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The principle of least action is one of the most fundamental physical principle. It says that among all possible motions connecting two points in a phase space, the system will exhibit those motions which extremise an action functional. Many qualitative features of dynamical systems, such as the presence of conservation laws and energy balance equations, are related to the existence of an action functional. Incorporating variational structure into learning algorithms for dynamical systems is, therefore, crucial in order to make sure that the learned model shares important features with the exact physical system. In this paper we show how to incorporate variational principles into trajectory predictions of learned dynamical systems. The novelty of this work is that (1) our technique relies only on discrete position data of observed trajectories. Velocities or conjugate momenta do {\em not} need to be observed or approximated and {\em no} prior knowledge about the form of the variational principle is assumed. Instead, they are recovered using backward error analysis. (2) Moreover, our technique compensates discretisation errors when trajectories are computed from the learned system. This is important when moderate to large step-sizes are used and high accuracy is required. For this, we introduce and rigorously analyse the concept of inverse modified Lagrangians by developing an inverse version of variational backward error analysis. (3) Finally, we introduce a method to perform system identification from position observations only, based on variational backward error analysis.

Lattice Boltzmann schemes rely on the enlargement of the size of the target problem in order to solve PDEs in a highly parallelizable and efficient kinetic-like fashion, split into a collision and a stream phase. This structure, despite the well-known advantages from a computational standpoint, is not suitable to construct a rigorous notion of consistency with respect to the target equations and to provide a precise notion of stability. In order to alleviate these shortages and introduce a rigorous framework, we demonstrate that any lattice Boltzmann scheme can be rewritten as a corresponding multi-step Finite Difference scheme on the conserved variables. This is achieved by devising a suitable formalism based on operators, commutative algebra and polynomials. Therefore, the notion of consistency of the corresponding Finite Difference scheme allows to invoke the Lax-Richtmyer theorem in the case of linear lattice Boltzmann schemes. Moreover, we show that the frequently-used von Neumann-like stability analysis for lattice Boltzmann schemes entirely corresponds to the von Neumann stability analysis of their Finite Difference counterpart. More generally, the usual tools for the analysis of Finite Difference schemes are now readily available to study lattice Boltzmann schemes. Their relevance is verified by means of numerical illustrations.

We develop an \textit{a posteriori} error analysis for the time of the first occurrence of an event, specifically, the time at which a functional of the solution to a partial differential equation (PDE) first achieves a threshold value on a given time interval. This novel quantity of interest (QoI) differs from classical QoIs which are modeled as bounded linear (or nonlinear) functionals. Taylor's theorem and an adjoint-based \textit{a posteriori} analysis is used to derive computable and accurate error estimates for semi-linear parabolic and hyperbolic PDEs. The accuracy of the error estimates is demonstrated through numerical solutions of the one-dimensional heat equation and linearized shallow water equations (SWE), representing parabolic and hyperbolic cases, respectively.

Given polynomials $f_0,\dots, f_k$ the Ideal Membership Problem, IMP for short, asks if $f_0$ belongs to the ideal generated by $f_1,\dots, f_k$. In the search version of this problem the task is to find a proof of this fact. The IMP is a well-known fundamental problem with numerous applications, for instance, it underlies many proof systems based on polynomials such as Nullstellensatz, Polynomial Calculus, and Sum-of-Squares. Although the IMP is in general intractable, in many important cases it can be efficiently solved. Mastrolilli [SODA'19] initiated a systematic study of IMPs for ideals arising from Constraint Satisfaction Problems (CSPs), parameterized by constraint languages, denoted IMP($\Gamma$). The ultimate goal of this line of research is to classify all such IMPs accordingly to their complexity. Mastrolilli achieved this goal for IMPs arising from CSP($\Gamma$) where $\Gamma$ is a Boolean constraint language, while Bulatov and Rafiey [ArXiv'21] advanced these results to several cases of CSPs over finite domains. In this paper we consider IMPs arising from CSPs over `affine' constraint languages, in which constraints are subgroups (or their cosets) of direct products of Abelian groups. This kind of CSPs include systems of linear equations and are considered one of the most important types of tractable CSPs. Some special cases of the problem have been considered before by Bharathi and Mastrolilli [MFCS'21] for linear equation modulo 2, and by Bulatov and Rafiey [ArXiv'21] to systems of linear equations over $GF(p)$, $p$ prime. Here we prove that if $\Gamma$ is an affine constraint language then IMP($\Gamma$) is solvable in polynomial time assuming the input polynomial has bounded degree.

While many works exploiting an existing Lie group structure have been proposed for state estimation, in particular the Invariant Extended Kalman Filter (IEKF), few papers address the construction of a group structure that allows casting a given system into the IEKF framework, namely making the dynamics group affine and the observations invariant. In this paper we introduce a large class of systems encompassing most problems involving a navigating vehicle encountered in practice. For those systems we introduce a novel methodology that systematically provides a group structure for the state space, including vectors of the body frame such as biases. We use it to derive observers having properties akin to those of linear observers or filters. The proposed unifying and versatile framework encompasses all systems where IEKF has proved successful, improves state-of-the art "imperfect" IEKF for inertial navigation with sensor biases, and allows addressing novel examples, like GNSS antenna lever arm estimation.

The modeling of optical wave propagation in optical fiber is a task of fast and accurate solving the nonlinear Schr\"odinger equation (NLSE), and can enable the research progress and system design of optical fiber communications, which are the infrastructure of modern communication systems. Traditional modeling of fiber channels using the split-step Fourier method (SSFM) has long been regarded as challenging in long-haul wavelength division multiplexing (WDM) optical fiber communication systems because it is extremely time-consuming. Here we propose a linear-nonlinear feature decoupling distributed (FDD) waveform modeling scheme to model long-haul WDM fiber channel, where the channel linear effects are modelled by the NLSE-derived model-driven methods and the nonlinear effects are modelled by the data-driven deep learning methods. Meanwhile, the proposed scheme only focuses on one-span fiber distance fitting, and then recursively transmits the model to achieve the required transmission distance. The proposed modeling scheme is demonstrated to have high accuracy, high computing speeds, and robust generalization abilities for different optical launch powers, modulation formats, channel numbers and transmission distances. The total running time of FDD waveform modeling scheme for 41-channel 1040-km fiber transmission is only 3 minutes versus more than 2 hours using SSFM for each input condition, which achieves a 98% reduction in computing time. Considering the multi-round optimization by adjusting system parameters, the complexity reduction is significant. The results represent a remarkable improvement in nonlinear fiber modeling and open up novel perspectives for solution of NLSE-like partial differential equations and optical fiber physics problems.

Influence maximization is the task of selecting a small number of seed nodes in a social network to maximize the spread of the influence from these seeds, and it has been widely investigated in the past two decades. In the canonical setting, the whole social network as well as its diffusion parameters is given as input. In this paper, we consider the more realistic sampling setting where the network is unknown and we only have a set of passively observed cascades that record the set of activated nodes at each diffusion step. We study the task of influence maximization from these cascade samples (IMS), and present constant approximation algorithms for this task under mild conditions on the seed set distribution. To achieve the optimization goal, we also provide a novel solution to the network inference problem, that is, learning diffusion parameters and the network structure from the cascade data. Comparing with prior solutions, our network inference algorithm requires weaker assumptions and does not rely on maximum-likelihood estimation and convex programming. Our IMS algorithms enhance the learning-and-then-optimization approach by allowing a constant approximation ratio even when the diffusion parameters are hard to learn, and we do not need any assumption related to the network structure or diffusion parameters.

Optimal transport distances have found many applications in machine learning for their capacity to compare non-parametric probability distributions. Yet their algorithmic complexity generally prevents their direct use on large scale datasets. Among the possible strategies to alleviate this issue, practitioners can rely on computing estimates of these distances over subsets of data, {\em i.e.} minibatches. While computationally appealing, we highlight in this paper some limits of this strategy, arguing it can lead to undesirable smoothing effects. As an alternative, we suggest that the same minibatch strategy coupled with unbalanced optimal transport can yield more robust behavior. We discuss the associated theoretical properties, such as unbiased estimators, existence of gradients and concentration bounds. Our experimental study shows that in challenging problems associated to domain adaptation, the use of unbalanced optimal transport leads to significantly better results, competing with or surpassing recent baselines.

Interpretation of Deep Neural Networks (DNNs) training as an optimal control problem with nonlinear dynamical systems has received considerable attention recently, yet the algorithmic development remains relatively limited. In this work, we make an attempt along this line by reformulating the training procedure from the trajectory optimization perspective. We first show that most widely-used algorithms for training DNNs can be linked to the Differential Dynamic Programming (DDP), a celebrated second-order trajectory optimization algorithm rooted in the Approximate Dynamic Programming. In this vein, we propose a new variant of DDP that can accept batch optimization for training feedforward networks, while integrating naturally with the recent progress in curvature approximation. The resulting algorithm features layer-wise feedback policies which improve convergence rate and reduce sensitivity to hyper-parameter over existing methods. We show that the algorithm is competitive against state-ofthe-art first and second order methods. Our work opens up new avenues for principled algorithmic design built upon the optimal control theory.

In this paper, we study the optimal convergence rate for distributed convex optimization problems in networks. We model the communication restrictions imposed by the network as a set of affine constraints and provide optimal complexity bounds for four different setups, namely: the function $F(\xb) \triangleq \sum_{i=1}^{m}f_i(\xb)$ is strongly convex and smooth, either strongly convex or smooth or just convex. Our results show that Nesterov's accelerated gradient descent on the dual problem can be executed in a distributed manner and obtains the same optimal rates as in the centralized version of the problem (up to constant or logarithmic factors) with an additional cost related to the spectral gap of the interaction matrix. Finally, we discuss some extensions to the proposed setup such as proximal friendly functions, time-varying graphs, improvement of the condition numbers.