Learning the temporal evolution of multivariate densities via normalizing flows

Lu, Yubin; Maulik, Romit; Gao, Ting; Dietrich, Felix; Kevrekidis, Ioannis G.; Duan, Jinqiao

doi:10.48550/arxiv.2107.13735

Cited by 4 publications

(4 citation statements)

References 51 publications

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“…• Our approach is based on PDE-loss functions, and does not need sample paths generated from stochastic differential equations. This is different from previous works such as [21] where sample paths of the corresponding stochastic dynamics are used.…”

Section: Introductionmentioning

confidence: 82%

Solving time dependent Fokker-Planck equations via temporal normalizing flow

Xing¹,

Zeng²,

Zhou³

2021

Preprint

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In this work, we propose an adaptive learning approach based on temporal normalizing flows for solving time-dependent Fokker-Planck (TFP) equations. It is well known that solutions of such equations are probability density functions, and thus our approach relies on modelling the target solutions with the temporal normalizing flows. The temporal normalizing flow is then trained based on the TFP loss function, without requiring any labeled data. Being a machine learning scheme, the proposed approach is mesh-free and can be easily applied to high dimensional problems. We present a variety of test problems to show the effectiveness of the learning approach.

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Section: Introductionmentioning

confidence: 82%

Solving time dependent Fokker-Planck equations via temporal normalizing flow

Xing¹,

Zeng²,

Zhou³

2021

Preprint

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“…Based on normalizing flows as explained in the previous session, we know it is worthwhile to simulate high-dimensional joint probability of the complex trading environment. In [33], the authors give a link between temporal normalizing folws of an SDE and and the solution of corresponding. For our case here, the density P x (x) is the solution of Fokker-Plank equation corresponding to equation (5).…”

Section: Proposed Modelmentioning

confidence: 99%

Model Based Reinforcement Learning with Non-Gaussian Environment Dynamics and its Application to Portfolio Optimization

Huang¹,

Gao²,

Li³

et al. 2023

Preprint

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With the fast development of quantitative portfolio optimization in financial engineering, lots of promising algorithmic trading strategies have shown competitive advantages in recent years. However, the environment from real financial markets is complex and hard to be fully simulated, considering non-stationarity of the stock data, unpredictable hidden causal factors and so on. Fortunately, difference of stock prices is often stationary series, and the internal relationship between difference of stocks can be linked to the decision-making process, then the portfolio should be able to achieve better performance. In this paper, we demonstrate normalizing flows is adopted to simulated high-dimensional joint probability of the complex trading environment, and develop a novel model based reinforcement learning framework to better understand the intrinsic mechanisms of quantitative online trading. Second, we experiment various stocks from three different financial markets (Dow, NASDAQ and S&P 500) and show that among these three financial markets, Dow gets the best performance results on various evaluation metrics under our back-testing system. Especially, our proposed method even resists big drop (less maximum drawdown) during COVID-19 pandemic period when the financial market got unpredictable crisis. All these results are comparatively better than modeling the state transition dynamics with independent Gaussian Processes. Third, we utilize a causal analysis method to study the causal relationship among different stocks of the environment. Further, by visualizing high dimensional state transition data comparisons from real and virtual buffer with t-SNE, we uncover some effective patterns of better portfolio optimization strategies. Our methodology will be beneficial to decision making process in many automatic trading systems.

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“…For a random vector z ∈ R D , which satisfies z ∼ p z (z). The main idea of normalizing flow is to find a transformation T such that: z = T (x), where z ∼ p z (z), (24) where p z (z) is a prior density. When the transformation T is invertible and both T and T −1 are differentiable, the density p x (x) can be expressed by a change of variables:…”

Section: Algorithm For Identification Of the Drift Termmentioning

confidence: 99%

Extracting stochastic dynamical systems with $α$-stable Lévy noise from data

Li,

Lu,

et al. 2021

Preprint

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With the rapid increase of valuable observational, experimental and simulated data for complex systems, much efforts have been devoted to identifying governing laws underlying the evolution of these systems. Despite the wide applications of non-Gaussian fluctuations in numerous physical phenomena, the data-driven approaches to extract stochastic dynamical systems with (non-Gaussian) Lévy noise are relatively few so far. In this work, we propose a data-driven method to extract stochastic dynamical systems with α-stable Lévy noise from short burst data based on the properties of α-stable distributions. More specifically, we first estimate the Lévy jump measure and noise intensity via computing mean and variance of the amplitude of the increment of the sample paths. Then we approximate the drift coefficient by combining nonlocal Kramers-Moyal formulas with normalizing flows. Numerical experiments on one-and two-dimensional prototypical examples illustrate the accuracy and effectiveness of our method. This approach will become an effective scientific tool in discovering stochastic governing laws of complex phenomena and understanding dynamical behaviors under non-Gaussian fluctuations.

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Learning the temporal evolution of multivariate densities via normalizing flows

Cited by 4 publications

References 51 publications

Solving time dependent Fokker-Planck equations via temporal normalizing flow

Solving time dependent Fokker-Planck equations via temporal normalizing flow

Model Based Reinforcement Learning with Non-Gaussian Environment Dynamics and its Application to Portfolio Optimization

Extracting stochastic dynamical systems with $α$-stable Lévy noise from data

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