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The probability density function of stochastic differential equations is governed by the Fokker-Planck (FP) equation. A novel machine learning method is developed to solve the general FP equations based on deep neural networks. The proposed algorithm does not require any interpolation and coordinate transformation, which is different from the traditional numercial methods. The main novelty of this paper is that penalty factors are introduced to overcome the local optimization for the deep learning approach, and the corresponding setting rules are given. Meanwhile, we consider a normalization condition as a supervision condition to effectively avoid that the trial solution is zero. Several numerical examples are presented to illustrate performances of the proposed algorithm, including one-and twodimensional systems. All the results suggest that the deep learning is quite feasible and effective to calculate the FP equation. Further, influences of the number of hidden layers, the penalty factors, and the optimization algorithm are discussed in detail. These results indicate that the performances of the machine learning technique can be improved through constructing the neural networks appropriately.

A bistable toggle switch is a paradigmatic model in the field of biology. The dynamics of the system induced by Gaussian noise has been intensively investigated, but Gaussian noise cannot incorporate large bursts typically occurring in real experiments. This paper aims to examine effects of variations from one protein imposed by a non-Gaussian Lévy noise, which is able to describe even large jumps, on the coherent switch and the on/off switch via the steady-state probability density, the joint steady-state probability density, and the mean first passage time. We find that a large burst of one protein due to the Lévy noises can induce coherent switches even with small noise intensities in contrast to the Gaussian case which requires large intensities for this. The influences of the stability index, skewness parameter and noise intensity on the on/off switch are analyzed, leading to an adjustment of the concentrations of both proteins and a decision which stable point to stay most. The mean first passage times show complex effects under Lévy noise, especially the stability index and skewness parameter. Our results also imply that the presence of non-Gaussian Lévy noises has fundamentally changed the escape mechanism in such a system compared with Gaussian noise.

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Since the last IAEA Fusion Energy Conference in 2018, significant progress of the experimental program of HL-2A has been achieved on developing advanced plasma physics, edge localized mode (ELM) control physics and technology. Optimization of plasma confinement has been performed. In particular, high-N H-mode plasmas exhibiting an internal transport barrier have been obtained (normalized plasma pressure N reached up to 3). Injection of impurity improved the plasma confinement. ELM control using resonance magnetic perturbation (RMP) or impurity injection has been achieved in a wide parameter regime, including Types I and III. In addition, the impurity seeding with supersonic molecular beam injection (SMBI) or laser blow-off (LBO) techniques has been successfully applied to actively control the plasma confinement and instabilities, as well as the plasma disruption with the aid of disruption prediction. Disruption prediction algorithms based on deep learning are developed. A prediction accuracy of 96.8% can be reached by assembling convolutional neural network (CNN). Furthermore, transport resulted from a wide variety of phenomena such as energetic particles and magnetic islands have been investigated. In parallel with the HL-2A experiments, the HL-2M mega-ampere class tokamak was commissioned in 2020 with its first plasma. Key features and capabilities of HL-2M are briefly presented.

In this paper, we consider the non‐Lipschitz stochastic differential equations and stochastic functional differential equations with delays driven by Lévy noise, and the approximation theorems for the solutions to these two kinds of equations will be proposed respectively. Non‐Lipschitz condition is much weaker condition than the Lipschitz one. The simplified equations will be defined to make its solutions converge to that of the corresponding original equations both in the sense of mean square and probability, which constitute the approximation theorems. Copyright © 2014 John Wiley & Sons, Ltd.