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1. Deep Fictitious Play for Finding Markovian Nash Equilibrium in Multi-Agent Games

   Han, Jiequn; Hu, Ruimeng (January 2020, Proceedings of Machine Learning Research)

   null (Ed.)

   We propose a deep neural network-based algorithm to identify the Markovian Nash equilibrium of general large 𝑁-player stochastic differential games. Following the idea of fictitious play, we recast the 𝑁-player game into 𝑁 decoupled decision problems (one for each player) and solve them iteratively. The individual decision problem is characterized by a semilinear Hamilton-Jacobi-Bellman equation, to solve which we employ the recently developed deep BSDE method. The resulted algorithm can solve large 𝑁-player games for which conventional numerical methods would suffer from the curse of dimensionality. Multiple numerical examples involving identical or heterogeneous agents, with risk-neutral or risk-sensitive objectives, are tested to validate the accuracy of the proposed algorithm in large group games. Even for a fifty-player game with the presence of common noise, the proposed algorithm still finds the approximate Nash equilibrium accurately, which, to our best knowledge, is difficult to achieve by other numerical algorithms. 

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2. Dynamic optical spectroscopy and pyrometry of static targets under optical and x-ray laser heating at the European XFEL

   https://doi.org/10.1063/5.0142196  

   Ball, O B; Prescher, C; Appel, K; Baehtz, C; Baron, M A; Briggs, R; Cerantola, V; Chantel, J; Chariton, S; Coleman, A L; et al (August 2023, Journal of Applied Physics)

   Experiments accessing extreme conditions at x-ray free electron lasers (XFELs) involve rapidly evolving conditions of temperature. Here, we report time-resolved, direct measurements of temperature using spectral streaked optical pyrometry of x-ray and optical laser-heated states at the High Energy Density instrument of the European XFEL. This collection of typical experiments, coupled with numerical models, outlines the reliability, precision, and meaning of time dependent temperature measurements using optical emission at XFEL sources. Dynamic temperatures above 1500 K are measured continuously from spectrally- and temporally-resolved thermal emission at 450–850 nm, with time resolution down to 10–100 ns for 1–200 μs streak camera windows, using single shot and integrated modes. Targets include zero-pressure foils free-standing in air and in vacuo, and high-pressure samples compressed in diamond anvil cell multi-layer targets. Radiation sources used are 20-fs hard x-ray laser pulses at 17.8 keV, in single pulses or 2.26 MHz pulse trains of up to 30 pulses, and 250-ns infrared laser single pulses. A range of further possibilities for optical measurements of visible light in x-ray laser experiments using streak optical spectroscopy are also explored, including for the study of x-ray induced optical fluorescence, which often appears as background in thermal radiation measurements. We establish several scenarios where combined emissions from multiple sources are observed and discuss their interpretation. Challenges posed by using x-ray lasers as non-invasive probes of the sample state are addressed. 

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3. Modulational instability of small amplitude periodic traveling waves in the Novikov equation

   https://doi.org/10.1063/5.0240100  

   Ehrman, Brett; Johnson, Mathew A; Lafortune, Stéphane (September 2025, Journal of Mathematical Physics)

   We study the spectral stability of smooth, small-amplitude periodic traveling wave solutions of the Novikov equation, which is a Camassa-Holm type equation with cubic nonlinearities. Specifically, we investigate the L2(R)-spectrum of the associated linearized operator, which in this case is an integro-differential operator with periodic coefficients, in a neighborhood of the origin in the spectral plane. Our analysis shows that such small-amplitude periodic solutions are spectrally unstable to long-wavelength perturbations if the wave number is greater than a critical value, bearing out the famous Benmajin-Feir instability for the Novikov equation. On the other hand, such waves with wave number less than the critical value are shown to be spectrally stable. Our methods are based on applying spectral perturbation theory to the associated linearization. 

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4. Nonlinear self-calibrated spectrometer with single GeSe-InSe heterojunction device

   https://doi.org/10.1126/sciadv.adn6028  

   Darweesh, Rana; Yadav, Rajesh Kumar; Adler, Elior; Poplinger, Michal; Levi, Adi; Lee, Jea-Jung; Leshem, Amir; Ramasubramaniam, Ashwin; Xia, Fengnian; Naveh, Doron (May 2024, Science Advances)

   Computational spectrometry is an emerging field that uses photodetection in conjunction with numerical algorithms for spectroscopic measurements. Compact single photodetectors made from layered materials are particularly attractive since they eliminate the need for bulky mechanical and optical components used in traditional spectrometers and can easily be engineered as heterostructures to optimize device performance. However, such photodetectors are typically nonlinear devices, which adds complexity to extracting optical spectra from their response. Here, we train an artificial neural network to recover the full nonlinear spectral photoresponse of a single GeSe-InSe p-n heterojunction device. The device has a spectral range of 400 to 1100 nm, a small footprint of ~25 × 25 square micrometers, and a mean reconstruction error of 2 × 10⁻⁴for the power spectrum at 0.35 nanometers. Using our device, we demonstrate a solution to metamerism, an apparent matching of colors with different power spectral distributions, which is a fundamental problem in optical imaging. 

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5. Standing and traveling waves in a minimal nonlinearly dispersive lattice model

   Parker, R; Germain, P; Cuevas_Maraver, J; Aceves, A; Kevrekidis, PG (June 2024, Physica D Nonlinear phenomena)

   In the work of Colliander et al. (2020) a minimal lattice model was constructed describing the transfer of energy to high frequencies in the defocusing nonlinear Schrödinger equation. In the present work, we present a systematic study of the coherent structures, both standing and traveling, that arise in the context of this model. We find that the nonlinearly dispersive nature of the model is responsible for standing waves in the form of discrete compactons. On the other hand, analysis of the dynamical features of the simplest nontrivial variant of the model, namely the dimer case, yields both solutions where the intensity is trapped in a single site and solutions where the intensity moves between the two sites, which suggests the possibility of moving excitations in larger lattices. Such excitations are also suggested by the dynamical evolution associated with modulational instability. Our numerical computations confirm this expectation, and we systematically construct such traveling states as exact solutions in lattices of varying size, as well as explore their stability. A remarkable feature of these traveling lattice waves is that they are of ‘‘antidark’’ type, i.e., they are mounted on top of a non-vanishing background. These studies shed light on the existence, stability and dynamics of such standing and traveling states in 1 + 1 dimensions, and pave the way for exploration of corresponding configurations in higher dimensions. 

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