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1. Six wave interaction equations in finite-depth gravity waves with surface tension

   https://doi.org/10.1017/jfm.2023.128  

   Ablowitz, Mark J.; Luo, Xu-Dan; Musslimani, Ziad H. (April 2023, Journal of Fluid Mechanics)

   Three wave resonant triad interactions in two space and one time dimensions form a well-known system of first-order quadratically nonlinear evolution equations that arise in many areas of physics. In deep water waves, they were first derived by Simmons in 1969 and later shown to be exactly solvable by Ablowitz & Haberman in 1975. Specifically, integrability was established by introducing a system of six wave interactions whose symmetry reduction leads to the well-known three wave equations. Here, it is shown that the six wave interaction and classical three wave equations satisfying triad resonance conditions in finite-depth gravity waves can be derived from the non-local integro-differential formulation of the free surface gravity wave equation with surface tension. These quadratically nonlinear six wave interaction equations and their reductions to the classical and non-local complex as well as real reverse space–time three wave interaction equations are integrable. Limits to infinite and shallow water depth are also discussed. 

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2. Data‐enabled extremum seeking: A cooperative concurrent learning‐based approach

   https://doi.org/10.1002/acs.3189  

   Poveda, Jorge I.; Benosman, Mouhacine; Vamvoudakis, Kyriakos G. (October 2020, International Journal of Adaptive Control and Signal Processing)

   Summary This paper introduces a new class of feedback‐based data‐driven extremum seeking algorithms for the solution of model‐free optimization problems in smooth continuous‐time dynamical systems. The novelty of the algorithms lies on the incorporation of memory to store recorded data that enables the use of information‐rich datasets during the optimization process, and allows to dispense with the time‐varying dither excitation signal needed by standard extremum seeking algorithms that rely on a persistence of excitation (PE) condition. The model‐free optimization dynamics are developed for single‐agent systems, as well as for multi‐agent systems with communication graphs that allow agents to share their state information while preserving the privacy of their individual data. In both cases, sufficient richness conditions on the recorded data, as well as suitable optimization dynamics modeled by ordinary differential equations are characterized in order to guarantee convergence to a neighborhood of the solution of the extremum seeking problems. The performance of the algorithms is illustrated via different numerical examples in the context of source‐seeking problems in multivehicle systems. 

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3. On Bayesian data assimilation for PDEs with ill-posed forward problems

   https://doi.org/10.1088/1361-6420/ac7acd  

   Lanthaler, S; Mishra, S; Weber, F (July 2022, Inverse Problems)

   Abstract We study Bayesian data assimilation (filtering) for time-evolution Partial differential equations (PDEs), for which the underlying forward problem may be very unstable or ill-posed. Such PDEs, which include the Navier–Stokes equations of fluid dynamics, are characterized by a high sensitivity of solutions to perturbations of the initial data, a lack of rigorous global well-posedness results as well as possible non-convergence of numerical approximations. Under very mild and readily verifiable general hypotheses on the forward solution operator of such PDEs, we prove that the posterior measure expressing the solution of the Bayesian filtering problem is stable with respect to perturbations of the noisy measurements, and we provide quantitative estimates on the convergence of approximate Bayesian filtering distributions computed from numerical approximations. For the Navier–Stokes equations, our results imply uniform stability of the filtering problem even at arbitrarily small viscosity, when the underlying forward problem may become ill-posed, as well as the compactness of numerical approximants in a suitable metric on time-parametrized probability measures. 

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4. Fault detection and accommodation of positive real infinite dimensional systems using adaptive RKHS-based functional estimation

   https://doi.org/10.23919/ACC53348.2022.9867405  

   Demetriou, Michael A. (June 2022, 2022 American Control Conference)

   This paper presents an adaptive functional estimation scheme for the fault detection and diagnosis of nonlinear faults in positive real infinite dimensional systems. The system is assumed to satisfy a positive realness condition and the fault, taking the form of a nonlinear function of the output, is assumed to enter the system at an unknown time. The proposed detection and diagnostic observer utilizes a Reproducing Kernel Hilbert Space as the parameter space and via a Lyapunov redesign approach, the learning scheme for the unknown functional is used for the detection of the fault occurrence, the diagnosis of the fault and finally its accommodation via an adaptive control reconfiguration. Results on parabolic PDEs with either boundary or in-domain actuation and sensing are included. 

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5. Learning Traveling Solitary Waves Using Separable Gaussian Neural Networks

   https://doi.org/10.3390/e26050396  

   Xing, Siyuan; Charalampidis, Efstathios G (May 2024, Entropy)

   In this paper, we apply a machine-learning approach to learn traveling solitary waves across various physical systems that are described by families of partial differential equations (PDEs). Our approach integrates a novel interpretable neural network (NN) architecture, called Separable Gaussian Neural Networks (SGNN) into the framework of Physics-Informed Neural Networks (PINNs). Unlike the traditional PINNs that treat spatial and temporal data as independent inputs, the present method leverages wave characteristics to transform data into the so-called co-traveling wave frame. This reformulation effectively addresses the issue of propagation failure in PINNs when applied to large computational domains. Here, the SGNN architecture demonstrates robust approximation capabilities for single-peakon, multi-peakon, and stationary solutions (known as “leftons”) within the (1+1)-dimensional, b-family of PDEs. In addition, we expand our investigations, and explore not only peakon solutions in the ab-family but also compacton solutions in (2+1)-dimensional, Rosenau-Hyman family of PDEs. A comparative analysis with multi-layer perceptron (MLP) reveals that SGNN achieves comparable accuracy with fewer than a tenth of the neurons, underscoring its efficiency and potential for broader application in solving complex nonlinear PDEs. 

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