The math behind even the simplest ocean waves is notoriously uncooperative. A team of Italian mathematicians has made major advances toward understanding it. The best perk of Alberto Maspero's job, he says, is the view from his window. Situated on a hill above the ancient port city of Trieste, Italy, his office at the International School for Advanced Studies overlooks a broad bay at the northern tip of the Adriatic Sea. "It's very inspiring," the mathematician said. "For sure the most beautiful view I've ever had." When the bora is strong enough, it drives the waves into reverse. But they never actually get there.

We revisit the optimal transport problem over angular velocity dynamics given by the controlled Euler equation. The solution of this problem enables stochastic guidance of spin states of a rigid body (e.g., spacecraft) over hard deadline constraint by transferring a given initial state statistics to a desired terminal state statistics. This is an instance of generalized optimal transport over a nonlinear dynamical system. While prior work has reported existence-uniqueness and numerical solution of this dynamical optimal transport problem, here we present structural results about the equivalent Kantorovich a.k.a. optimal coupling formulation. Specifically, we focus on deriving the ground cost for the associated Kantorovich optimal coupling formulation. The ground cost equals to the cost of transporting unit amount of mass from a specific realization of the initial or source joint probability measure to a realization of the terminal or target joint probability measure, and determines the Kantorovich formulation. Finding the ground cost leads to solving a structured deterministic nonlinear optimal control problem, which is shown to be amenable to an analysis technique pioneered by Athans et. al. We show that such techniques have broader applicability in determining the ground cost (thus Kantorovich formulation) for a class of generalized optimal mass transport problems involving nonlinear dynamics with translated norm-invariant drift.

This paper presents a data-driven framework for learning optimal second-order total variation diminishing (TVD) flux limiters via differentiable simulations. In our fully differentiable finite volume solvers, the limiter functions are replaced by neural networks. By representing the limiter as a pointwise convex linear combination of the Minmod and Superbee limiters, we enforce both second-order accuracy and TVD constraints at all stages of training. Our approach leverages gradient-based optimization through automatic differentiation, allowing a direct backpropagation of errors from numerical solutions to the limiter parameters. We demonstrate the effectiveness of this method on various hyperbolic conservation laws, including the linear advection equation, the Burgers' equation, and the one-dimensional Euler equations. Remarkably, a limiter trained solely on linear advection exhibits strong generalizability, surpassing the accuracy of most classical flux limiters across a range of problems with shocks and discontinuities. The learned flux limiters can be readily integrated into existing computational fluid dynamics codes, and the proposed methodology also offers a flexible pathway to systematically develop and optimize flux limiters for complex flow problems.

This paper presents and evaluates a framework for the coupling of subdomain-local projection-based reduced order models (PROMs) using the Schwarz alternating method following a domain decomposition (DD) of the spatial domain on which a given problem of interest is posed. In this approach, the solution on the full domain is obtained via an iterative process in which a sequence of subdomain-local problems are solved, with information propagating between subdomains through transmission boundary conditions (BCs). We explore several new directions involving the Schwarz alternating method aimed at maximizing the method's efficiency and flexibility, and demonstrate it on three challenging two-dimensional nonlinear hyperbolic problems: the shallow water equations, Burgers' equation, and the compressible Euler equations. We demonstrate that, for a cell-centered finite volume discretization and a non-overlapping DD, it is possible to obtain a stable and accurate coupled model utilizing Dirichlet-Dirichlet (rather than Robin-Robin or alternating Dirichlet-Neumann) transmission BCs on the subdomain boundaries. We additionally explore the impact of boundary sampling when utilizing the Schwarz alternating method to couple subdomain-local hyper-reduced PROMs. Our numerical results suggest that the proposed methodology has the potential to improve PROM accuracy by enabling the spatial localization of these models via domain decomposition, and achieve up to two orders of magnitude speedup over equivalent coupled full order model solutions and moderate speedups over analogous monolithic solutions.

In 1991, Moore [20] raised a question about whether hydrodynamics is capable of performing computations. Similarly, in 2016, Tao [25] asked whether a mechanical system, including a fluid flow, can simulate a universal Turing machine. In this expository article, we review the construction in [8] of a "Fluid computer" in dimension 3 that combines techniques in symbolic dynamics with the connection between steady Euler flows and contact geometry unveiled by Etnyre and Ghrist. In addition, we argue that the metric that renders the vector field Beltrami cannot be critical in the Chern-Hamilton sense [9]. We also sketch the completely different construction for the Euclidean metric in $\mathbb R^3$ as given in [7]. These results reveal the existence of undecidable fluid particle paths. We conclude the article with a list of open problems.

In this paper, we put forward a neural network framework to solve the nonlinear hyperbolic systems. This framework, named relaxation neural networks(RelaxNN), is a simple and scalable extension of physics-informed neural networks(PINN). It is shown later that a typical PINN framework struggles to handle shock waves that arise in hyperbolic systems' solutions. This ultimately results in the failure of optimization that is based on gradient descent in the training process. Relaxation systems provide a smooth asymptotic to the discontinuity solution, under the expectation that macroscopic problems can be solved from a microscopic perspective. Based on relaxation systems, the RelaxNN framework alleviates the conflict of losses in the training process of the PINN framework. In addition to the remarkable results demonstrated in numerical simulations, most of the acceleration techniques and improvement strategies aimed at the standard PINN framework can also be applied to the RelaxNN framework.

In this paper, we introduce an improved version of the fifth-order weighted essentially non-oscillatory (WENO) shock-capturing scheme by incorporating deep learning techniques. The established WENO algorithm is improved by training a compact neural network to adjust the smoothness indicators within the WENO scheme. This modification enhances the accuracy of the numerical results, particularly near abrupt shocks. Unlike previous deep learning-based methods, no additional post-processing steps are necessary for maintaining consistency. We demonstrate the superiority of our new approach using several examples from the literature for the two-dimensional Euler equations of gas dynamics. Through intensive study of these test problems, which involve various shocks and rarefaction waves, the new technique is shown to outperform traditional fifth-order WENO schemes, especially in cases where the numerical solutions exhibit excessive diffusion or overshoot around shocks.

Researchers have long been numerically solving the partial differential equations that govern important fluid phenomena such as weather, fusion plasmas, and aerodynamics. Of course, the accuracy of the results is always limited by the finite precision and spatial resolution of computer representations of the equations. Computers have also become a powerful tool for exact, rigorous mathematics. Proof assistants, for example, instill confidence a logical argument is sound and all cases have been considered. Programs can tirelessly examine superhumanly large libraries of combinations, such as those underlying the four-color map theorem proof in 1976.

The theoretical structure of deep neural network (DNN) has been clarified gradually. Imaizumi-Fukumizu (2019) and Suzuki (2019) clarified that the learning ability of DNN is superior to the previous theories when the target function is non-smooth functions. However, as far as the author is aware, none of the numerous works to date attempted to mathematically investigate what kind of DNN architectures really induce pointwise convergence of gradient descent (without any statistical argument), and this attempt seems to be closer to the practical DNNs. In this paper we restrict target functions to non-smooth indicator functions, and construct a deep neural network inducing pointwise convergence provided by gradient descent process in ReLU-DNN. The DNN has a sparse and a special shape, with certain variable transformations.

In this paper, a modification of A* algorithm is considered for the shortest path problem. A weightage is introduced in the heuristic part of the A* algorithm to improve its efficiency. An application of the algorithm is considered for UAV path planning wherein velocity is taken as the weigtage to the heuristic. At the outset, calculus of variations based Lagrange's equation was used to identify velocity as the decisive factor for the dynamical system. This approach would be useful for other problems as well to improve the efficiency of algorithms in those areas.