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Large-scale dynamics of the oceans and the atmosphere are governed by primitive equations (PEs). Due to the nonlinearity and nonlocality, the numerical study of the PEs is generally challenging. Neural networks have been shown to be a promising machine learning tool to tackle this challenge. In this work, we employ physics-informed neural networks (PINNs) to approximate the solutions to the PEs and study the error estimates. We first establish the higher-order regularity for the global solutions to the PEs with either full viscosity and diffusivity, or with only the horizontal ones. Such a result for the case with only the horizontal ones is new and required in the analysis under the PINNs framework. Then we prove the existence of two-layer tanh PINNs of which the corresponding training error can be arbitrarily small by taking the width of PINNs to be sufficiently wide, and the error between the true solution and its approximation can be arbitrarily small provided that the training error is small enough and the sample set is large enough. In particular, all the estimates area priori, and our analysis includes higher-order (in spatial Sobolev norm) error estimates. Numerical results on prototype systems are presented to further illustrate the advantage of using the$$H^s$$$H^s$norm during the training.

 

Abstract We analyze the systemic risk for disjoint and overlapping groups of financial institutions by proposing new models with realistic game features.Specifically, we generalize the systemic risk measure proposed in[F. Biagini, J.-P. Fouque, M. Frittelli and T. Meyer-Brandis, On fairness of systemic risk measures, Finance Stoch. 24 (2020), 2, 513–564]by allowing individual banks to choose their preferred groups instead of being assigned to certain groups.We introduce the concept of Nash equilibrium for these new models, and analyze the optimal solution under Gaussian distribution of the risk factor.We also provide an explicit solution for the risk allocation of the individual banks and study the existence and uniqueness of Nash equilibrium both theoretically and numerically.The developed numerical algorithm can simulate scenarios of equilibrium, and we apply it to study the banking structure with real data and show the validity of the proposed model. 

Real-world data can be multimodal distributed, e.g., data describing the opinion divergence in a community, the interspike interval distribution of neurons, and the oscillators’ natural frequen- cies. Generating multimodal distributed real- world data has become a challenge to existing generative adversarial networks (GANs). For ex- ample, it is often observed that Neural SDEs have only demonstrated successful performance mainly in generating unimodal time series datasets. In this paper, we propose a novel time series gen- erator, named directed chain GANs (DC-GANs), which inserts a time series dataset (called a neigh- borhood process of the directed chain or input) into the drift and diffusion coefficients of the di- rected chain SDEs with distributional constraints. DC-GANs can generate new time series of the same distribution as the neighborhood process, and the neighborhood process will provide the key step in learning and generating multimodal dis- tributed time series. The proposed DC-GANs are examined on four datasets, including two stochas- tic models from social sciences and computa- tional neuroscience, and two real-world datasets on stock prices and energy consumption. To our best knowledge, DC-GANs are the first work that can generate multimodal time series data and con- sistently outperforms state-of-the-art benchmarks with respect to measures of distribution, data sim- ilarity, and predictive ability.