A Machine-Learning-Based Importance Sampling Method to Compute Rare Event Probabilities

We develop a novel computational method for evaluating the extreme excursion probabilities arising from random initialization of nonlinear dynamical systems. The method uses excursion probability theory to formulate a sequence of Bayesian inverse problems that, when solved, yields the biasing distribution. Solving multiple Bayesian inverse problems can be expensive; more so in higher dimensions. To alleviate the computational cost, we build machine-learning-based surrogates to solve the Bayesian inverse problems that give rise to the biasing distribution. This biasing distribution can then be used in an importance sampling procedure to estimate the extreme excursion probabilities.

[1]  Dirk P. Kroese,et al.  Efficient Monte Carlo simulation via the generalized splitting method , 2012, Stat. Comput..

[2]  Eric Vanden-Eijnden,et al.  Rogue waves and large deviations in deep sea , 2017, Proceedings of the National Academy of Sciences.

[3]  Agnès Lagnoux,et al.  RARE EVENT SIMULATION , 2005, Probability in the Engineering and Informational Sciences.

[4]  Peter W. Glynn,et al.  Stochastic Simulation: Algorithms and Analysis , 2007 .

[5]  Tim Hesterberg,et al.  Monte Carlo Strategies in Scientific Computing , 2002, Technometrics.

[6]  Christian P. Robert,et al.  Monte Carlo Statistical Methods (Springer Texts in Statistics) , 2005 .

[7]  A. Rollett,et al.  The Monte Carlo Method , 2004 .

[8]  Geoffrey E. Hinton,et al.  Learning representations by back-propagating errors , 1986, Nature.

[9]  H. Kahn,et al.  Methods of Reducing Sample Size in Monte Carlo Computations , 1953, Oper. Res..

[10]  Themistoklis P. Sapsis,et al.  Sequential sampling strategy for extreme event statistics in nonlinear dynamical systems , 2018, Proceedings of the National Academy of Sciences.

[11]  Hoon Kim,et al.  Monte Carlo Statistical Methods , 2000, Technometrics.

[12]  Jimmy Ba,et al.  Adam: A Method for Stochastic Optimization , 2014, ICLR.

[13]  Nitish Srivastava,et al.  Dropout: a simple way to prevent neural networks from overfitting , 2014, J. Mach. Learn. Res..

[14]  Lambros S. Katafygiotis,et al.  Bayesian post-processor and other enhancements of Subset Simulation for estimating failure probabilities in high dimensions , 2011 .

[15]  Lambros S. Katafygiotis,et al.  A two-stage Subset Simulation-based approach for calculating the reliability of inelastic structural systems subjected to Gaussian random excitations , 2005 .

[16]  Heikki Haario,et al.  DRAM: Efficient adaptive MCMC , 2006, Stat. Comput..

[17]  Francesco Fedele,et al.  On Oceanic Rogue Waves , 2015, 1501.03370.

[18]  Lambros S. Katafygiotis,et al.  Geometric insight into the challenges of solving high-dimensional reliability problems , 2008 .

[19]  Ling Li,et al.  Bayesian Subset Simulation , 2016, SIAM/ASA J. Uncertain. Quantification.

[20]  James L. Beck,et al.  Hybrid Subset Simulation method for reliability estimation of dynamical systems subject to stochastic excitation , 2005 .

[21]  Gerardo Rubino,et al.  Introduction to Rare Event Simulation , 2009, Rare Event Simulation using Monte Carlo Methods.

[22]  Eric Vanden-Eijnden,et al.  Extreme Event Quantification in Dynamical Systems with Random Components , 2018, SIAM/ASA J. Uncertain. Quantification.

[23]  Stefano Giordano,et al.  Rare event simulation , 2002, Eur. Trans. Telecommun..

[24]  S. Rice.  Mathematical analysis of random noise , 1944 .

[25]  James L. Beck,et al.  Reliability Estimation for Dynamical Systems Subject to Stochastic Excitation using Subset Simulation with Splitting , 2005 .

[26]  Carl E. Rasmussen,et al.  Gaussian processes for machine learning , 2005, Adaptive computation and machine learning.

[27]  J. Beck,et al.  Estimation of Small Failure Probabilities in High Dimensions by Subset Simulation , 2001 .

[28]  Tom Ross,et al.  A climatology of 1980-2003 extreme weather and climate events , 2003 .

[29]  R. Adler.  The Geometry of Random Fields , 2009 .

[30]  W. L. Dunn,et al.  Exploring Monte Carlo Methods , 2011 .

[31]  Mihai Anitescu,et al.  Efficient computation of extreme excursion probabilities for dynamical systems , 2020, ArXiv.

[32]  Tapan Kumar Saha,et al.  Power system blackouts - literature review , 2009, 2009 International Conference on Industrial and Information Systems (ICIIS).