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Koopman operator theory is receiving increased attention due to its promise to linearize nonlinear dynamics. Neural networks that are developed to represent Koopman operators have shown great success thanks to their ability to approximate arbitrarily complex functions. However, despite their great potential, they typically require large training data-sets either from measurements of a real system or from high-fidelity simulations. In this work, we propose a novel architecture inspired by physics-informed neural networks, which leverage automatic differentiation to impose the underlying physical laws via soft penalty constraints during model training. We demonstrate that it not only reduces the need of large training data-sets, but also maintains high effectiveness in approximating Koopman eigenfunctions., Comment: 26 pages, 5 figures

For a linear complementarity problem, we present a relaxaiton accelerated two-sweep matrix splitting iteration method. The convergence analysis illustrates that the proposed method converges to the exact solution of the linear complementarity problem when the system matrix is an $H_+$-matrix and the convergence conditions are given. Numerical experiments show that the proposed method is more efficient than the existing ones.

Nonlinear differential equations rarely admit closed-form solutions, thus requiring numerical time-stepping algorithms to approximate solutions. Further, many systems characterized by multiscale physics exhibit dynamics over a vast range of timescales, making numerical integration computationally expensive due to numerical stiffness. In this work, we develop a hierarchy of deep neural network time-steppers to approximate the flow map of the dynamical system over a disparate range of time-scales. The resulting model is purely data-driven and leverages features of the multiscale dynamics, enabling numerical integration and forecasting that is both accurate and highly efficient. Moreover, similar ideas can be used to couple neural network-based models with classical numerical time-steppers. Our multiscale hierarchical time-stepping scheme provides important advantages over current time-stepping algorithms, including (i) circumventing numerical stiffness due to disparate time-scales, (ii) improved accuracy in comparison with leading neural-network architectures, (iii) efficiency in long-time simulation/forecasting due to explicit training of slow time-scale dynamics, and (iv) a flexible framework that is parallelizable and may be integrated with standard numerical time-stepping algorithms. The method is demonstrated on a wide range of nonlinear dynamical systems, including the Van der Pol oscillator, the Lorenz system, the Kuramoto-Sivashinsky equation, and fluid flow pass a cylinder; audio and video signals are also explored. On the sequence generation examples, we benchmark our algorithm against state-of-the-art methods, such as LSTM, reservoir computing, and clockwork RNN. Despite the structural simplicity of our method, it outperforms competing methods on numerical integration., Comment: 22 pages, 10 figures

We propose a multi-resolution convolutional autoencoder (MrCAE) architecture that integrates and leverages three highly successful mathematical architectures: (i) multigrid methods, (ii) convolutional autoencoders and (iii) transfer learning. The method provides an adaptive, hierarchical architecture that capitalizes on a progressive training approach for multiscale spatio-temporal data. This framework allows for inputs across multiple scales: starting from a compact (small number of weights) network architecture and low-resolution data, our network progressively deepens and widens itself in a principled manner to encode new information in the higher resolution data based on its current performance of reconstruction. Basic transfer learning techniques are applied to ensure information learned from previous training steps can be rapidly transferred to the larger network. As a result, the network can dynamically capture different scaled features at different depths of the network. The performance gains of this adaptive multiscale architecture are illustrated through a sequence of numerical experiments on synthetic examples and real-world spatial-temporal data., Comment: 20 pages, 11 figures

We consider a general piecewise deterministic Markov process (PDMP) $X=\{X_t\}_{t\geqslant 0}$ with measure-valued generator $\mathcal{A}$, for which the conditional distribution function of the inter-occurrence time is not necessarily absolutely continuous. A general form of the exponential martingales is presented as $$M^f_t=\frac{f(X_t)}{f(X_0)}\left[\mathrm{Sexp}\left(\int_{(0,t]}\frac{\mathrm{d}L(\mathcal{A}f)_s}{f(X_{s-})}\right)\right]^{-1}.$$ Using this exponential martingale as a likelihood ratio process, we define a new probability measure. It is shown that the original process remains a general PDMP under the new probability measure. And we find the new measure-valued generator and its domain.

This work presents a systematic investigation of the influence of weather conditions on the calibration errors by using Gaussian fitness, least chi-square linear fitness and wavelet transform to analyze the calibration coefficients from observations of the Chinese Solar Broadband Radio Spectrometers (at frequency bands of 1.0-2.0 GHz, 2.6-3.8 GHz, and 5.2-7.6 GHz) during 1997-2007. We found that calibration coefficients are influenced by the local air temperature. Considering the temperature correction, the calibration error will reduce by about $10\%-20\%$ at 2800 MHz. Based on the above investigation and the calibration corrections, we further study the radio emission of the quiet-Sun by using an appropriate hybrid model of the quiet-Sun atmosphere. The results indicate that the numerical flux of the hybrid model is much closer to the observation flux than that of other ones., Comment: Accepted by Apj, 24 pages, 10 pages, 3 tables

On August 9, 2011, there was an X6.9 flare event occurred near the west limb of solar disk. From the observation obtained by the spectrometer of the Chinese Solar Broadband Radio Spectrometer in Huairou (SBRS/Huairou) around the flare, we find that this powerful flare has only a short-duration microwave burst of about only 5 minutes, and during the short-duration microwave burst, there are several kinds of fine structures on the spectrogram. These fine structures include very short-period pulsations, millisecond spike bursts, and type III bursts. The most interesting is that almost all of the pulses of very short-period pulsation (VSP) are structured by clusters of millisecond timescales of spike bursts or type III bursts. And there exists three different kinds of frequency drift rates in the VSP: the frequency drift rates with absolute value of about 55 - 130 MHz s^{-1} in the pulse groups, the frequency drift rates with absolute value of about 2.91 - 16.9 GHz s^{-1} on each individual pulse, and the frequency drift rates with absolute value of about 15 - 25 GHz s$^{-1}$) at each individual spike burst or type III burst., Comment: 5 pages, 3 figures

Microwave observations of quasi-periodic pulsations (QPP) in multi-timescales are confirmed to be associated with an X3.4 flare/CME event at Solar Broadband Radio Spectrometer in Huairou (SBRS/Huairou) on 13 December 2006. It is most remarkable that the timescales of QPPs are distributed in a broad range from hecto-second (very long period pulsation, VLP, the period P>100 s), deca-second (long period pulsation, LPP, 10

Based on the joint-observations of the radio broadband spectral emissions of solar eclipse on August 1, 2008 at Jiuquan (total eclipse) and Huairou (partial eclipse) at the frequencies of 2.00 -- 5.60 GHz (Jiuquan), 2.60 -- 3.80 GHZ (Chinese solar broadband radiospectrometer, SBRS/Huairou), and 5.20 -- 7.60 GHz (SBRS/Huairou), the authors assemble a successive series of broadband spectrum with a frequency of 2.60 -- 7.60 GHz to observe the solar eclipse synchronously. This is the first attempt to analyze the solar eclipse radio emission under the two telescopes located at different places with broadband frequencies in the periods of total and partial eclipse. With these analyses, the authors made a new semiempirical model of the coronal plasma density of the quiet Sun and made a comparison with the classic models., Comment: 10 pages, 4 figures, published on Sci. China Ser. G, 2009, Vol.52, page 1765-1772