRT-NG's superior performance compared to LRT, CAPD, and Flow*., Comment: 12 pages, 14 figures

Topological data analysis aims to extract topological quantities from data, which tend to focus on the broader global structure of the data rather than local information. The Mapper method, specifically, generalizes clustering methods to identify significant global mathematical structures, which are out of reach of many other approaches. We propose a classifier based on applying the Mapper algorithm to data projected onto a latent space. We obtain the latent space by using PCA or autoencoders. Notably, a classifier based on the Mapper method is immune to any gradient based attack, and improves robustness over traditional CNNs (convolutional neural networks). We report theoretical justification and some numerical experiments that confirm our claims., Comment: 12 pages, accepted to IEEE ICMLA 2019

This work is motivated by the following question in data-driven study of dynamical systems: given a dynamical system that is observed via time series of persistence diagrams that encode topological features of solutions snapshots, what conclusions can be drawn about solutions of the original dynamical system? In this paper we provide a definition of a persistence diagram for a point in $\mathbb{R}^N$ modeled on piecewise monotone functions. We then provide conditions under which time series of persistence diagrams can be used to guarantee the existence of a fixed point of the flow on $\mathbb{R}^N$ that generates the time series. To obtain this result requires an understanding of the preimage of the persistence map. The main theorem of this paper gives conditions under which these preimages are contractible simplicial complexes., Comment: 12 pages, 3 figures

We explore the chaotic dynamics of Rayleigh-B\'enard convection using large-scale, parallel numerical simulations for experimentally accessible conditions. We quantify the connections between the spatiotemporal dynamics of the leading-order Lyapunov vector and different measures of the flow field pattern's topology and dynamics. We use a range of pattern diagnostics to describe the spatiotemporal features of the flow field structures which includes many of the traditional diagnostics used to describe convection as well as some diagnostics tailored to capture the dynamics of the patterns. Using precision-recall curves, we quantify the complex relationship between the pattern diagnostics and the regions where the magnitude of the leading-order Lyapunov vector is significant.

We introduce continuous Lagrangian reachability (CLRT), a new algorithm for the computation of a tight and continuous-time reachtube for the solution flows of a nonlinear, time-variant dynamical system. CLRT employs finite strain theory to determine the deformation of the solution set from time $t_i$ to time $t_{i+1}$. We have developed simple explicit analytic formulas for the optimal metric for this deformation; this is superior to prior work, which used semi-definite programming. CLRT also uses infinitesimal strain theory to derive an optimal time increment $h_i$ between $t_i$ and $t_{i+1}$, nonlinear optimization to minimally bloat (i.e., using a minimal radius) the state set at time $t_i$ such that it includes all the states of the solution flow in the interval $[t_i,t_{i+1}]$. We use $\delta$-satisfiability to ensure the correctness of the bloating. Our results on a series of benchmarks show that CLRT performs favorably compared to state-of-the-art tools such as CAPD in terms of the continuous reachtube volumes they compute.

We introduce LRT, a new Lagrangian-based ReachTube computation algorithm that conservatively approximates the set of reachable states of a nonlinear dynamical system. LRT makes use of the Cauchy-Green stretching factor (SF), which is derived from an over-approximation of the gradient of the solution flows. The SF measures the discrepancy between two states propagated by the system solution from two initial states lying in a well-defined region, thereby allowing LRT to compute a reachtube with a ball-overestimate in a metric where the computed enclosure is as tight as possible. To evaluate its performance, we implemented a prototype of LRT in C++/Matlab, and ran it on a set of well-established benchmarks. Our results show that LRT compares very favorably with respect to the CAPD and Flow* tools., Comment: Accepted to CAV 2017

We present a computer-assisted proof of heteroclinic connections in the one-dimensional Ohta-Kawasaki model of diblock copolymers. The model is a fourth-order parabolic partial differential equation subject to homogeneous Neumann boundary conditions, which contains as a special case the celebrated Cahn-Hilliard equation. While the attractor structure of the latter model is completely understood for one-dimensional domains, the diblock copolymer extension exhibits considerably richer long-term dynamical behavior, which includes a high level of multistability. In this paper, we establish the existence of certain heteroclinic connections between the homogeneous equilibrium state, which represents a perfect copolymer mixture, and all local and global energy minimizers. In this way, we show that not every solution originating near the homogeneous state will converge to the global energy minimizer, but rather is trapped by a stable state with higher energy. This phenomenon can not be observed in the one-dimensional Cahn-Hillard equation, where generic solutions are attracted by a global minimizer.

The paper studies the issue of stability of solutions to the Navier-Stokes and damped Euler systems in periodic boxes. We show that under action of fast oscillating-in- time external forces all two dimensional regular solutions converge to a time periodic flow. Unexpectedly, effects of stabilization can be also obtained for systems with stationary forces with large total momentum (average of the velocity). Thanks to the Galilean transformation and space boundary conditions, the stationary force changes into one with time oscillations. In the three dimensional case we show an analogical result for weak solutions to the Navier- Stokes equations.

The paper aims at constructing two different solutions to an elliptic system $$ u \cdot \nabla u + (-\Delta)^m u = \lambda F $$ defined on the two dimensional torus. It can be viewed as an elliptic regularization of the stationary Burgers 2D system. A motivation to consider the above system comes from an examination of unusual propetries of the linear operator $\lambda \sin y \partial_x w + (-\Delta)^{m} w$ arising from a linearization of the equation about the dominant part of $F$. We argue that the skew-symmetric part of the operator provides in some sense a smallness of norms of the linear operator inverse. Our analytical proof is valid for a particular force $F$ and for $\lambda > \lambda_0$, $m> m_0$ sufficiently large. The main steps of the proof concern finite dimension approximation of the system and concentrate on analysis of features of large matrices, which resembles standard numerical analysis. Our analytical results are illustrated by numerical simulations.

We describe a topological method to study the dynamics of dissipative PDEs on a torus with rapidly oscillating forcing terms. We show that a dissipative PDE, which is invariant with respect to Galilean transformations, with a large average initial velocity can be reduced to a problem with rapidly oscillating forcing terms. We apply the technique to the Burgers equation, and the incompressible 2D Navier-Stokes equations with a time-dependent forcing. We prove that for a large initial average speed the equation admits a bounded eternal solution, which attracts all other solutions forward in time. For the incompressible 3D Navier-Stokes equations we establish existence of a locally attracting solution.

We prove the existence of globally attracting solutions of the viscous Burgers equation with periodic boundary conditions on the line for some particular choices of viscosity and non-autonomous forcing. The attract- ing solution is periodic if the forcing is periodic. The method is general and can be applied to other similar partial differential equations. The proof is computer assisted., Comment: 38 pages, 1 figure

We present a computer assisted method for proving the existence of globally attracting fixed points of dissipative PDEs. An application to the viscous Burgers equation with periodic boundary conditions and a forcing function constant in time is presented as a case study. We establish the existence of a locally attracting fixed point by using rigorous numerics techniques. To prove that the fixed point is, in fact, globally attracting we introduce a technique relying on a construction of an absorbing set, capturing any sufficiently regular initial condition after a finite time. Then the absorbing set is rigorously integrated forward in time to verify that any sufficiently regular initial condition is in the basin of attraction of the fixed point., Comment: To appear in Topological Methods in Nonlinear Analysis, 2014