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We study the 2D Navier-Stokes equation with transport noise subject to periodic boundary conditions. Our main result is an error estimate for the time-discretisation showing a convergence rate of order (up to) 1/2. It holds with respect to mean square error convergence, whereas previously such a rate for the stochastic Navier-Stokes equations was only known with respect to convergence in probability. Our result is based on uniform-in-probability estimates for the continuous as well as the time-discrete solution exploiting the particular structure of the noise.

We propose MIMOC: Motion Imitation from Model-Based Optimal Control. MIMOC is a Reinforcement Learning (RL) controller that learns agile locomotion by imitating reference trajectories from model-based optimal control. MIMOC mitigates challenges faced by other motion imitation RL approaches because the references are dynamically consistent, require no motion retargeting, and include torque references. Hence, MIMOC does not require fine-tuning. MIMOC is also less sensitive to modeling and state estimation inaccuracies than model-based controllers. We validate MIMOC on the Mini-Cheetah in outdoor environments over a wide variety of challenging terrain, and on the MIT Humanoid in simulation. We show cases where MIMOC outperforms model-based optimal controllers, and show that imitating torque references improves the policy's performance.

We introduce three new stopping criteria that balance algebraic and discretization errors for the conjugate gradient algorithm applied to high-order finite element discretizations of Poisson problems. The current state of the art stopping criteria compare a posteriori estimates of discretization error against estimates of the algebraic error. Firstly, we propose a new error indicator derived from a recovery-based error estimator that is less computationally expensive and more reliable. Secondly, we introduce a new stopping criterion that suggests stopping when the norm of the linear residual is less than a small fraction of an error indicator derived directly from the residual. This indicator shares the same mesh size and polynomial degree scaling as the norm of the residual, resulting in a robust criterion regardless of the mesh size, the polynomial degree, and the shape regularity of the mesh. Thirdly, in solving Poisson problems with highly variable piecewise constant coefficients, we introduce a subdomain-based criterion that recommends stopping when the norm of the linear residual restricted to each subdomain is smaller than the corresponding indicator also restricted to that subdomain. Numerical experiments, including tests with anisotropic meshes and highly variable piecewise constant coefficients, demonstrate that the proposed criteria efficiently avoid both premature termination and over-solving.

Prior beliefs about the latent function to shape inductive biases can be incorporated into a Gaussian Process (GP) via the kernel. However, beyond kernel choices, the decision-making process of GP models remains poorly understood. In this work, we contribute an analysis of the loss landscape for GP models using methods from physics. We demonstrate $\nu$-continuity for Matern kernels and outline aspects of catastrophe theory at critical points in the loss landscape. By directly including $\nu$ in the hyperparameter optimisation for Matern kernels, we find that typical values of $\nu$ are far from optimal in terms of performance, yet prevail in the literature due to the increased computational speed. We also provide an a priori method for evaluating the effect of GP ensembles and discuss various voting approaches based on physical properties of the loss landscape. The utility of these approaches is demonstrated for various synthetic and real datasets. Our findings provide an enhanced understanding of the decision-making process behind GPs and offer practical guidance for improving their performance and interpretability in a range of applications.

We introduce Reactive Action and Motion Planner (RAMP), which combines the strengths of search-based and reactive approaches for motion planning. In essence, RAMP is a hierarchical approach where a novel variant of a Model Predictive Path Integral (MPPI) controller is used to generate trajectories which are then followed asynchronously by a local vector field controller. We demonstrate, in the context of a table clearing application, that RAMP can rapidly find paths in the robot's configuration space, satisfy task and robot-specific constraints, and provide safety by reacting to static or dynamically moving obstacles. RAMP achieves superior performance through a number of key innovations: we use Signed Distance Function (SDF) representations directly from the robot configuration space, both for collision checking and reactive control. The use of SDFs allows for a smoother definition of collision cost when planning for a trajectory, and is critical in ensuring safety while following trajectories. In addition, we introduce a novel variant of MPPI which, combined with the safety guarantees of the vector field trajectory follower, performs incremental real-time global trajectory planning. Simulation results establish that our method can generate paths that are comparable to traditional and state-of-the-art approaches in terms of total trajectory length while being up to 30 times faster. Real-world experiments demonstrate the safety and effectiveness of our approach in challenging table clearing scenarios.

In this paper, we propose a Poisson-Nernst-Planck-Navier-Stokes-Cahn-Hillard (PNP-NS-CH)model for an electrically charged droplet suspended in a viscous fluid subjected to an external electric field. Our model incorporates spatial variations of electric permittivity and diffusion constants, as well as interfacial capacitance. Based on a time scale analysis, we derive two approximations of the original model, namely a dynamic model for the net charge and a leaky-dielectric model. For the leaky-dielectric model, we conduct a detailed asymptotic analysis to demonstrate the convergence of the diffusive-interface leaky-dielectric model to the sharp interface model as the interface thickness approaches zero. Numerical computations are performed to validate the asymptotic analysis and demonstrate the model's effectiveness in handling topology changes, such as electrocoalescence. Our numerical results of these two approximation models reveal that the polarization force, which is induced by the spatial variation of electric permittivity in the direction perpendicular to the external electric field, consistently dominates the Lorentz force, which arises from the net charge. The equilibrium shape of droplets is determined by the interplay between these two forces along the direction of the electric field. Furthermore, in the presence of the interfacial capacitance, a local variation of effective permittivity leads to an accumulation of counter-ions near the interface, resulting in a reduction in droplet deformation. Our numerical solutions also confirm that the leaky dielectric model serves as a reasonable approximation of the original PNP-NS-CH model when the electric relaxation time is sufficiently short. The Lorentz force and droplet deformation both decrease when the diffusion of net charge is significant.

The reconfigurable intelligent surface (RIS) is an emerging technology that changes how wireless networks are perceived, therefore its potential benefits and applications are currently under intense research and investigation. In this letter, we focus on electromagnetically consistent models for RISs inheriting from a recently proposed model based on mutually coupled loaded wire dipoles. While existing related research focuses on free-space wireless channels thereby ignoring interactions between RIS and scattering objects present in the propagation environment, we introduce an RIS-aided channel model that is applicable to more realistic scenarios, where the scattering objects are modeled as loaded wire dipoles. By adjusting the parameters of the wire dipoles, the properties of general natural and engineered material objects can be modeled. Based on this model, we introduce a provably convergent and efficient iterative algorithm that jointly optimizes the RIS and transmitter configurations to maximize the system sum-rate. Extensive numerical results show the net performance improvement provided by the proposed method compared with existing optimization algorithms.

Wind power is becoming an increasingly important source of renewable energy worldwide. However, wind farm power control faces significant challenges due to the high system complexity inherent in these farms. A novel communication-based multi-agent deep reinforcement learning large-scale wind farm multivariate control is proposed to handle this challenge and maximize power output. A wind farm multivariate power model is proposed to study the influence of wind turbines (WTs) wake on power. The multivariate model includes axial induction factor, yaw angle, and tilt angle controllable variables. The hierarchical communication multi-agent proximal policy optimization (HCMAPPO) algorithm is proposed to coordinate the multivariate large-scale wind farm continuous controls. The large-scale wind farm is divided into multiple wind turbine aggregators (WTAs), and neighboring WTAs can exchange information through hierarchical communication to maximize the wind farm power output. Simulation results demonstrate that the proposed multivariate HCMAPPO can significantly increase wind farm power output compared to the traditional PID control, coordinated model-based predictive control, and multi-agent deep deterministic policy gradient algorithm. Particularly, the HCMAPPO algorithm can be trained with the environment based on the thirteen-turbine wind farm and effectively applied to larger wind farms. At the same time, there is no significant increase in the fatigue damage of the wind turbine blade from the wake control as the wind farm scale increases. The multivariate HCMAPPO control can realize the collective large-scale wind farm maximum power output.

We consider the numerical approximation of a sharp-interface model for two-phase flow, which is given by the incompressible Navier-Stokes equations in the bulk domain together with the classical interface conditions on the interface. We propose structure-preserving finite element methods for the model, meaning in particular that volume preservation and energy decay are satisfied on the discrete level. For the evolving fluid interface, we employ parametric finite element approximations that introduce an implicit tangential velocity to improve the quality of the interface mesh. For the two-phase Navier-Stokes equations, we consider two different approaches: an unfitted and a fitted finite element method, respectively. In the unfitted approach, the constructed method is based on an Eulerian weak formulation, while in the fitted approach a novel arbitrary Lagrangian-Eulerian (ALE) weak formulation is introduced. Using suitable discretizations of these two formulations, we introduce two finite element methods and prove their structure-preserving properties. Numerical results are presented to show the accuracy and efficiency of the introduced methods.

Transportation of probability measures underlies many core tasks in statistics and machine learning, from variational inference to generative modeling. A typical goal is to represent a target probability measure of interest as the push-forward of a tractable source measure through a learned map. We present a new construction of such a transport map, given the ability to evaluate the score of the target distribution. Specifically, we characterize the map as a zero of an infinite-dimensional score-residual operator and derive a Newton-type method for iteratively constructing such a zero. We prove convergence of these iterations by invoking classical elliptic regularity theory for partial differential equations (PDE) and show that this construction enjoys rapid convergence, under smoothness assumptions on the target score. A key element of our approach is a generalization of the elementary Newton method to infinite-dimensional operators, other forms of which have appeared in nonlinear PDE and in dynamical systems. Our Newton construction, while developed in a functional setting, also suggests new iterative algorithms for approximating transport maps.